Note: Descriptions are shown in the official language in which they were submitted.
CA 02475778 2004-12-30
AN IMAGE CODING METHOD AND ENCODER USING POSITIVE AND
NEGATIVE ROUNDING METHODS WHILE PERFORMING MOTION
COMPENSATION
This is a division of co-pending Canadian Patent
Application Serial No. 2,240,118, filed June 8, 1998.
BACKGROUND OF THE INVENTION
Field of the Invention
The present invention relates to an image sequence coding
and decoding method which performs interframe prediction using
quantized values for chrominance or luminance intensity.
Description of Related Art
In high efficiency coding of image sequences, interframe
prediction (motion compensation) by utilizing the similarity
of adjacent frames over time, is known to be a highly
effective technique for data compression. Today's most
frequently used motion compensation method is block matching
with half pixel accuracy, which is used in international
standards H.263, MPEG1, and MPEG2. In this method, the image
to be coded is segmented into blocks and the horizontal and
vertical components of the motion vectors of these blocks are
estimated as integral multiples of half the distance between
adjacent pixels. This process is described using the following
equation:
[Equation 1]
P(x,y) - R(x+u~,y+vi) (x,y) EBi,O~i<N ...(1)
where P(x,y) and R(x,y) denote the sample values (luminance or
chrominance intensity) of pixels located at coordinates (x, y)
in the predicted image P of the current frame and the
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CA 02475778 2004-08-25
reference image (decoded image of a frame which has been
encoded before the current frame) R, respectively. x and
y are integers, and it is assumed that all the pixels are
located at points where the coordinate values are integers .
Additionally it is assumed that the sample values of the
pixels axe quantized to non-negative integers . N, Bi , and
( ui , vi ) denote the number of blocks in the image , the set
of pixels included in the i-th black of the image, and the
motion vectors of the i-th block, respectively.
When the values for ui and vi are not integers , it is
necessary to find the intensity value at the point where no
pixels actually exist in the reference image. Currently,
bilinear interpolation using the adjacE~nt four pixels is the
most frequently used method for this process. This
interpolation method is described using the following
equation:
[Equation 2)
R(x+ ~,y+ ~)=y(d -q)((d -P)R(x~Y)+P~~(x+I,Y)~.
+ c~.((d - p)R(x, y + 1) + pR(x + I, y + I))~ / /d 2 ~ ~ ~ (2)
where d is a positive integer, and p and q are smaller than
d but not smaller than 0. "//" denotes integer division
which rounds the result of normal division ( division using
real numbers) to the nearest integer.
An example of the structure of an H.263 video encoder
is shown in Fig. 1. As the coding algorithm, H.263 adopts
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CA 02475778 2004-08-25
a hybrid coding method (adaptive interframe/intraframe
coding method ) which is a combination of block matching and
DCT (discrete cosine transform). A subtractor 102
calculates the difference between the input image ( current
frame base, image) 101 and the output image 113 (related
Later) of the interframe/intraframe coding selector 119, and
then outputs an error image 103. This error image is
quantized in a quantizer 105 after being converted into DCT
coefficients in a DCT converter 104 and then forms quantized
DCT coefficients 106. These quantized DCT coefficients are
transmitted through the communication channel while at the
same time used to synthesize the interframe predicted image
in the encoder. The procedure for synthesizing the
predicted image is explained next. The above mentioned
quantized DCT coefficients 106 forms th.e reconstructed error
image_110 (same as the reconstructed error image on the
receive side) after passing through a dequantizer 108 and
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inverse DCT converter 109. This reconstructed error image
and the output image 113 of the interframe /intraframe coding
2o selector 119 is added at the adder 111 and the decoded image
112 of the current frame (same image as the decoded image
of current frame reconstructed on the receiver side) is
obtained. This image is stored in a frame memory 114 and
delayed for a time equal to the frame interval. Accordingly,
at the current point, the frame memory 114 outputs the
decoded image 115 of the previous framae. This decoded image
of the previous frame and the original image 101 of the
current frame are input to the bloc; matching section 116
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and block matching is performed between these images. In the
block matching process, the original image of the current
frame is segmented into multiple blocks , and the predicted
image 117 of the current frame is synthesized by extracting
the section most resembling these blocks from the decoded
image of the previous frame. In this process, it is
necessary to estimate the motion between the prior frame and
the current frame for each block. The motion vector for each
block estimated in the motion estimation process is
transmitted to the receiver side as motion vector data 120.
On the receiver side, the same prediction image as on the
transmitter side is synthesized using the motion vector
information and the decoding image of 'the previous frame .
The prediction image 117 is input along with a "0" signal
118 to the interframe /intraframe coding selector 119. This
switch 119 selects interfrarne coding or intraframe coding
by selecting either of these inputs . Interframe coding is
performed when the prediction image 117 is selected ( this
case is shown in Fig. 2). On the other hand when the "0"
2o signal is selected, intraframe coding is performed since the
input image itself is converted, to a DCT coefficients and
output to the communication channel. In order for the
receiver side to correctly reconstruct the coded image, the
reciever must be informed whether intraframe coding or
interframe coding was performed on the transmitter side.
Consequently, an identifier flag 121 is o~ltput to the
communication circuit. Finally, an H.263 coded bitstream
123 is acquired by multiplexing the quantized DCT
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coefficients, motion vectors, and the
interframe/intraframe identifier flag information in a
multiplexer 122.
The structure of a decoder 200 for receiving the coded
bit stream output from the encoder of Fig. 1 is shown in Fig.
2. The H.263 coded bit stream 217 that is received is
demultiplexed into quantized DCT coefficients 201, motion
vector data 202, and a interframe/intraframe identifier flag
203 in the demultiplexer 216. The quantized DCT
coefficients 201 become a decoded error :image 206 after being
processed by an inverse quantizer 204 and inverse DCT
converter 205. This decoded error image is added to the
output image '215 of the interframe /intraframe coding
selector 214 in an adder 207 and the sum of these images is
output as the decoded image 208. The output of the
interframe /intraframe coding selector is switched
according to the interframe/intraframe identifier flag 203.
A prediction image 212 utilized when performing interframe
encoding is synthesized in the prediction image synthesizer
211 . In this synthesizer, the position of the blacks in the
decoded image 210 of the prior frame stored in frame memory
209 is shifted according to the motion vector data 202. On
the other hand, for intraframe coding, the interframe
/intraframe coding selector outputs 'the "0" signal 213 as
is .
SUMMARY OF THE INVENTION
The image encoded by H.263 is comprised of a luminance
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plane (Y plane) containing luminance information, and two
~chrominance planes (U plane and V plane) containing
chrominance information. Atthistime,characteristically,
when the image has 2m pixels in the horizontal direction and
2n pixels in the vertical direction (m and n are positive
integers), the Y plane has 2m pixels :horizontally and 2n
pixels vertically, the U and V planes have m pixels
horizontally and n pixels vertically. The low resolution
on the chrominance plane is due to the fact that the human
visual system has a comparatively dull visual faculty with
respect to spatial variations in chrominance. Having such
image as an input, H. 263 performs cooling and decoding in
block units referred to as macroblocks. The structure of
a macroblock is shown in Fig . 3 . The macroblock is comprised
of three blocks; a Y block, U block and V block. The size
of the Y block 301 containing the luminance. information is
16 X 16 pixels , and the size of the U block 302 and V block
303 containing the chrominance information is 8 X 8 pixels .
In H. 263, half pixel accuracy block matching is
2o applied to each block. Accordingly, when the estimated
motion vector is defined as ( a , v ) , a and v are both integral
multiples of half the distance between pixels. In other
words, 1/2 is used as the minimum unit. The configuration
of the interpolation method used for 'the intensity values
(hereafter the intensity values for "luminance" and
"chrominance" are called by the general term "intensity
value") is shown in Fig. 4. When performing the
interpolation described in equation ',, the quotients of
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division are rounded off to the nearest integer, and further,
when the quotient has a half integer value ( :i . a . 0 . 5 added
to an integer) , rounding off is performed to the next integer
in the direction away from zero. In other words, in Fig.
4, when the intensity values for 401, 402, 403, 404 are
respectively La, Lb, Lc, and Ld (La, Lb, Lc, and Ld are
non-negative integers), the interpolated intensity values
Ia, Ib, Ic, and Id (Ia, Ib, Ic, and Id are non-negative
integers) at positions 405, 406, 407, 408 are expressed by
1o the following equation:
[Equation 3]
la = La
Ib=~(La+Lba-1)/2,
Ic = ~(La + Lc + 1) / 2, ~ ~ ~ (3)
Id = ~(La + Lb + Lc ~+- Ld + 2) / 4~
where " f ] " denotes truncation to the nearest integer towards
0 (i.a. the fractional part is discarded) . The expectation
of the errors caused by this rounding to integers is
estimated as follows : It is assumed that i~he probability that
2o the intensity value at positions 405 , 4:06 , 407 , and 408 of
Fig . 4 is used is all 25 percent . When finding the intensity
value Ia for position 405 , the rounding error will clearly
be zero. Also, when finding the intensity value Ib for
position 406 , the error will be zero wren La+Lb is an even
number, and when an odd number the error is 1/2. If the
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CA 02475778 2004-08-25
probability that La-~-Lb will be an even number and an odd
number is both 50 percent, then the expectation for the error
will be 0 x 1 / 2 + 1 / 2 X 1 / 2 = 1 / 4. Further; when
finding the intensity value Ic for position 407, the
expectation for the error is 1 / 4 as fo~~ Ib. When finding
the intensity value Id for position 408 , the error when the
residual of La+Lb+Lc+Ld divided by four are 0 , 1, 2 , and 3
are respectively 0, -1/4, 1/2, and 1/4. If we assume that
the probability that the residual is 0 , l , 2 , and 3 is all
equal (i.e. 25 percent), the expectation for the error is
0 x 1 / 4 - 1 / 4 x 1 / 4 + 1 / 2 x 1 / 4 + 1 / 4 x 1 / 4
- 1 / 8. As described above, assuming that the possibility
that the intensity value at positions 405 - 408 being used
are all equal, the ffinal expectation far the error is O x
16 1 / 4 + 1 / 4 x 1 / 4 + 1 / 4 x 1 / 4 +~ 1 / 8 x 1 / 4 = 5
/ 32. This indicates that each time motion compensation is .
performed by means of block matching , an error of 5 / 32 occurs
in the pixel intensity value . Generally in low rate coding;
sufficient number of bits cannot be used for the encoding
of the interframe error difference so that the quantized step
size of the DCT coefficient is prone to be large.
Accordingly, errors occurring due to motion compensation are
corrected only when it is very large. When interframe
encoding is performed continuously without performing
intraframe coding under such environment, the errors tend.
to accumulate and cause bad effects on the reconstructed
image.
Just as explained above, the number of pixels is about
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CA 02475778 2004-08-25
half in both the vertical and horizontal direction on the
chrominance plane. Therefore, for the motion vectors of the
U block and V block, half the value of the motion vector for
the Y block is used for the vertical ,and horizontal
components. Since the horizontal and vertical components
of the motion vector for the Y block motion vector are
integral multiples of 1/2, the motion vE~ctor components for
the U and V blacks will appear as integral multiples of 1/4
(quarter pixel accuracy) if ordinary division is implemented.
However, due to the high computational complexity of the
intensity interpolation process for motion vectors with
quarter pixel accuracy, the motion vectors for U and V blocks
are rounded to half pixel accuracy in H.263. The rounding
method utilized in H.263 is as follows: According to the
definition described above, (u, v) denotes the motion vector
of the macroblock (which is equal to the motion vector for
the Y block). Assuming that r is an integer and s is an
non-negative integer smaller than 4 , a / 2 can be rewritten
as a / 2 = r + s / 4 . When s is 0 or 2 , no rounding is required
2o since a / 2 is already an integral multiple of 1 / 2. However
when s is equal to 1 or 3, the value of s is rounded to 2.
By increasing the possibility that s takes the value of 2
using this rounding method, the filtering effect of motion
compensation can be emphasized. When the probability that
the value of s prior to rounding is 0 , 1 , 2 , and 3 are all
25 percent, the probability that s will be 0 or 2 after
rounding will respectively be 25 percent and °75 percent . The
above explained process related to the:horizontal component
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CA 02475778 2004-08-25
a of the motion vector is also applied to the vertical
component v . Accordingly , in the U block and V block , the
probability for using the intensity value of the 401 position
is 1 / 4 X 1 / 4 = 1 / 16 , and the probability for using the
intensity value of the 402 and 403 positions is both 1 / 4
X 3 / 4 = 3 / 16, while the probability for using the intensity
value of position 404 is 3 / 4 X 3 / 4 = 9 / 16. By utilizing
the same method as above, the expectation for the error of
the intensity value is 0 X 1 / 16 + 1 / 4 X 3 / 16 + 1 / 4
1o X 3 / 16 + 1 / 8 X 9 / 16 = 21 / 128. . Just as explained above
for the Y block, when interframe encoding is continuously
performed, the problem of accumulated errors occurs.
As related above, for image sequence coding and
decoding methods in which interframe prediction is performed
i5 and luminance or chrominance intensity is quantized, the
problem of accumulated rounding errors occurs . This rounding
error is generated when the luminance or chrominance
intensity value is quantized during the generation of the
interframe prediction image.
In view of the above problems, it is therefore an object
of this invention, to improve the quality of the
reconstructed image by preventing error accumulation.
In order to achieve the above object, the accumulation
of errors is prevented by limiting the occurrence of errors
or performing an operation to cancel out errors that have
occurred.
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In accordance with one aspect of the present
invention there is provided an image coding method
comprising the steps of: synthesizing a prediction image
by performing motion compensation; and multiplexing
information related to a difference image between an
input image and the prediction iznage, and information
related to motion vectors estimated in performing the
motion compensation; wherein the motion compensation is
performed using a positive rounding method and a negative
rounding method in interpolating intensity values of
pixels.
In accordance with another aspect of the present
invention there is provided an image encoder comprising:
a DCT converter which performs DCT conversion of a
difference image between an input image of a current
frame and a prediction image of the current frame to
obtain DCT coefficients, the prediction image having been
synthesized by performing motion compensation; a
quantizer which quantizes the DCT coefficients to obtain
quantized DCT coefficients; a frame memory which stores a
decoded image of a reference frame; a block matching
section which synthesizes the prediction image of the
current frame by performing motion compensation between
the decoded image of the reference frame and the input
image of the current frame using pixel value
interpolation, and estimates motion vectors in performing
the motion compensation; and a multiplexer which
multiplexes information related to the quantized DCT
coefficients, and information related to the motion
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CA 02475778 2004-08-25
vectors; wherein the block matching section sometimes
uses a positive rounding method in performing the pixel
value interpolation, and sometimes uses a negative
rounding method in performing the pixel value
interpolation; wherein the positive rounding method is a
rounding method wherein results of the pixel value
interpolation are rounded to a nearest integer, with
half-integer values (integer values plus 0.5) being
rounded away from zero; and wherein the negative rounding
method is a rounding method wherein results of the pixel
value interpolation are rounded to a nearest integer,
with half-integer values (integer values plus 0.5) being
rounded towards zero.
BRIEF DESCRIPTION OF THE DRAWINGS
The present invention will be described in
conjunction with the invention described in co-pending
Canadian Patent Application Serial No. 2,240,118, filed
on June 8, 1998, with the aid of the accompanying
drawings, in which:
Figure 1 is a block diagram showing the layout of
the H.263 image encoder.
Figure 2 is a block diagram showing t:he layout of
the H.263 image decoder.
Figure 3 is a drawing showing the structure of the
macro block.
Figure 4 is a drawing showing the interpolation
process of intensity values for block matching with half
pixel accuracy.
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CA 02475778 2004-08-25
Figure 5 is a drawing showing a coded image
sequence.
Figure 6 is a block diagram showing a software image
encoding device.
Figure 7 is a block diagram showing a software image
decoding device.
Figure 8 is a flow chart showing an example of
processing in the software image encoding device.
Figure 9 is a flow chart showing an example of the
coding mode decision processing for the software image
encoding device.
Figure 10 is a flow chart showing an example of
motion estimation and motion compensation processing in
the software image encoding device.
Figure 11 is a flow chart showing the processing in
the software image decoding device.
Figure 12 is a flow chart showing an example of
motion compensation processing in the software image
decoding device.
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Figure 13 is a drawing showing an example of a storage
media on which an encoded bit stream generated by a.n encoding
method that outputs bit streams including I , P+ and P- frames
is recorded.
Figure 14 is a set of drawings showing specific
examples of devices using an encoding method where P+ and
P- frames coexist.
Figure 15 is a drawing showing an example of a storage
media on which an encoded bit stream generated by an encoding
method the outputs bit streams including I, B, P+, and P-
frames is recorded.
Figure 16 is a block diagram showing an example of a
block matching unit included in a device using an encoding
method where P+ and P- frames coexist.
Figure 17 is a block diagram showing the prediction
image synthesizer included in a device for decoding bit
streams encoded by ar.~ encoding method where P+ and P- frames
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coexist.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
First, in which circumstancesthe accumulated rounding
errors as related in the "Prior art" occur must be considered.
An example of an image sequences encoded by coding methods
which can perform both unidirectional prediction and
bidirectional prediction such as in MPEG . 1 , MPEG . 2 and H . 2 6 3
is shown in Fig. 5. An image 501 is a frame coded by means
of intraframe coding and is referred to as an I frame. In
contrast, images 503 , 505 , 507 , 509 are called :P frames and
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CA 02475778 2004-08-25
are coded by unidirectional interframe coding by using the
previous I or P frame as the reference image. Accordingly,
when for instance encoding image 505, image 503 is used as
the reference image and interframe prediction is performed.
Images 502, 504, 506 and 508 are called B frames and
bidirectional interframe prediction is performed utilizing
the previous and subsequent I or P frame. The B frame is
characterized by not being used as a reference image when
interframe prediction is performed. Since motion
io compensation is not performed in I frames, the rounding error
caused by motion compensation will not occur. In contrast ,
not only is motion compensation performed in the P frames
but the P frame is also used as a reference image by other
P or B frames so that it may be a cause leading to accumulated
rounding errors . In the B frames on the other hand, motion
compensation is performed so that the effect of accumulated
rounding errorsappears in the reconstructed image. However,
due to the fact that B frames are not used as reference images ,
B frames cannot be a source of accumulated rounding errors .
2o Thus, if accumulated rounding errors can be prevented in the
P frame, then the bad effects of rounding errors can be
alleviated in the overall image sequence . In H . 263 a frame
for coding a P frame and a B frame exists and is called a
PB frame (For instance, frames 503 and 504 can both be encoded
as a PB frame. ) If the combined two frames are viewed as
separate frames, then the same principle as above can be
applied. In other words, if countermeasures are taken
versus rounding errors for the P frame part within a PB frame,
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CA 02475778 2004-08-25
then the accumulation of errors can be prevented.
Rounding errors occur during interpolation of
intensity values when a value obtained from normal division
( division whose operation result is a real number ) is a half
integer {0.5 added to an integer) and this result is then
rounded up to the next integer in the direction away from
zero. For instance, when dividing by 4 to find an
interpolated intensity value is perfarmed, the rounding
errors fvr the cases when the residual is 1 and 3 have equal
1o absolute values but different signs. Consequently, the
rounding errors caused by these two cases are canceled when
the expectation for the rounding errors is calculated (in
more general words, when dividing by a positive integer d'
is performed, the rounding errors caused by the cases when
the residual is t and d' -t are cancelled) . However, when
the residual is 2 , in other words when the result of normal
division is a half integer, the rounding error cannot be
canceled and leads to accumulated errors. To solve this
problem, a method that allows the usage of tvao rounding
methods can be used. The two rounding methods used here are:
a rounding method that rounds half integers away from 0 ; and
a rounding method that rounds half integers towards O. By
combining the usage of these two rounding methads, the
rounding errors can be canceled. Hereafter, the rounding
method that rounds the result of normal division to the
nearest integer and rounds half integer values away from 0
is called "positive rounding". 'Additionally, the rounding
method that rounds the result of normal division to the
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CA 02475778 2004-08-25
nearest integer and rounds half integer values towards 0 is
called "negative rounding". The process of positive
rounding used in block matching with half pixel accuracy is
shown in Equation 3. When negative rounding is used instead,
this~equation can be rewritten as shown below.
[Equation 4)
Ia=Lu
Ib = ~(Lcz + Lb) / 2~
Ie = [(Lea + Lc) / 2~ . .. (4)
Id = ~(La + Lb + .Lc + Ld + 1) / 4~
io, Hereafter motion compensation methods that performs
positive and negative rounding for the synthesis of
interframe prediction images are called "motion
compensation using positive rounding" and "motion
compensation using negative rounding", respectively.
Furthermore, for P frames which use block matching with half
pixel accuracy for motion compensation, a frame that uses
positive rounding is called a "P+ frame°° and a-frame that
uses negative rounding is called a "P- frame °° ( under this
definition, the P frames in H. 263 are all P+ frames) . The
2o expectation for the rounding errors in P+ and P- frames have
equal absolute values but different signs. Accordingly,
the accumulation of rounding errors can be prevented when
P+ frames and P- frames are alternately located along the
time axis . In the example in Fig . 5 , if the frames 503 and
507 are set as P+ frames and the frames 505 and 509 are set
CA 02475778 2004-08-25
as P- frames, then this method can be implemented. The
alternate occurrence of P+ frames and P- frames leads to the
usage of a P+ frame and a P- frame in the bidirectional
prediction for B frames. Generally, the average of the
forward prediction image (i.e. the prediction image
synthesized by using frame 503 when frame 504 in Fig. 5 is
being encoded ) and the backward prediction image ( i . a . the
prediction image synthesized by using frame 505 when frame
504 in Fig. 5 is being encoded} is frequently used for
1o synthesizing the prediction image for B frames . This means
that using a P+ frame {which has a positive value for the
expectation of the rounding error} and a P- frame (which has
a negative value for the expectation of the rounding error)
in bidirectional prediction for a B frame is effective in
canceling out the effects of rounding errors. Just.as
related above, the rounding process in the B frame will not
be a cause of error accumulation. Accordingly, no problem ~
will occur even if the same rounding method is applied to
all the B frames. For instance, no serious degradation of
2o decoded images is caused even if motion compensation using
positive rounding is performed for all of the B frames 502 ,
504, 506, and 508 in Fig. 5. Preferably only one type of
rounding is performed for a B frame, in order to simplify
the B frame decoding process.
A block matching section 1600 of an image encoder
according to the above described motion compensation method
utilizing multiple rounding methods is shown in Fig. 16.
Numbers identical to those in other drawings indicate the
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CA 02475778 2004-08-25
same part. By substituting the block matching section 116
of Fig. 1 with 1600, multiple rounding methods can be used.
Motion estimation processing between the input image l0I and
the decoded image of the previous frame is performed in a
motion estimator 1601. As a result, motion information 120
is output. This motion information is utilized in the
synthesis of the prediction image in a prediction image
synthesizer 1603. A rounding method determination device
1602 determines whether to use positive rounding or negative
1o rounding as the rounding method.for the frame currently being
encoded. Lnformation 1604 relating to the rounding method
that was determined is input to the prediction image
synthesizer 1603. In this prediction image synthesizer
1603 , a prediction image 117 is synthesized and output based
on the rounding method determined by means of information
1604. In the block matching section 116 in Fig. 1, there
are no items equivalent to 1602, 1604 of Fig. 16, and the
redaction ima a is s nthesized onl b
p g y y y positive rounding.
Also, the rounding method 1605 determined at the block
2o matching section can be output, and this information can then
be multiplexed into the bit stream and be transmitted.
A prediction image synthesizer 1700 of an image decoder
which can decode bit streams generated by a coding method
using multiple rounding methods is shown in Fig. 17.
Numbers identical to those in other drawings indicate the
same part. By substituting the prediction image
'synthesizer 211 of Fig. 2 by 1700, multiple rounding methods
can be used. In the rounding method determination device
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CA 02475778 2004-08-25
1701, the rounding method appropriate for prediction image
synthesis in the decoding process is determined. In order
to carry out decoding correctly, the rounding method
selected here must be the same as the rounding method that
was selected for encoding. For instance the following rule
can be shared between the encoder and decoder: When the
current frame is a P frame and the number of P frames
( including the current frame ) counted from the most recent
I frame is odd, then the current frame is a P+ frame . When
io this number is even, then the current frame is a P- frame.
If the rounding method determination device on the encoding
side (For instance,- 1602 in Fig. 16) and the rounding method
determination device 1701 conform to this common rule, then
the images can correctly be decoded. The prediction image
is synthesized in the prediction image synthesizer 1703
using motion information 202 , decoding image 210 of the prior
frame, and information 1702 related to the rounding method
determined as dust described. This prediction image 212 .is '
output and then used for the synthesis of the decoded image.
2o As an alternative to the above mentioned case, a case. where
the information related to the rounding method is
multiplexed in the transmitted bit stream can also be
considered ( such bit stream can be generated at the encoder
by outputting the information 1605 related to the rounding
2s method from the blocDc matching section depicted in Fig . 16 ) .
In such case, the rounding method determiner device 1701 is
not used, and information 1704 related to the rounding method
extracted from the encoded bit stream is used at the
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CA 02475778 2004-08-25
prediction image synthesizer 1703.
Besides the image encoder and the image decoder
utilizing the custom circuits and custom chips of the
conventional art as shown in Fig . 1 and Fig . 2 , this invention
can also be applied to software image encoders and software
image decoders utilizing general-purpose procea sors. A
software image encoder 600 and a software image decoder 700
are shown in Fig . 6 and Fig . 7 . In the software image encoder
600, an input image 601 is first stored in the input frame
1o memory 602 and the general-purpose processor 603 loads
information from here and performs encoding. The program
for driving this general-purpose processor is loaded from
a storage device 608 'which can be a hard disk, floppy disk,
etc. and stored in a program memory 604. This general-
purpose processor also uses a process memory 608 to perform
the encoding. The encoding information output; by the
general-purpose processor is temporarily stored in the
output buffer 606 and then output as an encoded. bit stream
607.
A flowchart for the encoding software (recording
medium readable by computer) is shown in Fig. 8. The process
starts in 801 , and the value 0 is assigned to variable N in
802. Next, in 803 and 804, the value 0 is assigned to N when
the value for N is 100. N is a counter for the number of
frames. 1 is added for each one frame whose processing is
complete, and values from 0 to 99 are allowed when performing.
coding. When the value for N is 0, the current frame is an
I frame. When N is an odd number, the current frame is a
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CA 02475778 2004-08-25
p+ frame , and when an even number other than 0 , t:he current
frame is a P- frame. When the upper limit for th.e value of
N is 99, it means that one I frame is coded after 99 P frames
( P+ frames or P- frames ) are coded. By always inserting one
I frame in a certain number of coded frames , the following
benefits can be obtained: (a) Error accumulation due to a
mismatch between encoder and decoder processing can be
prevented (for instance, a mismatch in the computation of
DCT); and (b) The processing load for acquiring the
1o reproduced image of the target frame from the coded data
( random access ) is reduced . The optimal N value varies when
the encoder performance or the environment where the encoder
is used are changed. It does not mean, therefore, that the
value of N must always be 100. The process for determining
the rounding method and coding mode for each frame is
performed in 805 and the flowchart with details of this
operation is shown in Fig. 9. First of all, whether N is
a 0 or not is checked in 901 . If N is 0 , then ' I' is output
as distinction information of the prediction mode, to the
output buffer in 902. This means that the image to be coded
will be coded as an I frame. Here, '°output to the output
buffer" means that after being stored in the output buffer,
the information is output to an external device as a portion
of the coded bit stream. When N is not 0, then whether N
is an add or even number is identified in 904. When N is
an odd number, '+° is output to the output buffer as the
distinction information for the rounding method in 905 , and
the image to be coded will be coded as a P+ frame. On the
2 0
CA 02475778 2004-08-25
other hand, when N is an even number, '-' is output to the
output buffer as the distinction information for the
rounding method in 906, and the image to be coded will be
coded as a P- frame. The process again returns to Fig. 8,
where after determining the coding mode in 805, the input
image is stored in the frame memory A in 806. The frame
memory A referred to :here signifies a portion of the memory
zone { for instance , the memory zone maintained in the memory
of 605 in Fig. 6) of the software encoder. In 807, it is
checked whether the frame currently being coded is an I frame .
When not identified as an I frame, motion estimation and
motion compensation is performed in 808. The flowchart in
Fig. i0 shows details of this process performed in 808.
First of all, in 1001, motion estimation is performed between
the images stored in frame memories A and B ( just as written
in the final part of this paragraph, the decoded image of
the prior frame is stored in frame memory B). The motion
vector for each block is found, and this motion vector is
sent to the output buffer. Next, in 1002, whether or not
2o the current frame is a P+ frame is checked . When the current
frame is a P+ frame, the prediction image is synthesized in
1003 utilizing positive rounding and this prediction image
is stored in frame memory C. On the other hand, when the
current frame is a P - frame, the prediction image is
synthesized in 1004 utilizing negative rounding and this
prediction image is stored in the frame memory C . Next , in
1005, the differential image between frame memories A and
C is found and stored in frame memory A. Here; the process
21
CA 02475778 2004-08-25
again returns to Fig. 8. Prior to starting the processing
in 809, the input image is stored in frame memory A when the
current frame is an T frame, and the differential image
between the input image and the prediction image is stored
in frame memory A when the current frame is a P frame ( P+
or P- frame).. Tn 809, DCT is applied to the image stored
in frame memory A, and the DCT coefficients calculated here
are sent to the output buffer after being quantized. Tn 810,
inverse quantization is performed to the quantized DCT
1o coefficients and inverse DCT is applied. The image obtained
by applying inverse DCT is stored in frame memory B. Next
in 811, it is checked again whether the current frame is
an I frame. When the current frame is not an I frame, the
images stored in frame memory B and C are added and the result
i5 is stored in frame memory B. The coding process of a frame
ends here, and the image stored in frame memory B before going
into 813 is the reconstructed image of this frame ( this image ~
is identical with the one obtained at the decoding side).
In 8I3 , it is checked whether the frame whose coding has just
2o finished is the final frame in the sequence. If this is true,
the coding process ends . If this frame is not the final frame,
1 is added to N in 814 , and the process again returns to 803
and the coding process for the next frame starts.
A software decoder 700 is shown in Fig. 7. After the
25 coded bit stream 701 is temporarily stored in the input
buffer 702, this bit stream is then loaded into the
general-purpose processor 703. The program for driving this
general-purpose processor is loaded from a storage device
22
CA 02475778 2004-08-25
708 which can be a hard disk, floppy disk, etc, and stored
in a program memory 704. This general-purpose processor
also uses a process memory 605 to perform the decoding. The
decoded image obtained by the decoding process is
temporarily stored in the output frame memory 706 and then
sent out as the output image 707.
A flowchart of the decoding software for the software
decoder 700 shown in Fig. 7 is shown in Fig. 11. The process
starts in 1101, and it is checked in 1102 whether input
io information is present . If there is no input information ,
the decoding process ends in 1103. When input information
is present, distinction information of the prediction mode
is input in 1104. The word "input" used here means that the
information stored in the input buffer ( for instance 702 of
Fig. 7 ) is loaded by the general-purpose processor. In 1105 ,
it is checked whether the encoding mode distinction
information is "I". When not "I", the distinction
information for the rounding method is input and synthesis
of the interframe prediction image is performed in 1107. A
Zo flowchart showing details of the operation in :1107 is shown
in Fig . I2 . In 1201 , a motion vector is input for each block .
Then, in 1202, it is checked whether the distinction
information for the rounding method loaded in 1106 is a "+" .
When this information is "+", the frame currently being
~25 decoded is a P+ frame. In this case, the prediction image
is synthesized using positive rounding in 1203, and the
prediction image is stored in frame memory D. Here, frame
memory D signifies a portion of the memory zone of 'the
23
CA 02475778 2004-08-25
software decoder (for instance, this memory zone is obtained
in the processing memory 705 in Fig. 7). When the
distinction information of the rounding method is not "+'° ,
the current frame being decoded is a P- frame. The
prediction image is synthesized using negative rounding in
1204 and this prediction image is stored in frame memory
D. At this point, if a P+ frame is decoded as a P- frame
due to some type of error, or conversely if a P- frame is
decoded as a P+ frame, the correct prediction image is not
1o synthesized in the decoder and the quality of the decoded
image deteriorates. After synthesizing the prediction
image, the operation returns to Fig. 11 and the quantized
DCT coefficients is input in 1108. Inverse quantization and
inverse DCT is then applied to these coefficients and the
resulting image is stored in frame memory E. In 1109, it
is checked again whether the frame currently being decoded
is an I frame. If the current frame is not an I frame, images
stored in frame memory D and E are added in 1110 and the
resulting sum image is stored in frame memory E . The image
2o stored in frame memory E before starting the process in 1111
is the reconstructed image. This image stored in frame
memory E is output to the output frame memory (for instance,
706 in Fig . 7 ) in 1111, and then output from the decoder as
the reconstructed image. The decoding process for a frame
is completed here and the process for the next frame starts
by returning to 1102.
When a software based on the flowchart shown in Figs .
8 - 12 is run in the software image encoders or decoders,
Z4
CA 02475778 2004-08-25
the same effect as when custom circuits and custom chips are
utilized are obtained.
A storage media (recording media) with the bit stream
generated by the software encoder 601 of Fig. 6 being
recorded is shown in Fig. 13. It is assumed that the
algorithms shown in the flowcharts of Figs. 8 - 10 is used
in the software encoder. Digital information a_s recorded
concentrically on a recording disk 1301 capable of recording
digital information (for instance magnetic disks, optical
1o disk, etc. ) . A portion 1302 of the information recorded on
this digital disk includes: prediction mode distinction
information 1303, 1305, 1308, 1311, and 1314; rounding
method distinction information 1306, 1309, 1312, and 1315;
and motion vector and DCT coefficient information 1304 , 1307 ,
1310, 1313, and 1316. Information representing 'T' is
recorded in 1303 , ' P' is recorded in 1305 , 1308 , 1311 , and
1314 , ' +' is recorded in 1306 , and 1312 , and ' -' is recorded ~
in 1309, and 1315. In this case, 'I° and '+' can: be
represented by a single bit of 0, and 'P' and '-' can be
2o represented by a single bit of 1. Using this representation,
the decoder can correctly interpret the recorded information
and the correct reconstructed image is synthesized. By
storing a coded bit stream in a storage media using the method
described above, the accumulation of rounding errors is
prevented when the bit stream is read and decoded.
A storage media with the bit stream of the coded data
of the image sequence shown in Fig . 5 being recorded is shown
in Fig. 15. The recorded bit stream includes information
CA 02475778 2004-08-25
related to P+ , P- , and H frames . In the same way as in 1301
of Fig. 13, digital information is recorded concentrically
on a record disk 1501 capable for recording digital
information(for instance, magnetic disks, optical disks,
etc. ) . A portion 1502 of the digital information recorded
on this digital disk includes: prediction mode distinction
information 1503, 1505, 1508, 1510, and 1513; rounding
method distinction information 1506, and 1512; and motion
vector and DCT coefficient information 1504, 1.507, 1509,
1511, and 1514. Information representing 'I' is recorded
in 1503 , 'P' is recorded in 1505 , and 1510 , ' B' :is recorded
in 1508 , and 1513 , ' + ° is recorded in 1505 , and ' - ° is
recorded
in 1511. In this case, ' I' , ' P' and ' B' can be represented
respectively by two bit values 00 , Ol , and 10 , and ' +' and
Z5 ' -' can be represented respectively by one bit values 0 and
1. Using this representation, the decoder can correctly
interpret the recorded information and the correct
reconstructed is synthesized. In Fig. 15, information
related to frame 501 ( I frame ) in Fig . 5 is 1503 and 1504 ,
information related to 502 (B frame) is 1508 and 1509,
information related to frame 503 ( P+ frame ) is 1505 and 1507 ,
information related to frame 504 ( B frame ) is 1513 and 1514 ,
and information relhted to frame 505 ( P- frame ) is 1510 and
1512. When coding image sequences are coded using B frames,
the transmission order and display order of frames are
usually different. This is because the previous and
subsequent reference images need to be coded before the
prediction image for the B frame is synthesized.
26
CA 02475778 2004-08-25
Consequently, in spite of the fact that the frame 502 is
' displayed before frame 503 ; information related to frame 503
is transmitted before information related to frame 502 . As
described above, there is no need to use multiple rounding
methods for B frames since motion compensation in B frames
do not cause accumulation of rounding errors. Therefore,
as shown in this example, information that specifies
rounding methods ( a , g . ° +' and ' -' ) is not transmitted for
B frames. Thus for instance, even if only positive rounding
io is applied to B frames, the problem of accumulated rounding
errors does not occur. By storing coded bit streams
containing information related to B frames in a storage media
in the way described above, the occurrence of accumulated
rounding errors can be prevented when this bit stream is read
and decoded.
Specific examples of coders and decoders using the
coding method described in this specification is shown in ~
Fig. 14. The image coding and decoding methad can be
utilized by installing image coding and decoding software
into a computer 1401 » This software is recorded in some kind
of storage media (CD-ROM, floppy disk, hard disk, etc. ) 1412 ,
loaded into a computer and then used. Additionally, the
computer can be used as an image communication terminal by
connecting the computer to a communication lines . It is also
possible to install the decoding method described in this
specification into a player device 1403 that reads and
decodes the coded bit stream recorded in a storage media 1402 .
In this case, the reconstructed image signal can be displayed
27
CA 02475778 2004-08-25
on a television monitor 1404. The device 2403 can be used
only for reading the coded bit stream, and in this case, the
decoding device can be installed in the television monitor
1404. It is well known that digital data transmission can
be realized using satellites and terrestrial waves. A
decoding device can also be installed in a television
receiver 1405 capable of receiving such digital
transmissions. Also, a decoding device can also be
installed inside a set top box 1409 connected to a
satellite/terrestrial wave antenna, or a cable 1408 of a
cable television system, so that the reconstructed images
can be displayed on a television monitor 1410. :In this case,
the decoding device can be incorporated in the television
monitor rather than in the set top box, as in the case of
1404. The layout of a digital satellite broadcast system
is shown in 1413, 1414 and 1415. The video information in
the coded bit stream is transmitted from a broadcast station
1413 to a communication or broadcast satellite 1414: The
satellite receives this information, sends it to a home 1415
2o having equipment for receiving satellite broadcast programs ,
and the video information is reconstructed and displayed in
this home using devices such as a television receiver or a
set top box. Digital image communication using mobile
terminals 1406 has recently attracted considerable
attention, due to the fact that image communication at very
low bit rates has become possible. Digital portable
terminals can be categorized in the following three types
a transceiver having both an encoder and decoder; a
28
CA 02475778 2004-08-25
transmitter having only an encoder; and a receiver having
only a decoder. An encoding device can be installed.in a
video camera recorder 1407 . The camera can also be used just
for capturing the video signal and this signal can be
supplied to a custom encoder 1411. All of the devices or
systems shown in this drawing can be equipped with the coding
or/and decoding method described in this specification. By
using this coding or/and decoding method in these devices
or systems, images of higher quality compared with those
obtained using conventional technologies can be obtained.
The following variations are clearly included within
the scope of this invention.
(i) A prerequisite of the above described principle
was the use of block matching as a motion compensation method.
However, this invention is further capable of being applied
to all image sequence coding and decoding methods in which
motion compensation is performed by taking a value for the
vertical and horizontal components of the pixel motion
vector that is other than an integer multiple of the sampling
2o period in the vertical and horizontal directions of the pixel ,
and then finding by interpolation, the intensity value of
a position where the. sample value is not present. Thus for
instance, the global. motion compensation listed in Japanese
Patent Application No. Hei 08-060572 and the warping
prediction listed in Japanese Patent Application No. Hei
08-249601 are applicable to the method of this invention.
(ii) The description of the invention only mentioned
the case where a value integral multiple of 1/2 was taken
29
'!
CA 02475778 2004-08-25
for the horizon~.al. and vertical components of the motion
vector. However, this invention is also generally
applicable to methods in which integral multiples of 1/d (d
is a positive integer and also an even number) are allowed
for the horizontal and vertical components of the motion
vector. However, when d becomes large, the divisor for
division in bilinear interpolation (square of d, see
Equation 2) also becomes large, so that in contrast, the
probability of results from normal division reaching a value
of 0.5 become low. Accordingly, when performing only
positive rounding, the absolute value of the expectation for
rounding errors becomes small and the bad effects caused by
accumulated errors become less conspicuous. Also
applicable to the method of this invention, is a motion
compensation method where for instance, the d value is
variable, both positive rounding and negative rounding are
used when d is smaller than a fixed value, and only positive ~
rounding or only negative rounding is used when the value
of d is larger than a fixed value.
(iii) As mentioned in the prior art, when DCT is
utilized as an error coding method, the adverse effects from
accumulated rounding errors are prone to appear when the
quantized step size of the DCT coefficient is large . However
a method is also applicable to the invention , in which, when
the quantization step size of DCT coefficients is larger than
a threshold value then both positive rounding and negative
rounding are used. When the quantization step size of the
DCT coefficients is smaller than the threshold value then
CA 02475778 2004-08-25
only positive rounding or only negative rounding is used.
(iv) In cases where error accumu~~atzons occur on the
luminance plane and cases where error accumulations occur
on the chrominance plane, the bad effects on the
reconstructed images are generally more, serious in the case
of error accumulations on the chrominance plane. This is
due to the fact that rather than cases where the image darkens
or lightens slightly, cases where overall changes in the
image color happen are more conspicuous. However, a method
is also applicable to this invention in which both positive
rounding and negative rounding are usedl for the chrominance
signal, and only positive rounding or negative rounding is
used for the luminance signal.
As described in the description of related art, 1/4
pixel accuracy motion vectors obtained by halving the 1/2
pixel accuracy motion vectors are rounded to 1/2 pixel
accuracy in H . 263 . However by adding certain changes to this
method, the absolute expectation value for rounding errors
can be reduced. In H. 263 that was mentioned in the prior
2o art, a value which is half the horizontal or vertical
components of the motion vector for the luminance plane is
expressed as r + s / 4 (r is an integer, s is an integer less
than 4 and not smaller than 0 ) , and when :~ is 1 or 3 , a rounding
operation is performed to obtain a 2. This operation can
be changed as follows : When s is 1 , a rounding operation is
performed to obtain a 0 , and when s is 3 a 1 is be added to
r to make s a 0 . By performing these operations , the number
of times that the intensity values at positions 406 - 408
31
1
CA 02475778 2004-08-25
in Fig . 4 is definitely reduced ( Probabi:lity that horizontal
and vertical components of motion vector will be an integer
become high. ) so that the absolute expectation value for the
rounding error becomes small. However,. even if the size of
the error occurring in this method ca:n be limited, the
accumulation of errors cannot be completely prevented.
( v ) The invention described in triis specification is
applicable to a method that obtains the final interframe
prediction image by averaging the prediction images obtained
by different motion compensation methods. For example, in
the method described in Japanese Patent Application No . Hei
8-2616, interframe predictidn images obtained by the
following two methods are averaged: block matching in which
a motion vector is assigned to each 16:x16 pixel block; and
block matching in which a motion vector is assigned to each
8x8 pixel blocks . In this method, rounding is also performed
when calculating the average of the two prediction images .
When only positive rounding is continuously performed in
this averaging operation, a new type of rounding error
2o accumulates . This problem can be solved by using multiple
rounding methods for this averaging operation. In this
method, negative rounding is performed in the averaging
operation when positive rounding is ~?erformed in block
matching. Conversely, positive rounding is used for the
averaging when negative rounding is used for block matching.
By using different rounding methods for averaging and block
matching, the rounding errors from two different sources is
cancelled within the same frame.
3~
CA 02475778 2004-08-25
(vi) When utilizing a method that alternately locates
p+ frames and P- frames along the time <ixis , the encoder or
the decoder needs to determine whether the currently
processed P frame is a P+ frame or a P- frame . The following
is an example of such identification method: A counter counts
the number of P frames after the most recently coded or
decoded I frame, and the current P frame is a P+ frame when
the number is odd, and a P- frame wher.~ the number is even
( this method is referred to as an implicit scheme ) . There
to is also a.method for instance, that writes into the header
section of the coded image information, information to
identify whether the currently coded P frame at the encoder
is a P+ frame or a P- frame ( this method is referred to as
an explicit scheme). Compared with t:he implicit method,
this method is well able to withstand i~ransmission errors,
since there is no need to count the number of P frames.
Additionally, the explicit method has the following
advantages : As described in "Descript.ion for Related Art" ,
past encoding standards (such as MPEG-1. or MPEG-2) use only
2o positive rounding for motion compensation. This means for
instance that the motion estimation/rnotion compensation
devices (for example equivalent to l~~s in Fig.l) f or
MPEG-1/MPEG-2 on the market are not compatible with coding
methods that uses both P+ frames and P- frames . It is assumed
that there is a decoder which can decode bit streams
generated by a coding method that uses P+ frames and P- frames .
In this case if the decoder is based on the above mentioned
implicit method, then it will be difficult to develop an
33
CA 02475778 2004-08-25
encoder that generates bit streams that can be correctly
decoded by the above mentioned decoder, using the above
mentioned motion estimation/compensation device for
MPEG-1/MPEG-2. However, if the decoder is based on the above
mentioned explicit method, this problem can be solved. An
encoder using an MPEG-1/MPEG-2 motion estimation/motion
compensation device can continuously send P+ frames, by
continuously writing rounding method distinction
information indicating positive rounding into the frame
information header. When this is performed, a decoder based
on the explicit method can correctly decode the bit stream
generated by this encoder. Of course, it should be more
likely in such case that the accumulation of rounding errors
occurs, since only P+ frames are present. However, error
accumulation is not a serious problem in cases where the
encoder uses only small values as the quantization step size
for the DCT coefficients (an example :for such coders is a ~
custom encoder used only for high rate coding) . In addition
to this interoperability between past standards, the
2o explicit method further have the following advantages s ( a }
the equipment cost for high rate custom encoders and coders
not prone to rounding error accumulation due to frequent
insertion of I frames can be reduced by installing only
positive or negative rounding as the pixel value rounding
:25 method for motion compensation; and(b) the above encoders
not prone to rounding error accumulation have the advantage
in that there is no need to decide whether to code the current
frame as a P+ or P- frame , and the processing is simplified .
34
CA 02475778 2004-08-25
(vii) The invention described in this specification
is applicable to coding and decoding methods that applies
filtering accompanying rounding to the interframe
prediction images. For instance, in t:he international
standard H. 261 for :image sequence coding, a low-pass filter
(called a loop filter) is applied to 'block signals whose
motion vectors are not 0 in interframe prediction images.
Also, in H. 263, filters can be used to smooth out
discontinuities on block boundaries (blocking artifacts).
1o All of these filtezs perform weighted averaging to pixel
intensity values and rounding is then performed on the
averaged intensity values. Even for these cases, selective
use of positive rounding and negative rounding is effective
for preventing error accumulation.
(viii) Besides I P+ P- P+ P- . . . , various methods for
mixing P+ frames and P- frames such as I P+ P+ P- P- P+ P+ . . . ,
or I P+ P- P- P+ P+ . . . are applicable to the method of this ~
invention. For instance, using a random number generator
that outputs 0 and 1 both at a probability of 50 percent,
2o the encoder can code a P+ and P- frame when the output is
0 and 1, respectively. In any case, the less the difference
in probability that P+ frames and P- frarnes occur in a certain
period of time, the less the rounding error accumulation is
prone to occur. Further, when the encoder is allowed to mix
P+ frames and P- frames by an arbitrary method, the encoder
and decoder must operate based on the explicit method and
not with the implicit method described above. Accordingly,
the explicit method is superior when viewed from the
CA 02475778 2004-08-25
perspective of allowing flexibility configuration for the
encoder and decoder.
(ix)The invention described in this specification does
not limit the pixel value interpolation method to bilinear
interpolation . Interpolation methodsfor intensity values
can generally be described by the following equation:
[Equation 5)
x x
R(x+r,y+s)=T(~ ~h(r-j,s-k)R(x+.j,y+k))
j._ac j~_x
1~
where , r and s are real numbers , h ( r , s ) is a function
for interpolating the real numbers , and T ( z ) is a function
for rounding the real number z. The definitions of R (x,
y), x, and y are the same as in Equation ~. Motion
compensation utilizing positive rounding is performed when
T (z) is a function representing positive rounding, and
r
motion compensation utilizing negative rounding is
performed when the function representing negative rounding.
This invention is applicable to interpolation methods that
can be described using Equation 5 . For instance , bilinear
interpolation can be described by defining h(r, s) as shown
below.
{Equation 6]
h(r,s) _ (1- IrI)(1-,s~), 0 s ~r~ s 1, 0 s s s 1, ~ ~ ~ (6)
0, otherwise.
36
CA 02475778 2004-08-25
However, if for instance h(r,s~ is defined as shown
below,
[Equation 7]
h(r,s) = 1- Irl - ~sl, 0 s irl + ~sl s l, rs < 0,
1-~r~, 'r~ z IsI, ~r' s l, rs a 0, ~-~(7)
1- IsI, lsl > Ir~, IsI s l, rs > Q,
0, otherwise.
then an interpolation method different from bilinear
interpolation is implemented but the invention is still
applicable.
(x) The invention described in this specification does
not limit the coding method for error images to DCT (discrete
cosine transform). For instance, wavelet transform (for
example, M. Antonioni, et, al,"Image Coding Using Wavelet
Transform" IEEE Trans. Image Processing, vol. l, no.2,
April 1992) and Walsh-Hadamard transform (for example, A.
N. Netravalli and B. G. Haskell, "Digital Pictures",
Plenum Press, 1998) are also applicable to this invention.
37
