Note: Descriptions are shown in the official language in which they were submitted.
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METHODS FOR IN-SCENE ATMOSPHERIC COMPENSATION BY
ENDMEMBER MATCHING
FIELD OF THE DISCLOSURE
[0001] The present disclosure relates generally to spectral imagery, and
more
particularly to atmospheric compensation of multispectral and/or hyperspectral
imagery.
BACKGROUND
[0002] Spectral imaging relates to the acquisition of information about
an
object, area, or phenomenon by analyzing data acquired by a sensing device not
in contact with the object, area, or phenomenon. From this point forward, the
term hyperspectral will be used generally to refer to either or both of
hyperspectral and multispectral.
[0003] Spectral imaging and analysis is used in a wide range of
applications.
These applications include but are not limited to surveillance, airborne
observation of the health and growth of agricultural crops, environmental
emissions monitoring, identifying minerals of interest in geological samples,
and
monitoring and sorting of food products.
[0004] Hyperspectral imaging involves acquiring and analyzing information
from among several different spectral bands across the electromagnetic
spectrum. A hyperspectral image typically consists of a set of images, each
image taken in a specific band of the spectrum. Figure 1 is a representative
drawing of an airborne spectral sensor 102 mounted on an aircraft 104 that
takes
a plurality of images 120, 122, 124, 126, 128 of a single scene, each image
being taken in a different spectral band. The set of images taken across
different
bands together forms a hyperspectral image 110, sometimes called a
hyperspectral cube.
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[0005] In some applications, a hyperspectral image is generated by an
airborne sensor. For example, a sensor may be mounted to a satellite or an
aircraft, and may take images of the surface of the earth or of the sky. In
other
applications, a sensor may be positioned in other ways or locations, for
example
on a tripod directed at a scene on the ground.
[0006] One of the major problems in the field of solar reflective (i.e.
Visible/Near Infrared/Short Wave Infrared bands or VIS/NIR/SWIR) remote
hyperspectral imaging is the influence of the environment, and to some degree
of
the instrument, on the measurements. Through different physical processes, the
environment can modify the apparent spectral signatures of the surfaces being
imaged and make them more difficult to identify or in a more general sense to
exploit. A role of hyperspectral imagery compensation, also known as
correction,
is to remove one or more of these effects. The images may then be converted so
that the spectra have reflectance units. Reflectance may be a more useful
quantity than radiance since it is intrinsic to the observed material and has
no
dependence on the environment. Reflectance may then be used for target
identification or surface characterization.
[0007] There are generally two different classes of algorithms that
attempt to
transform radiance or raw images into reflectance. One class consists of first
principles methods that rely on physical knowledge of how the environment
interacts with the surfaces to produce radiance. These are sometimes referred
to
as model based methods or radiative transfer methods. The other class consists
of empirical methods, which are sometimes referred to as in-scene methods.
Both classes of algorithms have benefits and drawbacks.
[0008] First principles methods typically use accurate models of radiative
transfer in the atmosphere in order to calculate the environmental effect on
the
target signature and to remove it. These are fundamental methods that are
based on the physical modeling of the environment and its radiative
interaction
with the measured surfaces. Once this interaction is well understood and can
be
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numerically quantified, it can be removed mathematically from the measured
signal to reveal the intrinsic surface characteristics, usually the surface
reflectance.
[0009] Examples of first principle techniques are Fast Line-of-sight
Atmospheric Analysis of Hypercubes (FLAASH Tm) and Atmospheric and
Topographic Correction (ATCORTm-4). Further examples of first principle
techniques include Atmospheric CORrection Now (ACORN) and Atmosphere
REMoval (ATREM).
[0010] First principles methods have a number of advantages, which may
include but are not limited to one or more of the following. It may be
possible to
attain very high precision levels of accuracy in cases where the optical
properties
of the sensor and environment are known, since the modeling of the
phenomenology is well understood and modeled. In addition, the underlying
physical understanding of the phenomena that modify the measured signal can
lead to the deduction of certain parameters required for the modeling, such as
the quantity of water vapor in the air column or the aerosol load.
Furthermore,
since the correction can be applied on a pixel-by-pixel basis, other local
effects
can sometimes also be removed. Also, the properties of the surfaces
themselves, such as the directionality of their reflectance (e.g.
bidirectional
reflectance distribution function - BRDF), may also be modeled accurately and
contribute to the accuracy or interpretation of the correction. Moreover,
prior
knowledge of scene elements may not be strictly necessary since atmospheric
modeling does not depend on them in a first approximation level, for example
if
the effect of neighboring surfaces (adjacency) is neglected.
[0011] Although first principles modeling may be a very powerful tool for
hyperspectral image compensation, there are one or more drawbacks to using
them in some circumstances. These disadvantages may include but are not
limited to one or more of the following. Accurate radiative transfer
calculations
can require one or both of a large amount of computer processing power and
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processing time. In addition, the accuracy of modeling results is sensitive to
the
characterization of atmospheric parameters such as pressure, temperature and
water vapor profiles. In addition, radiative transfer atmospheric compensation
techniques generally do not perform well in less than ideal environmental
conditions, for example partly to fully overcast conditions and dusty
conditions.
Furthermore, first principles methods concentrate on the modeling of the
atmosphere. Sensor characteristics are usually required to be well known and
accounted for before the correction can be applied. If this is not the case
large
errors are likely to occur.
[0012] Again, these are only some possible advantages and disadvantages of
first principle (i.e. radiative transfer) methods.
[0013] The other class of atmospheric compensation algorithms is based on
empirical methods. These algorithms are typically purely empirical in that
they
require no prior knowledge of the atmosphere or environment, and no physical
modeling of radiation transport processes is needed. Instead, they rely on the
presence of surfaces of known reflectance within the scene to calibrate the
sensor-atmosphere system. This usually implies finding a linear relationship
that
transforms the measured arbitrarily-calibrated measurement of these known
targets into their known reflectance. It is then assumed that this linear
relationship holds for all other pixels in the scene, giving rise to a
reflectance
image. Since there are two unknowns to be found ¨ the gain and the offset ¨ at
least two known targets that are sufficiently different in all spectral bands
must
be present in the scene.
[0014] This type of approach is known as an Empirical Line Method (ELM).
ELM is independent of any knowledge or pre-calibration of the sensor, since
the
effect of the sensor is implicitly included in the calibration target measured
signatures. In addition, the ELM algorithm is relatively compact and simple
and
does not depend on external models. This translates to much faster
calculations
that are easily implemented into parallel (multi-processor) or super-parallel
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(GPU) architectures. Furthermore, in ELM the knowledge of atmospheric
properties or aerosol load is not necessary since their effect is
automatically
considered in the calculation of the gain and offset parameters.
[0015] However, empirical methods have some drawbacks. These drawbacks
.. may include but are not limited to the following. One drawback is that the
assumption is made that the scene properties are uniform: the calibration in
gain
and offset is applied identically throughout the entire image. This excludes
directly taking into account broken clouds, adjacency effects or spatial
variability
in aerosol load or water vapor content.
[0016] Another disadvantage may be that surfaces of known reflectances that
are sufficiently different throughout their spectral signature must be present
within the image in order to find the required gains and offsets at each
spectral
band. In other words, knowledge of image elements in the scene must be known
a priori. Therefore empirical methods are often not well suited to
applications
.. where there is little or no prior information about a potential scene. This
condition
is rarely fulfilled in operational contexts, which may include certain
surveillance
and military operations.
[0017] In addition to the above mentioned class of atmospheric
compensation
methods that are based on empirical methods, a more recent family of
.. algorithms has been introduced in an attempt to alleviate some of the
drawbacks
of purely empirical methods. This family of algorithms may be referred to as
empirical-statistical algorithms. For example, the Quick Atmospheric
Correction
(QUACTM) attempts to eliminate the principal inconvenience of the traditional
ELM algorithm in operational contexts, that is, to eliminate the need for a
priori
.. knowledge of scene element reflectances.
[0018] The QUAC family of methods is based on a number of statistical
observations that seem to hold on a number of hyperspectral and multispectral
images: notably, that the mean reflectance of many scene endmembers (at least
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10) is constant (within a multiplicative constant) from scene to scene, as
long as
the scene is diversified enough. The QUAC method also relies on the presence
of a dark (very low reflectance) pixel somewhere in the scene. These two
hypotheses are then used to calculate the gain and offset parameters similarly
to
the ELM method. Vegetation pixels can be used to calculate the multiplicative
factor that transforms reflectance from relative to absolute, since vegetation
reflectance amplitude above the red edge is relatively predictable.
[0019] In a 2012 paper describing some revision to the QUAC algorithms,
it is
explained that although the above described observation on which QUAC is
.. based holds for a wide variety of scenes, there are exceptions in which the
observation does not hold. The identified exceptions include when there is
vegetation, mud, or very shallow water in the spectral image. In this
revision, the
proposed solution is to detect the situations in which the performance of QUAC
would degrade significantly (e.g. presence of mud, shallow water, etc.) and
adjust the universal reflectance spectrum code accordingly.
[0020] Although the more recent revision of QUAC attempts to identify and
account for the known situations in which QUAC would otherwise fail or suffer
degraded performance, the reality is that there will likely always be cases
where
QUAC will fail since it is difficult if not impossible to predict all of these
situations
.. in advance. For example, it has been observed by the present inventors that
the
version of QUAC that is commercially available in ENVI TM 5.0 (a product or
ExelisTM) performs poorly or fails for some winter scenes. This is only one
example of a type of scene for which a recent version of QUAC has poor
performance. It is very likely that QUAC will have difficulties with other
types of
scenes as well. Therefore there is a need for a robust atmospheric
compensation
technique that does not rely as much on the identification and adjustment for
problematic scenes or materials.
[0021] Therefore existing atmospheric compensation methods include first
principles methods, purely empirical methods, and QUAC.
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, =
[0022] In addition, some atmospheric correction techniques use
additional
instrumentation to measure and collect information that can be used to
compensate for atmospheric effects. For example, some hyperspectral sensors
are used in conjunction with a fiber optic downwelling irradiance sensor
(FODIS).
The FODIS collects additional information that is used in the atmospheric
compensation process.
[0023] Where the use or definition of a term or expression in a
reference
referred to in this disclosure is different or inconsistent with the use or
definition
of the term or expression as used herein, the use or definition of the term or
expression provided herein applies.
SUMMARY
[0024] The present disclosure is directed to a method for automatically
performing atmospheric compensation of a multi or hyper spectral image, the
method comprising transforming, at least two endmembers extracted from the
image into at-ground reflectance, finding a matching component in a spectral
library for each of the at least two extracted endmembers, each extracted
endmember and matching component together forming a matched extracted
endmember and spectral library component pair, calculating a gain value and an
offset value using the at least two matched extracted endmember and spectral
library component pairs, and compensating at least part of the image using the
calculated gain and offset values.
[0025] The present disclosure is also directed to a method for
automatically
performing atmospheric compensation of a multi or hyper spectral image, the
method comprising transforming, at least one endmember extracted from the
image into at-ground reflectance, finding a matching component in a spectral
library for each of the at least one endmember extracted from the image, each
extracted endmember and matching component together forming a matched
extracted endmember and spectral library component pair, calculating an offset
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value using a black level obtained from the image or from a radiative transfer
model, calculating a gain value using the at least one matched extracted
endmember and spectral library component pair and the black level, and
compensating at least part of the image using the calculated gain and offset
values.
[0026] The present disclosure is also directed to a method for use in
atmospheric compensation of a multi or hyper spectral image, the method
comprising designating pixels of the image into one of at least a lower water
content group and a higher water content group based on a retrieved water
vapor value of each pixel, calculating an average water vapor content value
for
each of the lower and higher water content groups of pixels, calculating a
gain
value and an offset value for each of the lower and higher water content
groups,
associating each of the average water vapor content values with the respective
gain and offset values of each of the lower and higher water content groups,
calculating gain and offset values for individual pixels of the image by
interpolating or extrapolating based on the two average water vapor content
values using the water vapor value of each pixel, and compensating at least
part
of the image using the calculated gain and offset values of the individual
pixels.
[0027] The present disclosure is also directed to a method for use in
shadow
compensation of a multi or hyper spectral image, the method comprising (a)
identifying a brightest pixel in the image in term of its norm and assuming
the
pixel is fully lit by both the sun and the sky, (b) determining a direct flux
(sun) and
a diffuse flux (sky) of the pixel, (c) determining a sunlit only surface
signature and
a skylit only surface signature of the pixel using the direct and diffuse
fluxes, (d)
determining an abundance of each of the sunlit surface and skylit surface
signatures in pixels of the image, and removing the sunlit surface and skylit
surface signatures from pixels of the image, (e) repeating (a) through (d) a
predetermined number of times or until the norm of the pixel last identified
in (a)
is below a predetermined threshold value, where the pixel identified in (a) in
the
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next iteration is the next brightest pixel in the image after the previous
iteration
sunlit and skylit only surface signatures have been removed, and (f)
compensating at least some pixels of the image, where a given pixel is
compensated using the direct and diffuse fluxes of the given pixel as well as
a
sun illumination factor and a sky illumination factor of the given pixel, the
sun
illumination factor being the sum of the abundances of the sunlit surface
signatures and the sky illumination factor being a sum of the abundances of
the
skylit surface signatures of all pixels selected in the iterative process for
the
given pixel.
[0028] The present disclosure is also directed to a non-transitory computer-
readable storage medium comprising instructions for execution on one or more
electronic devices, the instructions for a method for automatically performing
atmospheric compensation of a multi or hyper spectral image, the method
comprising transforming, at least two endmembers extracted from the image into
at-ground reflectance, finding a matching component in a spectral library for
each of the at least two extracted endmembers, each extracted endmember and
matching component together forming a matched extracted endmember and
spectral library component pair, calculating a gain value and an offset value
using the at least two matched extracted endmember and spectral library
component pairs, and compensating at least part of the image using the
calculated gain and offset values.
[0029] The present disclosure is also directed to a non-transitory
computer-
readable storage medium comprising instructions for execution on one or more
electronic devices, the instructions for a method for automatically performing
atmospheric compensation of a multi or hyper spectral image, the method
comprising transforming, at least one endmember extracted from the image into
at-ground reflectance, finding a matching component in a spectral library for
each of the at least one endmember extracted from the image, each extracted
endmember and matching component together forming a matched extracted
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endmember and spectral library component pair, calculating an offset value
using a black level obtained from the image or from a radiative transfer
model,
calculating a gain value using the at least one matched extracted endmember
and spectral library component pair and the black level, and compensating at
least part of the image using the calculated gain and offset values.
[0030] The present disclosure is also directed to a non-transitory
computer-
readable storage medium comprising instructions for execution on one or more
electronic devices, the instructions for a method for performing atmospheric
compensation of a multi or hyper spectral image, the method comprising
designating pixels of the image into one of at least a lower water content
group
and a higher water content group based on a retrieved water vapor value of
each
pixel, calculating an average water vapor content value for each of the lower
and
higher water content groups of pixels, calculating a gain value and an offset
value for each of the lower and higher water content groups, associating each
of
the average water vapor content values with the respective gain and offset
values of each of the lower and higher water content groups, calculating gain
and offset values for individual pixels of the image by interpolating or
extrapolating based on the two average water vapor content values using the
water vapor value of each pixel, and compensating at least part of the image
using the calculated gain and offset values of the individual pixels.
[0031] The present disclosure is also directed to a non-transitory
computer-
readable storage medium comprising instructions for execution on one or more
electronic devices, the instructions for a method for performing shadow
compensation of a multi or hyper spectral image, the method comprising (a)
identifying a brightest pixel in the image in term of its norm and assuming
the
pixel is fully lit by both the sun and the sky, (b) determining a direct flux
(sun) and
a diffuse flux (sky) of the pixel, (c) determining a sunlit only surface
signature and
a skylit only surface signature of the pixel using the direct and diffuse
fluxes, (d)
determining an abundance of each of the sunlit surface and skylit surface
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signatures in pixels of the image, and removing the sunlit surface and skylit
surface signatures from pixels of the image, (e) repeating (a) through (d) a
predetermined number of times or until the norm of the pixel last identified
in (a)
is below a predetermined threshold value, where the pixel identified in (a) in
the
next iteration is the next brightest pixel in the image after the previous
iteration
sunlit and skylit only surface signatures have been removed, and (f)
compensating at least some pixels of the image, where a given pixel is
compensated using the direct and diffuse fluxes of the given pixel as well as
a
sun illumination factor and a sky illumination factor of the given pixel, the
sun
illumination factor being the sum of the abundances of the sunlit surface
signatures and the sky illumination factor being a sum of the abundances of
the
skylit surface signatures of all pixels selected in the iterative process for
the
given pixel.
BRIEF DESCRIPTION OF THE DRAWINGS
[0032] The present disclosure will be better understood having regard to
the
drawings in which:
[0033] Figure 1 is a drawing representing an airborne spectral sensor
collecting a plurality of images each at a different spectral band;
[0034] Figure 2 is a drawing representing various light paths from the
atmosphere (e.g. the sun) to a spectral sensor;
[0035] Figure 3 is a flow diagram of a process for atmospheric
compensation
according to the present disclosure;
[0036] Figure 4 is a flow diagram of a process for atmospheric
compensation
according to the present disclosure;
[0037] Figures 5A to 5D are graphs representing the states of an example
pixel at various steps in an algorithm of the present disclosure;
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[0038] Figure 6 is a graph showing an example set of radiance endmembers
extracted from an AVIRIS image of a moderately developed rural area;
[0039] Figure 7 is a graph showing the extracted endmembers of Figure 6
after a first pass correction (e.g. transformed into approximate reflectance);
[0040] Figure 8 is a graph showing the library matches in an example
library
for the extracted and corrected reflectance endmembers shown in Figure 7;
[0041] Figure 9 is a graph representing a least square regression method
for
calculating gain and offset values;
[0042] Figure 10 is a graph representing a weighted average of endmembers
method for calculating gain and offset values;
[0043] Figure 11A is a portion of an AVIRIS sensor image compensated (or
"corrected") using an algorithm according to the present disclosure;
[0044] Figures 11B, 11C, 11D and 11E are graphs representing pixel
reflectance spectra for vegetation, dry vegetation, a road and a blue roof,
respectively, indicated in Figure 11A as corrected by QUAC, FLAASH and an
algorithm according to the present disclosure;
[0045] Figure 12 is a flow diagram of a process for correcting for the
effects
of water absorption;
[0046] Figure 13 is a flow diagram of a process for correcting for the
effects
of shadows in an image;
[0047] Figure 14 is an example image prior to shadow correction;
[0048] Figure 15 is the image of Figure 14 after shadow correction;
[0049] Figure 16 is the image of Figure 11 after shadow correction; and
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[0050] Figure 17 is simplified block diagram of a computing device
capable of
performing the one or more of the methods or functions of the present
disclosure.
DETAILED DESCRIPTION OF THE DRAWINGS
[0051] The present disclosure is directed to methods and techniques for use
in atmospheric compensation of images in the solar reflective wavelength range
(e.g. < 3000nm).
[0052] In one aspect, the present disclosure is directed to atmospheric
compensation algorithms for use with hyperspectral and multispectral images.
At
least one algorithm is a two-pass hybrid approach that applies a first
correction
that transforms image endmembers (usually in at-sensor radiance units) into
approximate reflectance. The resulting reflectance endmembers may then be
matched to elements in a spectra library, and a correction may then be
established in a manner similar to empirical line methods. For example, pixels
of
the input image may be corrected using calculated gain and offset values to
transform the pixels from radiance into at-ground reflectance. The algorithm
may
achieve results with similar accuracy as first principles methods with a speed
comparable to empirical methods. The new algorithm may therefore be useful in
operational contexts for near real time detection and classification purposes.
[0053] In another aspect, the present disclosure is directed to atmospheric
compensation comprising one or more of the following: transforming radiance
calibrated measurements into approximate reflectance using top-of-atmosphere
solar irradiance; masking anomalous pixels that may corrupt or otherwise
negatively affect the correction; extracting endmembers from the transformed
scene; correcting the endmembers, sometimes approximately, to at-ground
reflectance; matching resulting reflectance endmembers to known reflectances
contained in a reflectance library; using the matched library spectra along
with
the corresponding measured spectra to compute the gain and offset parameters;
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and transforming the original radiance image into a reflectance image using
the
calculated spectral gain and offset.
[0054] In another aspect, the present disclosure relates to atmospheric
compensation in which the effects of water absorption are reduced or removed
from a multi or hyper spectral image. Pixels of an input image may be divided
into at least two groups, a lower water content group and a higher water
content
group. Gain and offset values are determined for each of the groups. An
average
water vapor content value is also calculated for each of the groups. Gain and
offset values may then be determined for individual pixels of the input image
by
interpolating or extrapolating based on the two average water vapor content
values using the water vapor value of an individual pixel.
[0055] In another aspect, the present disclosure relates to methods for
performing shadow compensation on hyper and/or multi spectral images.
[0056] In a multispectral or hyperspectral image, the radiance at the
sensor
for each pixel and wavelength may be represented by the following equation:
R (A) = S B (2.) 13 (A) + C(A) S(A) (A) + P a (A)]
(1)
1¨padj(A)Scab (A) padi(A)Salb(A)
Where S, is the solar flux normal to the surface above the atmosphere, p is
the
actual surface reflectance, padj is the average reflectance of the surfaces
adjacent to the measured surface and Sat is the atmosphere spherical albedo ¨
the capacity of the atmosphere to reflect upwelling radiation from the ground
back towards the ground. The A, B and C coefficients represent the scattering
and extinction effects along different paths in the atmosphere as represented
by
Figure 2; A represents the radiance contribution of the sun directly scattered
by
the atmosphere while never encountering the surface, B is the transmittance
along the sun to surface to sensor path and C is the light that was reflected
off
other adjacent surfaces before being scattered by the atmosphere between the
sensor and the measured surface (giving rise to the adjacency effect).
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[0057] The
observed reflectance at the sensor is obtained by dividing the at-
sensor radiance by the incident solar flux:
Pobs(A) = (41? (2)
And the surface reflectance can be isolated to yield:
P (A) =
1¨pacijOOSalb(A)[ CO)
Pobs(A) A(A)
B (d) 1¨ p adi(A)Salb(A) P a ( (3)
[0058] Figure 2 is a diagram showing various paths of light coming from
the
atmosphere or space and ending up at an airborne sensor 202. The p is the
ground reflectance of object or target 204, pad; is the average reflectance in
the
area adjacent to the target, and poi, is the reflectance observed by sensor
202.
As shown in Figure 2, light may be reflected by target 204, other objects or
surfaces 206, such as the ground, or by the atmosphere 208 (e.g. after
atmospheric or solar light has been scattered). The A, B and C coefficients in
equation (3) and shown in Figure 2 represent the scattering and extinction
effects along different paths in the atmosphere.
[0059] In reality, the adjacent reflectances padi in the denominator of
the B
and C terms are not exactly the same as those in the numerator of the C term
because they do not stem from the same scattering geometry, but they can
usually be approximated as being equal. Furthermore, equation (3) can be re-
written in a simple linear correction form with a gain a and offset iq
P(A) = a(A)P0bs(2-) + fl(A) (4)
where
a(A) ,l-Paci.1:(A2))Salb(A) (4)
and
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¨a .(20Sath(AYI c (A)
fl (A) = A(A){1pd '8(A)
e (A) Padi ) = (5)
Or
13(A) = A(A)a(A) ¨ ¨
Bc((AA)) Padj(A) (7)
[0060] All relevant quantities, the A, B and C coefficients, Sea and the
kernel
required to obtain the average ground reflectance pad' are usually calculated
from existing radiative transfer models such as MODTRAN TM or 6S. Also note
that the gain and offset terms can be dependent on the pixel location
[0061] Aspects of the present disclosure are now described in more
detail.
[0062] Figure 3 shows a process for atmospheric compensation according to
one aspect of the present disclosure.
[0063] The process starts at block 302 and proceeds to block 304, where
at
least two endmembers are extracted from an image, in some embodiments an
input image. The extracted endmembers are generally in the form of extracted
pixel endmembers, although this is not necessarily a requirement in all
embodiments. The process then proceeds to block 306 where the at least two
extracted endmembers are compared to a spectral library. A match is selected
for each of the at least two extracted endmembers. The process then proceeds
to block 308 where gain and offset values are calculated using the matched
extracted endmember and library component pairs. The process then proceeds
to block 310 where a new or compensated image is generated by applying the
calculated gain and offset values to the input image. The process then
proceeds
to block 312 and ends.
[0064] The process shown in Figure 3 uses two or more extracted
endmembers to calculate gain and offset values. However, it may also be
possible to calculate gain and offset values using one (at least one)
extracted
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endmember and a black level. The gain value may be determined using a
matched extracted endmember and spectral library component pair, and a black
level obtained from the image or from a radiative transfer model. The offset
may
be determined using the black level. In at least some embodiments, the order
of
.. calculating gain and offset values is not necessarily crucial; the gain may
be
calculated before the offset or the offset may be calculated before the gain.
In
other embodiments, more than one extracted endmember may be used to
calculate a gain value.
[0065] Accordingly, since there are two unknowns, gain and offset, these
values may be calculated using at least two extracted endmember and spectral
library component pairs. These at least two pairs provide at least two
equations,
which may be solved for the two unknowns. In other embodiments, gain and
offset values may be determined using at least one extracted endmember and
spectral library component pair and a black level. The at least one extracted
endmember and spectral library component pair and the black level may be used
to determine the gain. The offset is determined based on the black level.
[0066] According to another aspect of the present disclosure, a method
for
atmospheric compensation is provided. For descriptive purposes, the method or
algorithm may be divided into four main steps: transforming an input image
into a
top of atmosphere (TOA) reflectance image; extracting one or more
endmembers from the TOA reflectance image; transforming (or "correcting") the
endmembers to at-ground (e.g. intrinsic) but approximate reflectances and
matching the spectra of these endmembers to spectra in a library; and
computing gain and offset values using the resulting matched endmembers and
library spectra, and applying these to one or more corresponding bands of the
input image to derive a compensated (or "corrected") reflectance image. Figure
4 shows a process for use in atmospheric compensation according to this aspect
of the present disclosure.
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, .
[0067] Although a specific number and order of steps (blocks) are
shown in
Figure 4, this is not meant to be limiting. Other embodiments may have fewer
or
more steps, and the steps may be performed in a different order as far as is
possible.
[0068] The process shown in Figure 4 starts at block 402 and proceeds to
block 404, where an input image or a portion thereof may be transformed into
top
of atmosphere (TOA) reflectance. In some embodiments, the input spectral
image may be calibrated in radiometric units, such as for example at-sensor
radiance. In other embodiments, the input spectral image may be uncalibrated.
Whether or not the input image is calibrated may depend on the one or more
techniques or algorithms used in one or more following steps in the process.
[0069] Endmember extraction algorithms are more effective when
image
spectra from pixel to pixel are as different as possible. Therefore in at
least some
embodiments, it is beneficial to remove common spectral elements such as the
solar spectrum. This may be achieved by normalizing the input image, or a part
thereof, by the extra-atmospheric solar irradiance. In addition, the image may
be
normalized by the extra-atmospheric solar irradiance that is resampled to the
sensor resolution and that may be normalized (or modeled or adjusted)
according to the sun-earth distance.
[0070] The input image may be normalized using any suitable method or
technique. For example, the image or a sub-portion thereof maybe normalized
using the methods described in Kurucz, R. L., The Solar Interior and
Atmospheric (A. N. Cox, W. C. Livingston, and M. Matthews, Eds.), University
of
Arizona Press, Tucson, pp. 1-663 (1992), or Kurucz, R. L. "The solar
irradiance
by computation", Proceedings of the 17th Annual Review Conference on
Atmospheric Transmission Models, 7 June, Air Force Geophys. Lab., Hanscom
AFB, MA. (1994).
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[0071] In at least one embodiment, only a portion of the image is
transformed
to top-of-atmosphere reflectance. The portion of the image that is transformed
may correspond to a limited number of spectral bands of the image that is
needed to find the required endmembers. Transforming only a limited number of
bands in the image may be advantageous. For example, most endmember
extraction algorithms require only as many bands as the desired number of
endmembers, or even fewer bands, in order to function correctly. Thus
transforming only a limited number of bands may have a partial or even
significant speed advantage over transforming all bands. In contrast,
transforming additional bands in excess of the number of bands needed may
unnecessarily take up more time and use processing resources.
[0072] In at least one embodiment, one or more bands that are
transformed
may correspond to one or more window regions of the atmosphere. A window
region of the atmosphere is a band where there is little molecular absorption
of
light by molecules in the atmosphere. In this sense, the atmosphere is more or
less transparent in a window region. By using and transforming bands that
correspond to window regions, an endmember extraction algorithm may be
made more effective since some common features amongst image pixels, such
as absorption features, are not present or are relatively low in the window
regions. Furthermore, as mentioned above, transforming only a limited number
of bands may be advantageous in terms of one or both of processing time and
use of processing resources compared to transforming more or all bands.
[0073] In at least one embodiment, the algorithm attempts to find bands
close
to one or more of 0.47, 0.55, 0.66, 0.86, 1.027, 1.246, 1.38, 1.612 and 2.15
pm
in the input image to be transformed to TOA reflectance. Most of these bands
correspond to window regions of the atmosphere (except for the 1.38 band,
which corresponds to a water absorption band used for cirrus cloud detection)
and are well suited for endmember extraction. This is merely an example. One
or
more bands at different wavelengths may be transformed.
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[0074] In at least one embodiment, the transformation may additionally
use
downwelling flux measurements in order to obtain more accuracy. These
measurements may be obtained from a downwelling irradiance sensor. The use
of downwelling flux measurements may be desirable in less than ideal
.. environmental conditions, for example where there are clouds (broken or
unbroken cloud cover) located above a target and the spectral sensor. If the
downwelling flux measurements contain the same sensor effects as the data and
is in the same units (as is often the case when using an at-sensor FODIS), the
approximate reflectance may be obtained with:
p=7 (8)
Imeas
where M is the measurement and / i the measured downwelling flux
-meas .s
including the sensor effects and with the same units as M. Using at sensor
downwelling irradiance or even better, irradiance collected at ground level
close
to the image location, has the benefit of removing at least part of the
transmission effects. In addition, since it is a measurement, it reflects the
actual
state of the atmosphere. As a result, difficult and even cloudy (broken or
not)
conditions may be transformed with more accuracy.
[0075] Data from a downwelling irradiance sensor may also be used in
clear
sky conditions. In addition, a downwelling irradiance sensor may be used when
the input spectral image is arbitrarily calibrated so long as the downwelling
irradiance sensor provides data with the same arbitrary calibration as the
input
spectral image.
[0076] The process then proceeds from block 404 to block 406 where one or
more pixels, such as bad, anomalous or unwanted pixels, may be masked or
removed. Masking may involve simply ignoring one or more pixels. Certain kinds
of pixels may negatively affect the correction and therefore it may be
desirable to
remove them. Since the correction coefficients (e.g. a and 13 in equations (5)
and
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(6) above) may be obtained from the relation between image (endmember) pixel
spectra and library spectra, bad or unwanted pixels that may negatively affect
the correction may be removed. Unwanted pixels may include one or more of
bad pixels, corrupted pixels, pixels that may have been added to the edge of
the
image by procedures such as orthorectification, sharp glints, hot objects and
clouds. Some of these unwanted pixels may not have a corresponding
component (or "entry") in a spectral library and may therefore skew the
overall
result of the atmospheric compensation. In addition, vegetation pixels may
also
negatively affect the overall results as they may have a sharp reflectance
characteristic (e.g. a red edge). In at least one embodiment, pixels
containing
vegetation or other materials having a sharp reflectance characteristic may be
masked or otherwise ignored.
[0077] Pixel masking may be less important in embodiments that use a
weighted algorithm (e.g. least squares, etc.) for calculating gain and/or
offset
values since pixels having either no match or a poor match in the spectral
library
will be assigned a lower weight and thus will have a diminished impact on the
overall accuracy and performance of the compensation algorithm.
[0078] The process proceeds from block 406 to block 408 where one or more
endmembers are extracted from the transformed image data. An endmember is
a spectrally distinct component of an image (e.g. soil type 1, soil type 2,
vegetation type 1, vegetation type 2, manmade material type 1, etc.).
[0079] In the present process, any suitable endmember extraction
algorithm
may be used as long as it is sufficiently accurate and yields physically
plausible
reflectance endmembers. In this sense, physically plausible means that if the
input reflectance image has no negative values, the resulting endmembers are
also positive in all bands, as reflectances normally should be. Sufficiently
accurate means that endmembers output by the algorithm are sufficiently
different from one another and represent "pure" spectra as much as possible.
It
is also desirable that endmembers be actual pixels from the image (these are
- 21 -
called pixel endmembers). This ensures that the atmospheric effect in these
endmember spectra is representative of the image.
[0080] Several endmember extraction algorithms currently exist,
including N-
FINDR, Convex Cone Analysis (CCA), Iterated Constrained Endmembers (ICE),
Vertex Component Analysis (VCA), etc. Some of these algorithms can be slow,
not easily parallelizable, or produce unphysical negative coefficients or
spectra,
so not all are necessarily well-suited for use in embodiments of the present
disclosure. Another endmember extraction method is the sequential maximum
angle convex cone endmember model, often referred to as "SMACC". The
SMACC algorithm is described in Gruninger, John H., Anthony J. Ratkowski, and
Michael L. Hoke, "The sequential maximum angle convex cone (SMACC)
endmember model" Proceedings of SPIE, Vol. 5425 (2004). The SMACC
algorithm is based on a convex cone model of the spectral vector data. It
selects
endmembers directly from the set of pixel spectra while simultaneously
generating positivity-constrained expansion coefficients for each pixel in the
image. SMACC and other similar algorithms (such as the augmented modified
Gram-Schmidt, or AMGS algorithm on which SMACC is based) are efficient and
produce physical pixel endmembers and may be suitable for use in some
embodiments of the present disclosure. In addition, the number of endmembers
is not limited by the number of bands leading to better efficiency since fewer
bands may be used. Because of this, in at least one embodiment, endmembers
may be extracted using the SMACC or a similar algorithm.
[0081] Although the endmembers may be obtained from the TOA reflectance
image, in at least one embodiment of the present disclosure the endmember
extraction algorithm is implemented to track the location (pixel) of each
endmember in the image. This location may be used to obtain the full
resolution
radiance endmembers from the original image, which may be used to calculate
the radiance to reflectance gains and offsets for the entire image and at all
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wavelengths from a weighed least square regression. The radiance of an
example of a set of 20 non-vegetation endmembers extracted from an Airborne
Visible / Infrared Imaging Spectrometer (AVIRIS) image of a moderately
developed rural area is shown in Figure 6.
[0082] In the present method, the number of endmembers extracted from the
image and used for atmospheric compensation may vary. In at least one
embodiment, the number of endmembers extracted and used for compensation
may be in the range of approximately 20 to 30 endmembers. However, this
range is not meant to be limiting. Other numbers of endmembers may be used.
Using more endmembers will typically statistically decrease the error in a
least
square regression used to calculate gain and/or offset values. However, too
many endmembers may hinder the correction, since repetition may start to
occur, giving too much weight to a given pixel type. Increasing the number of
endmembers also increases the amount of processing required, thus there is a
trade-off between the number of endmembers used and the amount of
processing required.
[0083] The process shown in Figure 4 then proceeds from block 408 to
block
410 where the one or more radiance endmembers extracted from the input
image may be transformed to at-ground reflectances. This transformation may
be considered to be a first pass endmember correction.
[0084] The extracted radiance endmembers may be transformed to at-ground
reflectances using any suitable method. In some embodiments, the
transformation may transform the endmembers to an approximate at-ground
reflectance.
[0085] In at least one embodiment, a radiative transfer (RT) technique may
be
used for the transformation. Any suitable radiative transfer technique may be
used. Examples include but are not limited to MODTRAN, 6S and FLAASH.
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[0086] It may be acceptable that the transformation into at-ground
reflectance
is only relatively accurate. Sufficient accuracy may be achieved when the
correction lets most reflectance endmembers be matched to their corresponding
library entry when such an entry exists. In such a case, tabulated extinction
coefficients and Beer's Law (multiplication of path segment transmittances)
may
be used to increase the efficiency of this step. Thus in some embodiments, it
is
not necessary that the transformation be void of errors or spectral artifacts,
or to
meet the accuracy of full features radiative transfer methods like FLAASH.
Again, it may be sufficient that the transformation allows for the chance of
finding
good matches for the endmembers in a spectral library.
[0087] In other embodiments, instead of a radiative transfer technique,
one or
more other techniques for transforming extracted endmembers into reflectances
or approximate reflectances may be used. In at least one embodiment, an
empirical or statistical-empirical technique such as the one used in QUick
Atmospheric Correction (QUAC) may be used. Other options exist including but
not limited to dividing by top-of-atmosphere irradiance and using input from a
fiber optic downwelling irradiance sensor (FODIS).
[0088] In some embodiments, the first pass correction based on radiative
transfer calculations may be made more accurate by the use or more of water
.. retrieval techniques, aerosol retrieval techniques, and measurements of
downwelling irradiance.
[0089] In at least one embodiment, a pixel-by-pixel based water retrieval
algorithm using a tree band ratio may be used. Such an algorithm may be based
on a method such as that of Kaufman, Y. J., and Gao B-C. "Remote sensing of
water vapor in the near IR from EOS/MODIS." Geoscience and Remote Sensing,
IEEE Transactions on 30.5 (1992): 871-884 and of FeIde, G. W., Anderson, G.
P., Gardner, J. A., Adler-Golden, S. M., Matthew, M. W., & Berk, A. (2004,
August). "Water vapor retrieval using the FLAASH atmospheric correction
- 24 -
algorithm." In Defense and Security (pp. 357-367). International Society for
Optics and Photonics.
[0090] Aerosol retrieval used in some embodiments may use methods such
as, but not limited to that described by Kaufman, Y. J., and Tame, D.
"Algorithm
for remote sensing of tropospheric aerosol from MODIS." NASA MODIS
Algorithm Theoretical Basis Document, Goddard Space Flight Center 85 (1998)
or presented by Bernstein, L.S., Adler-Golden, S.M., Sundberg, R.L., Levine,
R.Y., Perkins, T.C., Berk, A., Ratkowski, A.J., FeIde, G., Hoke, M.L., "A new
method for atmospheric correction and aerosol optical property retrieval for
VIS-
SWIR multi- and hyperspectral imaging sensors: QUAC (QUick atmospheric
correction)," Geoscience and Remote Sensing Symposium, IGARSS '05.
Proceedings. 2005 IEEE International, vol.5, pp. 3549- 3552, (July 2005).
[0091] Measured solar irradiance at sensor or ground level may be used
in
some embodiments to increase the accuracy of either the normalization of
radiance data to reflectance or of the first-pass correction. A method such as
Homolova, L., Alanko-Huotari, K., & Schaepman, M. E. (2009, August).
"Sensitivity of the ground-based downwelling irradiance recorded by the FODIS
sensor in respect of different angular positions." In Hyperspectral Image and
Signal Processing: Evolution in Remote Sensing, 2009. WHISPERS'09. First
Workshop on (pp. 1-4). IEEE, may be used,
[0092] In at least one embodiment that uses a fast radiative transfer
technique, the radiative technique may be based on lookup tables. The
technique may be a single layer analytical delta-Eddington multiple scattering
algorithm for downward irradiance calculations, which is described in more
detail
below. In addition, the technique may incorporate water vapor and aerosol
retrieval techniques.
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[0093] A method for calculating the correction offset from the black
level is
now described.
[0094] Although not necessary in all embodiments according to the present
disclosure, it may be useful and more accurate to use the path radiance
between
.. the sensor and ground directly to calculate the correction offset. This may
be
accomplished in several different ways.
[0095] One way is simply to find the minimum value for each band in the
input
image. For even a moderately diverse scene (many types of materials), it is
usually a good assumption that at least one material will have a low
reflectance
.. in each band. This is true even if no single pixel (material) is dark in
every band.
The black level curve may then be used directly as an anchor point in a least
square regression, which is described further below.
[0096] In some cases, either because of spectral artifacts that have not
been
detected by the masking procedures or because the image does not contain
.. sufficient materials with low spectral reflectance bands, finding the black
level
offset using the image minimum can fail. It may be more robust to use a true
but
approximate radiative transfer model to calculate the black level. For
example,
the total (direct and scattered) downwelling irradiance at ground level may be
measured or calculated.
[0097] Delta-Eddington single layer two-stream multiple scattering is now
described.
[0098] As previously mentioned, the more accurate the correction on the
endmembers is, the more likely the spectral matching will be accurate and so
will
the entire atmospheric correction. However, this increased accuracy may come
.. at the price of an increase in computation time. Because of this, a
simplified
multiple scattering algorithm may be implemented based on the delta-Eddington
theory. This theory is described in Joseph, J. H., W. J. Wiscombe, and J. A.
- 26 -
õ .
Weinman, "The delta-Eddington approximation for radiative flux transfer÷
Journal
of the Atmospheric Sciences vol 33, 12, 2452-2459 (1976); Meador, W. E., and
W. R. Weaver, "Two-stream approximations to radiative transfer in planetary
atmospheres- A unified description of existing methods and a new improvement",
Journal of the atmospheric sciences vol. 37, 3, pp. 630-643 (1980); and
Jimenez
Aquino, J. I., and J. R. Varela, "Two stream approximation to radiative
transfer
equation: An alternative method of solution", Revista mexicana de fisica
51.001
(2009).
[0099] A major simplification in the multiple scattering algorithm
may be that
the atmosphere modeled is composed of a single uniform but representative
atmospheric layer. In such a case, the radiative transfer equations can be
solved
using upper and lower boundary conditions to produce analytical expressions
for
the upwelling radiative flux F+ (t) and the downwelling radiative flux F(t) at
any
optical depth 1" within the layer (see previously mentioned references). The
lower
boundary condition is usually a lambert reflector with the average scene
reflectivity fi, while the upper boundary condition is either a diffuse flux F
- (0) or
a collimated source where the diffuse part is F - (0) = 0.
[00100] The path radiance may be calculated by integrating the multiple
scattering source function multiplied by the transmittance along the path. In
at
least one embodiment, the source function approximation used may be as
described in Isaacs, R. G., et al., "Multiple scattering LOVVTRAN and FASCODE
models", Applied optics 26.7 (1987): 1272-1281. This source function
approximation is:
wo
ims(T,P) = ¨(F+(T)[1 ¨ fls OM + F¨ (T)13s(,1)1 (9)
rr
where ps is the atmosphere back-scattering fraction and p is the cosine of the
sensor look zenith angle. The backscattering fraction is defined as the
integration
of the azimuthally averaged scattering phase function over the backward
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direction. In at least one embodiment, the following Henyey-Greenstein phase
function approximation may be used:
1 ¨ g2
P(i-tscat) = (10)
+ g2 2gti5Cat)312
where Ltscat - is the cosine of the scattering angle.
,-
[00101] In embodiments in which the approximation of equation (10) is used,
the backscattering fraction may be analytically integrated and the results
fitted to
a fifth degree polynomial in g and tabulated in it (linearly interpolated).
This is the
method used in MODTRAN.
[00102] The multiple scattering path radiance becomes
1
Inis( ) = ¨ f Jrnser, pt)ea-s-01 dt (11)
Ts
where the integration is between the sensor optical depth Ts and the ground
(total) optical depth T. Ims( ) can be solved by inserting the analytical
expressions for F+(/) and F(T) into equation (9), and since the expressions
for
fluxes mainly involve simple exponential function, equation (11) can be
integrated analytically.
[00103] In at least some embodiments, the sensor viewing angle is assumed to
be in the nadir direction so that pt = 1 in equations (11) and the resulting
solution.
However, this is not intended to be limiting.
[00104] In the case where the upper boundary flux is diffuse, the path
radiance
is given by equation (11), so that /( ) = /ms(p). On the other hand, if the
upper
boundary condition is a collimated source, equation (11) only results in the
multiple scattering contribution. The single scattering path radiance must be
added by using the Henyey-Greenstein phase function given in (10) so that
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/(y) = iins(y) + WOP(Rscat)Sn _T
e dbio erT;5-T1(tt0-1+1C1)11 (12)
(110-1 +11-1)
where uscat --
is the solar scattering angle, Po is the cosine of the collimated source
!1
scat
angle and Ti is the optical depth from the TOA to the ground. Again, it
may be assumed that the look angle is towards the nadir, so that II
r-SCat = Ito
and = 1 in equation (12). Note that using a more accurate representation of
the
.. phase function may produce more accurate results here.
[00105] We now return to the transformation of the radiance endmembers into
approximate at-ground reflectance. Again, this step is represented by block
410
in Figure 4.
[00106] The radiance endmembers may be transformed to at-ground (e.g.
intrinsic) reflectance by calculating the necessary gain and offset. In at
least one
embodiment, when the upper boundary flux is diffuse, the black level is the
path
radiance as calculated with equation (11). In the case where upper boundary
condition is a collimated source, then the black level may be the path
radiance
as calculated according to equation (12).
[00107] In a case where the downwelling flux is considered to be isotropic
(clouds), the gain may be calculated by
Jr
a = ____________________________________________________________ (13)
exp [¨ (T1 ¨ Ts)] (TO
which is the scattered downwelling flux at ground level multiplied by the
ground
to sensor transmittance (assuming a nadir look angle).
[00108] When the downwelling flux (TOA or FODIS) is a collimated beam
(sun), the gain in most instances includes the direct transmitted solar
irradiance
a = _____________________________________________________________ (14)
exp[¨(T1 ¨ Ts)] F- (TO + exp(To Tihio)Snito
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[00109] Note that contrarily to equations (1) through (7), the adjacency
corrections are not explicitly stated in equations (13) and (14) since the
delta-
Eddington multiple scattering calculates them implicitly by setting the bottom
condition as the (approximate) average image reflectance A.
[00110] The correction offset can be calculated from the black level b (as
obtained from the image or modeled) by
[3 = ¨ba (15)
[00111] Figure 7 shows the endmembers of Figure 6 after they have been
transformed (or corrected) into an approximate at-ground reflectance.
[00112] The resulting correction may generally not be of the same quality as
if
corrected by algorithms such as FLAASH because of the approximations used.
Nevertheless, the correction may be generally sufficient in order to enable
the
matching of the corrected endmembers to a spectral library.
[00113] The process shown in Figure 4 proceeds from block 410 to block 412,
in which at least one endmember that was extracted from the input image and
transformed to approximate reflectance is compared against a spectral library,
which may contain a wide variety of materials. In some embodiments, most or
all
extracted endmember are compared to the spectral library.
[00114] The spectral library may comprise of high quality reflectances (e.g.
in
terms of quality of the samples preparation, instrument calibration,
reflectance
standards used and of the measurement protocol) for commonly found surface
materials, which may include both natural and manmade materials. In some
embodiments, a generic library may be used as a default. However, better
overall performance may be achieved in some embodiments by using a library
that better reflects or corresponds to the types of endmembers likely to
appear in
the image to be taken. For example, the components of the library may better
represent the geographical location where the input image is taken. For
example,
- 30 -
a spectral library may contain more components for endmember spectra
commonly found in arid environments when the image to be taken in such an
environment. Furthermore, library components for endmember spectra that are
unlikely to be found in the environment in which the image is to be taken may
be
removed from the library. For instance, snow reflectances may be removed from
a library if an image is to be collected in an environment where there is no
snow.
In addition or alternatively, the library may be supplemented with in-situ
reflectances, or may even consist solely of in-situ reflectances. However, in
some circumstances, in-situ reflectances may not be available prior to the
time at
which an image is collected and therefore may not form part of the library.
[00115] In at least one embodiment, a spectral library may be composed of
spectra that are common enough and likely to be present in a hyperspectral
image in pure form in an unmixed pixel, are sufficiently different from one
another, and cover a wide range of reflectances in all spectral regions.
[00116] For example, in one embodiment that has been tested, the spectral
database was composed of 213 spectra that are a combination of manmade
material, soil, mineral, vegetation and snow spectra from the Johns Hopkins
University (JHU) Spectral Library and the U.S. Geological Survey (USGS)
Digital
Spectral Library libraries that are provided with the ENVI software,
complemented with dry vegetation spectra from Elvidge, C., D., "Visible and
near
infrared reflectance characteristics of dry plant materials", Remote Sensing
vol.11, 10, pp. 1775-1795 (1990). However, this is not intended to be
limiting.
Other libraries or subsets thereof may be used.
[00117] The one or more extracted and transformed endmembers are
compared to the components (or "entries") in a spectral library in an attempt
to
find a best match between a given endmember and a component in the spectral
library. In at least one embodiment, the comparison may be done automatically,
meaning without any user intervention. For example, in some existing
correction
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techniques, such as standard ELM, a human operator identifies (e.g. by
clicking
with a mouse, entering the coordinates, etc.) of pixels in an image that are
known to be associated with specific spectra (e.g. known targets). In
contrast, in
some embodiments of the present disclosure, the comparison and matching may
be done automatically.
[00118] The set of components in the library to which an endmember is
compared may be referred to as a comparison set. In some embodiments, the
comparison set comprises at least some of the components in the library. In
some embodiments, the comparison set comprises every component in the
library. A metric or score indicating the relative degree of the match between
an
endmember and the library component may be computed for each component in
the comparison set. A matching library component for the given endmember may
then be selected based on the metrics or scores that were computed for the
components in the comparison set. For example, the library component selected
as the match may be the component that has the highest or lowest metric or
score value. However, other ways of selecting the matching component are
possible.
[00119] The comparison between an endmember and a library component
may be made using any suitable endmember matching technique or metric. In at
least one embodiment, the following equation may be used:
F = Ds tan(0,) (16)
where F is the metric and Os is the spectral angle between an extracted
endmember and a spectral library component. A small 0, indicates that the two
spectra have a similar shape. D, is the Euclidean distance between the library
and data reflectance spectrum:
Ds = .\II[Pub Pdata]2 (17)
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and Os may be calculated by
, (kit)) Pdata))
Os = COS¨ õ (18)
iPub I IPdata
where ( ) denotes the dot product.
[00120] This F metric tends to match spectra that are similar in both shape
and
amplitude. Therefore the value of metric F decreases as the degree of matching
between the extracted endmember and a spectral library component increases.
[00121] As mentioned above, any suitable endmember matching technique or
metric may be used. Other possible metrics include but are not limited to
normalized spectral distance and spectral information divergence (SID).
[00122] As mentioned above, an endmember maybe compared to every library
component in the comparison set. Therefore in an embodiment where the above
F metric is used, the metric may be computed for every library component in
the
comparison set. One library component is then selected as the match for the
particular endmember. The endmember and the selected library component may
be referred to as an endmember-library component pair.
[00123] A matching procedure may be performed for each of one or more
endmembers extracted from the input image. In at least some embodiments,
every extracted endmember is compared against the spectral library.
[00124] Figure 8 shows best library matches in an example library for the
extracted and corrected reflectance endmembers shown in Figure 7.
[00125] The resulting metrics and endmember-spectral library component pairs
may then be passed to the next step.
[00126] Once the one or more extracted endmembers have been compared to
the spectral library and a best match is found for the one or more endmembers,
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the process proceeds from block 412 to block 414 where a gain and an offset
value are calculated. The gain and offset may be calculated using the one or
more matched endmember-spectral library component pairs. In addition, a gain
value and an offset value may be determined for each of a plurality of
spectral
bands in the image.
[00127] Gain and offset values may be calculated using a regression technique
in which a line is fitted to a data set comprising the matched endmember-
spectral
library component pairs. In at least one embodiment, a weighted least square
regression technique may be used. In order to calculate the final gain and
offset,
a weighed linear regression may be made at each image spectral band using the
radiance endmember data points and the corresponding set of matched library
spectral points.
[00128] A graph containing example data of matched endmember-spectral
library matched pairs is provided in Figure 9. The plot represents the library
reflectance versus the endmember measurement data for a given spectral band.
Each dot or circle 902 represents an endmember-library component pair. The
line represents the regression fit. The line slope represents the gain, while
the
abscissa intercept 904 represents the black level and is related to the offset
by
equation (15).
[00129] Each endmember-spectral library component pair may be assigned a
weight in any suitable way. In at least some embodiments, the weight assigned
to each endmember-spectral library component pair may be proportional to the
degree of matching between the two. In other words, well-matched pairs may be
assigned higher weights than poorly-matched pairs.
[00130] Weights may be assigned based on a metric used in the endmember
matching step. In the above example where the equation F = Ds tan(0) is used
in the matching, the weight to be assigned to each endmember-library
component pair may be inversely proportional of the 'F' metric value. In one
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embodiment, the assigned weight may be the inverse of the F metric value, that
is 1/F. As noted above, since the value of F decreases as the degree of
matching increases, higher matching endmember-library component pairs are
assigned higher weights. Similarly, lower matching endmember-library
component pairs are assigned lower weights. Thus foreign materials in the
scene, meaning materials that have a poor or no match in the spectral library,
are therefore less likely to affect the final correction.
[00131] In addition, the assignment of lower weights to poorer matches
between extracted endmembers and library components may reduce the
negative effect of any bad or unwanted pixels on the overall result. For
example,
in some embodiments, bad pixels are not masked. In other situations, one or
more bad pixels may be missed in the masking process. In either case, a bad
pixel is unlikely to have a good match or even any match at all in a spectral
library. Accordingly, using a weighting where poor matches are assigned a low
weight reduces the negative effect the one or more bad pixels has on the final
result.
[00132] Furthermore, in some embodiments, an image black level may be
determined. The darkest value in each spectral band of the input image to be
corrected may be calculated. In even moderately diverse scenes (e.g. having
.. many types of different materials), it is typically a good assumption that
at least
one material will have a low reflectance value in each band. This is typically
the
case even if no single pixel (material) is dark in every band.
[00133] The resulting black level signal corresponds to the path radiance. The
black level signal may be used to directly determine the offset instead of or
in
addition to calculating the offset another way, for example by using a least
square regression technique as described above. In another embodiment, the
black level signal may be used in association with another way of calculating
the
offset, for example to provide redundancy, improve accuracy, etc. For example,
offset values may be calculated in two or more ways and then used to derive a
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single value to be used in the compensation of the input image. For example,
two or more offset values may be averaged. However, other ways of arriving at
a
single offset value are possible.
[00134] A regression used to calculate gain and offset values may be
anchored at the abscissa intercept by adding the data-library value pair
(black
level, 0) and setting its weight to numerical infinity (or a very large
value). The
black level may be obtained in any suitable way, including from the image
minimum or using the results of the radiative transfer. In the situations
where a
proper black level may be obtained, doing this removes a degree of freedom
from the linear regression and usually produces better results. For example,
more accurate values may be obtained for low reflectance materials. On the
other hand, having a proper black level is not critical to the atmospheric
compensation algorithm functioning correctly, and thus is not necessary in all
embodiments.
[00135] In other embodiments, different methods for obtaining the gain may be
used. For example, a weighted average may be obtained for both the data and
library endmembers. The gain may simply be the negative of the black level,
and
the gain may be given by
library mean
Gain= (19)
data mean ¨ data black level
[00136] A graph showing example data in this embodiment is provided in
Figure 10. Similar to the graph of Figure 9, in Figure 10 each dot or circle
1002
represents an endmember-library component pair. The line represents the
regression fit. The line slope represents the gain, while the abscissa
intercept
1004 represents the black level related to the offset trough equation (15).
The
large circle represents weighted mean value of all of the endmember-library
component pairs.
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[00137] The process of Figure 4 proceeds from block 414 to block 416 where
the calculated gain and offset values are applied to pixels in one or more
bands
of the input image to generate a compensated (or "corrected") reflectance
image.
The gain and offset values may be applied using equation (4). In some
embodiments, the gain and offset may be applied to substantially all pixels in
the
input image.
[00138] The process then proceeds from block 416 to block 418 and ends.
[00139] Figure 11A shows a portion of an AVIRIS sensor image compensated
(or "corrected") using an algorithm according to the present disclosure.
Indicated
.. in Figure 11A are pixels of vegetation 1110, dry vegetation 1112, a road
1114
and a blue roof 1116. Figures 11B, IIC, 11D and 11E are graphs representing
reflectance spectra for the vegetation, dry vegetation, road and blue roof
pixels,
respectively, corrected by QUAC, FLAASH and an embodiment of the present
disclosure. As seen in Figures 11B-11E, the algorithm according to the present
disclosure produces a correction similar to FLAASH for the four different
pixels
1110, 1112, 1114 and 1116.
[00140] In some embodiments, for example where a least square regression is
used to obtain the gain and offset for each spectral band, a regression error
may
be computed for one or more bands of the input image. This spectral regression
error may be passed on to exploitation algorithms (e.g. target detection and
others). An exploitation algorithm may then be able to reject one or more
bands
where the correction is deemed less accurate from their treatment. For
example,
certain bands might be omitted from participating in a target detection
algorithm if
the regression error is above a certain threshold.
[00141] Some embodiments of the present disclosure may have an acceptable
performance with as few as two extracted endmembers having good matches in
the spectral library. In some embodiments, it is desirable that the at least
two
extracted endmembers are sufficiently different in most or all of the spectral
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bands of the image being corrected. In addition, an acceptable performance may
be achieved using one extracted endmember having a good library match and
an image black level. In contrast, some other existing atmospheric
compensation
algorithms require many endmembers. For example, some versions of QUAC
require at least 10 different endmembers.
[00142] In addition, in embodiments of the present disclosure where several
endmembers are used, errors in the compensation process may be averaged
out. It may be unreasonable to expect perfect matches between endmembers
and the library. Some discrepancies may lead to an overestimation of the
surface
reflectance while others to an underestimation of the reflectance. However, on
average these discrepancies may tend to cancel out.
[00143] The present disclosure provides advantages over existing atmospheric
compensation techniques.
[00144] At least some embodiments of the present disclosure do not require a
priori information of in-scene targets or materials.
[00145] In addition, at least some embodiments of the present disclosure have
a lower degree of computational complexity or require less processing time
compared to other atmospheric compensation techniques.
[00146] Figures 5A to 5D show the state of an example vegetation pixel at
various steps in an algorithm according to the present disclosure. Figure 5A
is a
graph of the at-sensor radiance of the pixel versus the wavelength. Figure 5B
is
a graph of the top of atmosphere reflectance of the pixel after the input
image
has been normalized using top of atmosphere irradiance. The points on the
graph indicated by circles 510 are points used for endmember extraction. While
Figure 5B illustrates using a small subset of all the bands (points), the
resulting
number of endmembers may be more than the number of bands used (if allowed
by the endmember extraction algorithm) or less than the number of bands used.
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Figure 5C is a graph of the at-ground (or at-surface) reflectance of the pixel
after
the endmembers have been transformed into approximate at-ground reflectance
using an approximate radiative transfer model correction. Figure 5D is a graph
of the at-ground reflectance of the pixel after the input image has been
corrected
using calculated gain and offset values determined using matched endmember
spectral library pairs.
[00147] Other features and aspects of the present disclosure are now
described.
[00148] In one aspect, the present disclosure provides methods for removing
the effect of water absorption from a multi or hyper spectral image as part of
an
atmospheric compensation method. In particular, methods for correctly removing
the effect of water absorption when the column amount of water varies across
the scene are provided. A column amount of water varying across the scene can
happen for example near a coastline where water vapor is more prevalent near
the shore, and is also found in uneven terrain, since water vapor is reduced
at
higher altitudes.
[00149] Therefore a method of the present disclosure for removing the effects
of water absorption on pixels in an image uses retrieved water vapor values
for
each pixel as well as an average water vapor value for the whole image. Pixels
may be separated or designated into at least two groups: those with water
vapor
content higher than the image average, and those with water vapor lower than
average. The pixels may be separated using water vapor values retrieved from
the image. Endmembers, black levels, aerosol properties, and finally gain and
offset may all be derived, for example using methods and techniques described
above, for each of the two groups of pixels.
[00150] In at least one embodiment, the gain and offset for the lower water
content group may be the gain and offset of a pixel having the average water
vapor content in that group. Similarly, the gain and offset for the high water
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content group may be the gain and offset of a pixel having the average water
vapor content in the high water content group. Gains and offsets may be
determined in any suitable way, for example using any of the methods and
techniques described herein. For any other water content value found in the
image, the gain and offset for that pixel may be interpolated/extrapolated
linearly
between the low and high average water contents.
[00151] A flow diagram of an example process for correcting for the effects of
water absorption according to the present disclosure is shown in Figure 12.
The
process begins at block 1202 and proceeds to block 1204 where pixels in a set
of pixels are divided into two separate groups: a lower water content group
and a
higher water content group. As mentioned above, the pixels may be divided
using a calculated average vapor water content value for the pixels of the
image.
However, other values and techniques may be used for dividing the pixels into
the two or more groups, for example a calculated median vapor water content
value.
[00152] Similar steps are then performed using the lower water content and
higher water content groups. In particular, the process proceeds to blocks
1206
and 1212, where the average retrieved water vapor content of the lower water
content group and of the higher water content group are determined,
respectively. The process then proceeds to blocks 1208 and 1214 where
endmembers, black levels and aerosol values are derived for each group,
respectively. From blocks 1208 and 1214, the process proceeds to blocks 1210
and 1216, respectively, where gain and offset values are calculated for each
of
the two groups. The steps performed in blocks 1208, 1210, 1214 and 1216 may
be performed in any suitable way, including using any of the methods,
algorithms
or embodiments described above.
[00153] In at least one embodiment, the calculated average water vapor
content for one or both of the lower and higher water content groups may be
used in a first pass correction (e.g. transforming endmembers to at-ground
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reflectance). Although not shown in Figure 12, the first pass correction may
be
performed between blocks 1208 and 1210 for the lower water content group,
and/or between blocks 1214 and 1216, for the higher water content group.
[00154] The gain and offset values calculated in blocks 1210 and 1216 may be
considered to be the gain and offset values of a pixel having the average
water
content of each of the lower and higher content groups, respectively.
[00155] The process then proceeds from blocks 1210 and 1216 to block 1218.
For pixels having water content values other than the lower and higher average
values, the gain and offset for that pixel are interpolated/extrapolated
linearly
between the lower and higher average water content values using the retrieved
water vapor content for the particular pixel. Thus gain and offset values may
be
determined for each of several bands of individual pixels in the image. This
is
unlike previously described atmospheric compensation methods that determine a
gain and offset value for each of several spectral bands for the whole image,
not
for each of several spectral bands of individual pixels of the image.
[00156] The process then proceeds to block 1220 where the pixels in the
original image are corrected using the calculated gain and offset values. The
process then proceeds to block 1222 and ends.
[00157] Compared to existing methods, a process according to the one shown
in Figure 12 does not rely on direct correction of the water vapor absorption
through modeled transmittance curves, which can introduce undesirable features
near water bands. The present process uses in-scene elements for the
correction, and thus may continue to benefit from the benefits of in-scene
corrections, while effectively managing cases where water vapor variation
exists.
At least some embodiments of the present process correct for water vapor on a
pixel by pixel basis. In addition, other atmospheric effects that are
correlated with
water vapor content along the line of sight may implicitly be corrected by
this
process. For example, aerosol optical properties are often correlated with
water
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vapor content of the atmosphere. Also, differences in optical path lengths
from
pixel to pixel (caused by uneven terrain, non-vertical look angles, wide
fields of
views) may also be corrected, since the column amounts of aerosols and other
absorbing gases will vary along with water vapor
[00158] The present process may also be used to correct other types of
variations across the scene, particularly when these variations are
quantifiable.
For example, the present process may be used to correct for the presence of
cirrus clouds, particularly when the amount of cirrus is quantifiable on a
pixel
basis.
[00159] In another aspect, the present disclosure provides methods for
correcting for shadows or other illumination variability in a scene.
[00160] Shadows and other illumination variability in a scene can occur for
example because of broken clouds, trees, buildings or terrain topography. This
variability can hamper target detection or surface characterization algorithms
since it will usually have some effect on the observed spectral signature.
This is
because shadows do not simply scale down the reflectance spectrum, but
typically add a blue cast because the surfaces are illuminated mostly by the
sky.
[00161] Correcting these effects is not necessarily a simple task. This is
especially true when no prior knowledge of the scene content, for example
height
of trees, presence of clouds, etc., or topography is known. In such cases, it
may
be difficult to distinguish between shadows and dark surfaces.
[00162] One technique often employed to detect shadows is the use of a
shadow endmember. Once most pixels in a scene can be described by a sum of
endmembers with associated abundances, the shadow endmember is basically
what is required for the sum of abundances to reach unity as should be the
case
for reflectances and when abundances are prevented from being negative. One
problem with this approach is that it does not give the source of illumination
for
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. .
what remains. In order to properly correct for shadows, it is desirable to
know
what proportion of the pixel remaining illumination is from the sky and what
proportion is from the sun. This is because the sun and sky have different
spectral signatures and therefore typically affect the signature of the
surface
differently.
[00163] According to the present disclosure, a method is provided that
attempts to solve this problem and correct (i.e. remove), as best as possible,
shadows in a scene. The method is based on the assumption that the
endmembers of a scene are fully lit by direct and diffused sunlight. Figure
13,
which is described in more detail below, shows a flow diagram representing a
method of at least one embodiment. In at least one embodiment, the method
relies on the augmented modified Gram Schmidt (AMGS) method for iterative
endmember extraction algorithm (basically SMACC without the positivity and
sum constraints) as described in Rice, J. R., Experiments on Gram-Schmidt
Orthogonalization, Math. Comp. 20 (1966), 325-328. The method starts by
finding the brightest pixel in the scene and sets it as the first endmember in
an
iterative endmember extraction process. A brightest pixel may be the pixel
with
the largest norm (square root of the sum of the squares of the spectra
values).
The signature from that pixel may be removed from all other pixels by any
suitable way of spectral unmixing, including but not limited to projection.
The
unmixing may also find or determine the abundance of that endmember in each
pixel. This procedure may be repeated for a desired number of times, or until
the
largest residual pixel signature reaches a given tolerance (e.g. until the
norm of
the pixel last identified as the brightest pixel remaining in the image is
below a
predetermined threshold value).
[00164] In at least one embodiment, the AMGS algorithm is modified in order
to find the proportion of sun and sky illumination within a pixel. The
brightest
pixel found at each pass is always assumed to be fully illuminated by both sun
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and sky. The signatures of that surface, if it was only illuminated by the sun
or
only illuminated by the sky may be found with the following
P sky = P ( Idiff I a_ (20)
'dir 'cliff
and
dir
Psun = P (i
Idir 'cliff
(21)
= p(1.!cliff )
,
Idir 4- Idyl
where psky is the surface signature illuminated only by the sky ("skylit only
surface signature"), psõ is the surface signature when only illuminated by the
sun ("sunlit only surface signature"), p is the signature when fully
illuminated by
both sun and sky ("fully lit pixel signature"), 'din- is the diffuse (sky)
flux and /dir
is the direct (solar) flux. In other embodiments, psky and n
sun may be calculated
by multiplying p by values that are proportional to ( I dir a` ) and
(1 ithff
+I"dif f Idir+Idiff
respectively.
[00165] Direct and diffuse fluxes may be calculated using any suitable method.
In at least one embodiment, direct and diffuse fluxes may be calculated by the
single layer multiple scattering algorithm described above, where the diffuse
flux
may be represented as
'cliff ¨ F-(Ti) (22)
[00166] Both sun and sky signatures may be removed from all other pixels by
projection and their abundances calculated in the same way as in the standard
AMGS algorithm. After the final iteration, the sum of the "sun" abundances
gives
the sun illumination fraction (or factor), while the sum of the "sky"
abundances
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CA 02836210 2013-12-11
gives the fraction (or factor) of sky illumination. All pixels having been
chosen as
endmembers will end up with sun and sky fractions of exactly 1.
[00167] In this and similar embodiments, SMACC may also be used instead of
AMGS but a problem may arise when the positivity constraint is enabled. This
is
because the constraint may be too strong, and negative abundances are
permanently set to 0 as soon as they are encountered. This unnecessarily
eliminates many endmembers from participating in the composition of a pixel's
signature, as often, the abundance would become positive again as other
endmembers are found. When the constraint is enabled, it is in fact so strong
that the weaker sky contribution usually becomes zero for all pixels.
[00168] Once the sky and sun illumination fractions fsky and Lim are known,
one or more pixels of the image may be corrected for illumination. A given
pixel
may be compensated using the direct and diffuse fluxes of the given pixel. The
compensation may alternatively or additionally use a solar illumination factor
and
a sky illumination factor of the given pixel, the solar illumination factor
being the
sum of the abundances of the sunlit surface signatures and the sky
illumination
factor being a sum of the abundances of the skylit surface signatures of all
pixels
selected in the iterative process for the given pixel. In at least one
embodiment,
the correction of one or more pixels may involve multiplying the given pixel
by a
value proportional to
(
'di, + f
Pcorr PpLxel (23)
/sun dir J sky' dif f
[00169] In at least one embodiment, the solar illumination factor may be the
greater of zero and the sum of the abundances of the sunlit surface signatures
and the sky illumination factor may be the greater of zero and a sum of the
abundances of the skylit surface signatures of all pixels selected in the
iterative
process for the given pixel. This may be desirable in embodiments where there
are no positivity constraints on abundances and thus some abundance may
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CA 02836210 2013-12-11
become negative. To protect from this, the sum may be set to zero if it
becomes
negative.
[00170] As mentioned above, Figure 13 is a flow diagram representing a
process for at least one embodiment of the present disclosure. The process
starts at block 1302 and proceeds to block 1304 where the brightest pixel in
terms of norm is identified. It may be assumed that this pixel is fully lit by
both the
sun and the sky. The process proceeds from block 1304 to block 1306 where
direct (sun) and diffuse (sky) fluxes are determined for the pixel using the
fluxes.
The process then proceed to block 1308 where a skylit only surface signature
and a sunlit only surface signature are determined for the pixel. The process
proceeds to block 1310 where an abundance of each of the sunlit and skylit
surface signatures in pixels in the image are determined. The sunlit and
skylit
surface signatures may also be removed from pixels in the image.
[00171] It is then determined at block 1312 whether to proceed to block 1304
or to block 1314. The process may proceed to block 1304 if the process has not
yet performed a predetermined number of iterations of blocks 1304 to 1310, or
until the norm of the pixel last identified in block 1304 is below a
predetermined
threshold value. Otherwise the process may proceed to block 1314 where sun
and sky illumination factors are determined by summing sun abundances and
summing sky abundances determined in block 1310 in all iterations for each
pixel. The process then proceeds to block 1316 where one or more pixels in the
image are compensated. A given pixel may be compensated using the direct and
diffuse fluxes of the given pixel, as well as the sun and sky illumination
factors for
the given pixel. The process then proceeds to block 1318 and ends.
[00172] Figure 14 is an example image prior to any shadow correction. Figure
15 is the image of Figure 14 corrected for shadows using an embodiment of the
algorithm described above. Area 1402 in Figure 14 indicates a road or path
through a wooded area. In the corrected image of Figure 15, details within the
tree line shadow in the same area, area 1502, have been restored. The
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CA 02836210 2013-12-11
improvement is more easily seen in the blown-up portions of the images of
Figure 14 and Figure 15.
[00173] In addition, areas indicated by 1404 and 1404 on the pre and post
corrected images also demonstrate how variations in illumination due to
terrain
topography may be reduced considerably.
[00174] As another example, Figure 16 shows the same image used in Figure
11 corrected for shadows. In this case, the correction performs even better
than
the correction shown in Figures 14 and 15, with the resulting image being much
flatter and practically void of shadows.
[00175] The above examples illustrate the application and results of the above
described method of shadow correction.
[00176] Embodiments of the present disclosure may be performed by a
computing device or group of computing devices. Reference is now made to
Figure 17, which shows an example simplified computing device that may be
used.
[00177] Computing device 1710 may be a single device or group of devices
and includes at least one processor 1720 to perform the one or more of the
methods and functions described above. Processor 1720 in the example of
Figure 17 is a logical unit and may be comprised of one or more physical
processors or chips.
[00178] Processor 1720 communicates with a memory 1740. Again, memory
1740 is logical and may be located within computing device 1710 or possibly
distributed remotely from device 1710. Memory 1740 may be configured to store
program code and image data, that when executed may perform one or more
methods or functions of the present disclosure.
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[00179] A communications subsystem 1730 allows computing device 1710 to
communicate with one or more other devices, including but not limited to other
computing devices, optical sensors and radiation sensors. Communication
subsystem 1730 may be any wired or wireless communications subsystem and
may allow communication either directly with the other devices or through any
local or wide area network, such as for example the Internet.
[00180] In addition, the various components of computing device 1710 may
communicate with each other, for example, through a bus 1750. However other
possibilities for communication between the components exist.
[00181] Although some embodiments are described in the form of methods
alone, this is not meant to be limiting. Apparatus for carrying out part or
all of
these methods are also contemplated.
[00182] In addition, one or more the steps, methods, algorithms or other
techniques according to the present disclosure may be performed automatically,
meaning without any user intervention.
[00183] The structure, features, accessories, and alternatives of specific
embodiments described herein and shown in the Figures are intended to apply
generally to all of the teachings of the present disclosure, including to all
of the
embodiments described and illustrated herein, insofar as they are compatible.
In
other words, the structure, features, accessories, and alternatives of a
specific
embodiment are not intended to be limited to only that specific embodiment
unless so indicated.
[00184] Furthermore, additional features and advantages of the present
disclosure will be appreciated by those skilled in the art.
[00185] In addition, the embodiments described herein are examples of
structures, systems or methods having elements corresponding to elements of
- 48 -
õ
the techniques of this application. This written description may enable those
skilled in the art to make and use embodiments having alternative elements
that
likewise correspond to the elements of the techniques of this application. The
intended scope of the techniques of this application thus includes other
structures, systems or methods that do not differ from the techniques of this
application as described herein, and further includes other structures,
systems or
methods with insubstantial differences from the techniques of this application
as
described herein.
[00186] Moreover, the previous detailed description is provided to enable any
person skilled in the art to make or use the present invention. Various
modifications to those embodiments will be readily apparent to those skilled
in
the art, and the generic principles defined herein may be applied to other
embodiments without departing from the spirit or scope of the invention
described herein. Thus, the present invention is not intended to be limited to
the
embodiments shown herein, but is to be accorded the full scope consistent with
the claims, wherein reference to an element in the singular, such as by use of
the article "a÷ or "an" is not intended to mean "one and only one" unless
specifically so stated, but rather "one or more". Moreover, nothing disclosed
herein is intended to be dedicated to the public regardless of whether such
disclosure is explicitly recited in the claims.
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