CA1089100A - Image coder-decoder using a matrix transform with weighted contribution of several points of the image to the formation of one point of the transform - Google Patents

Abstract

ABSTRACT OF THE DISCLOSURE
A matrix transform system for coding and decoding images. The system comprises means for sampling the lines and the columns of an image and forming with these samples a sample square matrix and storing the same. This sample square matrix is splitted into component square matrices of order 3N x 3N having a central part of order N x N. Each of these component matrices is multiplied by a first rectangular coefficient matrix of order N x 3N having a square central part in which the coefficients are equal to unity and two square lateral parts in which the coefficients are selec-tively equal to zero and a predetermined factor smaller than unity, which gives an intermediate matrix of order N x 3N. This inter-mediate matrix is multiplied by a second rectangular coefficient matrix of order 3Nx N which is the transpose of the first rec-tangular coefficient matrix, which gives an output matrix of order N x N which is the tranform of the central part of the component matrix. If N = 2, the output matrix is of order 2 x 2 ; one of its terms depends on four samples of the image, two of its terms depend on twelve samples of the image and the last of its terms depends on thirty six samples of the image.

Description

lO~9i~0 BACKGROUND OF THE INVENTION
The present invention relates to a system for coding and decoding images using a matrix transform, the said system allowing a compression of the image signals.
PRIOR ART
It is known to make a transform F~u,v) correspond to a bi-dimentional image function f(x,y) having N2 points, the ;~
transform F(u,v) being defined by the matrix product:
~(u,v~ = ~H(u,v~ ~(x,y~ LH(U,V)7 T (I) an equation in which H is a transform matrix of order N x N, ~nd LHJT is the corresponding transposed matrix. This matrix can, for example, be a Hadamard or a Haar matrix. In the case ~ is a ....: .
Hadamard matrix, the following references may be referred to: (i) "Ultilisation de la transformée de Hadamard pour le codage et la compression des signaux d'images" by Jacques Poncin, "Annales des ~; Télécommunications" Volume 26, No. 7-8, 1971; (ii) "Hadamard ~ 1 . . . ~
Transform Image Coding" by William K. PRATT, Julius KANE and Harry C. ANDREWS, Proceedings of the IEEE, Vol. 57, No. 1 January --., . . , ~.,~- . ~., .
1969; (iii) "Intraframe Image Coding by Cascaded Hadamard Trans~
~orms" by Takahiko FUNINIKI and Masachika MIYATA, IEEE Transâc- ~ -tions on Communicatlons, Vol. Com. 21, No. 3, March 1973. In the case where ~H~ is a Haar matrix, the following reference can ~, .~ .. .
~ be referred to:"A Generalized Technique for Spectral Analysis" ~ ~
.
~;~ by Harry C. ANDREWS and Kenneth L. CASPARI, IEEE Transactions j on Computers, Vol. C-l9, No. 1, January 1970, pages 16-25. ~-~
If the matrix ~ is orthogonal and orthonormal, the ~-product ~ T XfH¦is equal to N times the unit matrix. It is the same for the Hadamard and Haar matrices and, in this case, the inverse transform: :
~ 30 ~f (X y~ 7 = 1 ~(U v~ T . ~F(u,v~ . LH (U V~ (2) ~ ~
- j N
,,, '~~ I ' ~ ~
. ~ ;
~ ' .

lQ~91(i`0 uses the same transform matrix rH~ as the direct transform.
Since the Hadamard matrices are square matrices of order N x N = 2n x 2n, it is possible to treat by trans~ormation, either the complete image, or successively small component s~b-images.
In fact, in order to restrict the circuitry in the case of high definition images and, notably, the capacity of the random access memory containing the image samples and the read only memory containing the coefficients of the Hadamard matrix, the transfor-mation is generally applied to partial images having N2 points, the total number of image points being N 2, Thus, these transformations make a transform of N2 coef-ficients correspond to a partial image of NxN points, the said coefficients only taking into account the N2 points o origin.
In this way if, during the inverse transformation, the image is reconstructed by only taking into account cèrtain coefficients, a structure with false step-like contours appears. In particular, if a sub-image is reconstructed by inverse transformation from a .
single element representing the average luminance of the sub- ~
image, the overall image is formed by a juxtaposition of square-s ~ -~20 whose repetitive structure can mask the content of the imagë~itself. ~;
SUr~MARY OF THE INVENTION `~
The object of the present invention is a system for - coding and decoding images by means of a transformation consisting -of a sequence of matrix multiplications operating on partial ima-ges, allowing the alleviation to a great extent o~f the above~
desc~ibed faults of the images obtained by Hadamard direct trans-formation and Hadamard inverse transformation, and allowing a better compression of the rate in bits per second necessary for the transmission of the total image.
~30 According to the present invention, there is provided a matrix transform system for coding images with weighted contribu-

-2-''' ' ' ... . ... . .....

10891(~) tion of several points o the image to each,point of the coded image and for decoding coded images with weighted contribution of several points of the coded image to each point of the decoded image. ~ :~
The system comprises:
a) means for sampling the lines of an image to be coded and forming with the image samples a square matrix associated with the image, b) means for splitting the square matrix associated . ,;
with the image into a plurality of first input component matrices ~ ,~
of order 3N x 3N having a central part of,order N x N~
c) means for multiplying each of the first input compo~
nent matrices by a first rectangular coefficient matrix of order N x 3N having a square central part in which the coefficients are `.- ~ :
equal to unity and two æquare lateral parts in which ~he coeffi-cients are selectively equal to zero and a predetermined factor : smaller than unity, and forming first intermediate matrices of order N x 3N; :~;
., . ,,d) means for multiplying each of the first intermediate ':~ 20 matrices by a second rectangular coefficient matrix of order 3N x.N
which is the transpose of said first rectangular coefficient ma- '' ~.-trix, and forming first output matrices of order N x N, each of ~ , the fir'st output matrices being the transform of the central part ,. , ; ~ . of~a first input component matrix; `
, e) means for forming with the first output matrices'a ' ~i -; square matrix associated with the coded image;
: f) means for splitting the square matrix associated ~' , with the coded image into a plurality of second input component matrices of order 3N x 3N having a central part of order N x N; , ,: , 1 30 g) means for multiplying each of said second'input com- : ~
ponent matrices by a third rectangular coefficient matrix of order ~ :

' , N x 3N having a square central part of order N x N in which the coefficients are equal to unity and two square lateral parts of order N x N in which the coefficients are selectively equal to zero and a predetermined factor smaller than unity and forming second intermediate matrices of order N x 3N;
h) means for multiplying each of the second intermediate matrices by a fourth rectangular coefficient matrix of order 3N x N
which is the transpose of the third rectangular coefficient matrix and forming second output matrices of order N x N, each of the second output matrices being the transform of the central pajrt of :~ -a second input component matrix; and i) means for forming with th.e second output matrices a . ~.
square matrix associated with-the decoded image.
The tranæformation used in the present invention will be ~ ~
explained by comparing it with a classic Hadamard transformation. ~.
In the case of a unidimensional transformation, a certain ..~.
number of successive samples of a line of the sample matrix are :
considered, for example the six samples xi j to xi j+5. The two ; -.~ . central samples xi j+2 Xi j+3 are the two samples to be transformed, .:~
the four other samples,.two on the left xi j and xi j+l and~two .. -:
on the right xi j+4 and xi j+5 are samples acting on the trans~
formation. These six samples form a rectangular matrix of order l:x 6.
the multiplication matrix is a rectangular matrix of .
order 6 x 2 for which the central part is a Hadamard matrix of `
order 2. It follows that the product of the.multiplication is a .matrix of order 1 x 2. The matrix equation of the transformation . - is the following~
'~

- _ ~

', , , ' ' . '. ' ' ~' , ' ' , o ~
O O! ..
~ i,j Xi,j+l lXi,i+2 Xi,i+31 Xi,j+4 Xi,j+ ~X

tUi,j+2 Ui~j+3~ (3) this gives for the coefficients u obtained by matrix multiplica- ~ .
tion~
Ui j+2 = Xilj+2 + Xi,i~3 (2) .
Ui j 3 = -~Xi j -axi j+l + Xi,j+2 ~ Xi,j+3 + ~ i~j+4 i~,j+5 ~ .
where a designates a constant less than one. The coefficient .
Ui j+2 is the average value of the luminances of the two points Xi j+2 and xi j+3. As for the coefficient ui j+3 it can also be written : ~.
Ui j+3 = a ui j + x~ 2 - xi,j+3 ~ i~j+4 ln the formula (3), the parts which would have alone been present with a Hadamard transformation of order two have been surrounded in the sample matrix of order lx6 and in the multi~
plication coefficient matrix of order 6x2 of the first mem~er.
The inverse transformation allowing the reconstruction ; f Xi j+2 and xi j+3 takes into account not only the coefficients ::~
Ui j+2 and ui j+3, but also transformed coefficients of adjacent !~
sub-assemblies ui j~ ui,j+l and ui,j+4 ui,j+5.
The matrix equation is the.following: - . ;
a a ~
LXi,i+2 ~i,j+~= l2 [Ul,j Ui~+l~ui i+~ Ul,j+31Ui, j~4 Ui j+,~

which gives: -,. 2xi~i+2 = ~Ui,~ l+ Ui,j+2 + ui,j+3 - ~ui,j+4 ~7) ; -5-v~ ,, , ~ , ''10~ 0 2x j 3 = -~Ui j + Ui,j+2 ~ Ui,i+3 i,j+4 (8) By putting:
Wi,j+2 = ui,j + ui j+3 - aui j+4 the formula (7) and (B) can be written:
2xi j+2 = Ui~j+2 + Wi,j+2 2xijj+3 = Ui~j+2 ~ Wi,j+2 (8') It can be seen that the two consecutive samples of the transform are formed from six samples of the original image and l0 that two consecutive samples of the reconstructed image are formed ;
from six samples of the transform. In the formula (6), the parts which would have alone been present with an inverse Hadamard trans-formation of order two had been ringed, in the sample matrix of order lx6 and in the general matrix of order 6x2 of the second member.
In the case of a bidimensional transformati~n, a square sample matrix of order six is considered that is multiplied on the right by the coefficient matrix of equation (3) and on the ;
left by the coefficient matrix transpose, which gives a square matrix of order two. One has; ~-~ ,, ri~ --Xi~+5-~ 0,~, ~

r0 ~ 0 0 x Xi+2,j+2 Xi+2,i+3' ....... x ~ ~ ~
~ ~ ta -a ~1 a Xi+3,j+2 Xi+3,i+3 1 0 a ;~ ~
:~, . : ....... ...... O o~ .:~
+5,j-~ -- ----- Xi+5,j+5 _ _ ~i+2 "+2 Ui+2,j ~i+3,j+-2 Ui+3,j+~
This gives for the coefficients obtained by double matrix multi-plication:
Ui+2,j+2 = Xi+2,j+2 + Xi+2~i+3 + xi+3 j+2 + Xi 3 j (l0) . ~ : : .: , 91~0 U1+2 j+3 = -aXi+2 j - ~Xi+2,j+1 + Xi+2,i+2 Xi+2,i+3 + axi+2,j+4 + aXi+2,i+5 i+3,j i+3,j + Xi 3 j 2 ~ xi+3 j+3 + ~xi+3 j+4 + aXi+3 j+5 (11) Ui+3,j+2 = -~ ~ i,j+2 + xi,j+3 + Xi+l,j+2 + Xi+l,j+
+ Xi+2,j+2 + Xi+2,j+3 ~ Xi+3,j+2 Xi+3,j+3 ~ .
+~ ~i+4,j+2 + xi+4,j+3 + Xi+5,j+2 + Xi+5,j+~ (12) Ui~3,j+3 = a2LXi,j + Xi~i+l + Xi+l~j + Xi+l,j+l , ,, + Xi+4,j+4 + Xi+4,j+5 + Xi+5,j+4 + Xl+5~j+~ 1 -~2 LXi,j+4 + xi,j+5 + Xi+1,~+4 + Xi~l,j+5 +Xi+4,j + Xi+4,j+1' + Xi+5,j + Xi+5,j+~
+~ CXi,j+3 + Xi+l,j+3 + Xi+2,j+4 + Xi+2,j+5 '" + Xi+3,j + Xi+3,j+1 + Xi+4,j+2 + Xi+5,j~
~Xi,j+2 + Xi+l,j+2 + Xi+2 j + xi+2 j+l + xi+3,j+4 + Xi+3~;+s + xi+4,j+3 + xi+5,j+
+ Xi+2 j+2 - Xi+2 j+3 ~ Xi+3,j+2 + Xi+3,j+3 (13) In equation (9), the parts of the matrices which corres-ponded to the Hadamard transformation have been ringed. `
Referring for the moment to Figure l, four coefficients - . .
of the bidimensional transform Ui+2,j+2~ Ui+2,j+3, Ui+3,j+2, : .
Ui+3 j+3 are shown as well~as the regions formed by the samples of the image which lead to the~formation of the transform. It can be seen that Ui+2 j+2 is~only a function of four samples of the image -~
and is nothing more than the mean value of these samples xi+2 j+2 +2,j+3' Xi+3,j+2f xi+3,j+3. The coefficient Ui+2 j+3 and the coefficient Ui+3 j+2 are a function of twelve samples and the co-efficient Ui+3 J=3 is a function of thirty six samples. It can ~30 also be seen that certain samples have a complete contribution, . .
- . positive or negative, and other samples have a lesser contribution I
!
.",,. ~
, ' ' ' : ,' :', , '.
. ' ' , ' . ' ' ' ' ' ' ' ' ' --, 10~91~0 in ~ or in ~2, positive or negative.
The matrix equation of the inverse transformation in the bi-dimensional case is the following. A square sample matrix of order six is considered that is multiplied on the right by the .
coefficient matrix of equation (6) and on the left by the coef-ficient matrix transpose which gives a square matrix of order two.
U~ Ui,j+5 ~.
..... ......
Fi+2,j+2 Xi+2,j+~ o ~ -a ~ ,,,.~Ui+2,j+2 Ui+2,i+31 -i+3,j+2 xi+3,j+3J 4 ~ 0~ ,..... 1Ui+3,j+2 Ui+3~j+3 .... ...... ."
Ui+5,j '' Ui+5,i+5 ' '. . ~

: x O O ( 4) ~: . ~ - ~ -O O

~ The parts of the matriaes which correspond to the : inverse Hadamard transformation have been ringed in equation (14).
: It has.been supposed.up to now that the multiplication matrix was of order 6x2, its central parts being a Hadamard matrix . . ~ .- ~
of order two.
To extend the algorithm of the transformation of order two to order four, it is noticed that the iterative application of the matrix transformation resulting from equation (3) allows ~ .

the following coefficients of the transform to be obtained~

Ui j_4 U~ 3 U~ 2 Ui,j-l Ui,j Ui,i+l.Ui,i+
~' ~ . ' ." ' ;. ' . ~ -8~
:

Ui,j+3 Ui,j+4 Ui,j+5 Ui,j+6 Ui,j+7- The samples Ui j+l and Ui j+3 thus calculated are retained and a matrix of order lx6 having coefficients with an index j+p where p is even is now multiplied by the coefficient matrix of order 6x2 of equation (3), which gives:

O -c~ i O -a ~i j 4 Ui j 2 Ui j Ui,j+2 U"j+4 Ui i+~ ~ ~ Li j Vi j+~

The four coefficients of the transform are thus:
i,j; vi,j+l; vi,j+2 = Ui,j+l; vi j+3 = Ui j+3 It can be seen that the coefficients of the unidirec-tional transform are obtained from twelve image samples. This can be put in the form of the following matrix equation:

, . O -o~ O O
'. O -o~, - O O ..
O -a -a O
., . . . O -a -a 1 1 -1 -a ~ 4 Xi,i-3 ' Xi' ~ 1 1 -1 -a -Ei j Vi j+lvi j+2vi j~
1 -1 a 1 (16) 1 -1 ~,-1 O a O a O a O a O a O O . ~.
. ~_ a O O l The coding and decoding system of the inv~ntion gives a - bette~ quality of compression, sinae for a 2xl transformation, the probability of the coefficient ui j+l being zero or small is greater than is the case for a Hadamard transformation. The transformation according to the invention introduces an attenuation of the false square contours and a better statistic approximation of the signal.
. ~................. I .
_g _ , ,.. ,.... - : ~: .

1~91~0 In effect, during the calculation of the inverse transormation, a filtering operation is carried out on the coefficient U
BRIEF DESCRIPTION OF THE DRAWINGS
Other advantages will appear from a reading of the :~
description which follows and the attached drawings which illustra- ~:
te it and in which~
- Figure 1, which has been mentioned in the introduction shows the respective contributions of the samples of the image to the samples of the transform;
- Figures 2a, 2b and 3a, 3b shows respectively in the form of diagrams, the quickest method of forming samples of the transform from image samples and thernethod of formation of the : : ;
, samples of the reconstructed image from the samples of the :~
transform;
- Figure 4 shows in the form of a block diagram, the :~
coding and decoding system of images of the invention for a 2x2 ~.
dimension; :
- Figure S shows, also in the form of a block diagram, ~ ~:
a system for coding and decoding images for 4x4 dimensfons, -:
2a - Figure 6 is a diagram showing the development of the `
points of the transform from the points of the imagè in the case of the system of Figure 4.
, DETAILED DESCRIPTION:
Referring to Figure 2a, the manner of forming the -~:
samples u of the image transform from the sample x of the original ~: :
image has been shown in the case of the equations (4) and (5). The :samples xi 0 to xi 5 are applied to the terminals 100 to 105 dur-ing a first period, then the samples xi 2 to xi 7 are applied in a second period and so on by shifting the applied samples by two samples.
The terminals l00 and 101 are connected to the adder .. ' I .

, -, 10~ 0 circuit 110. The terminals 102 and 103 are connected to the adder circuit 112 and to the subtractor circuit 113. The terminals 104 and 105 are connected to the adder circuit 114. The output of the adder circuit 112 is connected to the output terminal 122 where the coefficients ui 2~ Ui 4~ ... are found successively. The outputs of the adder circuits 110 and 114 are connected to the subtractor circuit 115 which provide a signal (-ui 0 + ui 4), (-Ui 2 + Ui 6)~ . The output of the subtractor circuit 115 is connected to a multiplier by ~ circuit 116. The output of the multiplier 116 and the output of the subtractor circuit 113 are connected to the adder circuit 117. Finally, the output of this latter adder circuit is connected to the output terminal 123. It is clear that, when xi 0 to xi 5 are applied to the ter-minals 100 to 105, the coefficient ui 2 appears at the terminal ,122 and the coefficient ui 3 appears at the terminal 123.
It can be seen (Figure 2b) that it is possible to omit the adder circuit 110 and replace it by a memory in the form of a shift register 110', which stores ui 2p and reapplies it to the dalculator during the calculation of ui 2p+2. In Figure 2b, 13 is a shift register having four outputs (in fact there are as many shift registers in parallel as bits in the samples) receiving ,' the sa~ples in series and advancing two steps at,a time. The outputs 132-135 are respectively connected to the inputs 102-105 of the calculator. When ui 2p appears at the output terminal 122, it is also applied to ~he input of the shit register 110' which reapplies it to the subtractor circuit 115, a cycle later.
Referring to Figure 3a, the manner of forming the samples x of the reconstituted image from the coefficients u of the image transform in the case of equations (7) and (8) is shown. The coefficients ui 0 to ui 5 are applied to the terminals 200 to 205 during a first pfriod, then the coefficients ui 2 to ui 7 are ap-.

10~91~0 plied coefficients by two coefficients.
The terminals 202 and 203 are connected to the adder circuit 212 and the subtractor circuit 213. The terminals 200 and 204 are connected to the subtractor circuit 215. The output~ of this circuit 215 is connected to two multiplier circuits 216 and 216' having multiplication factors of ~ and -~. Finally, the outputs of the multiplier circuits 216 and 216,' are connected - . .
to two adder circuits 217 and 217', connected, in addition, res- -~
pectively to adder circuits 212 and subtractor circuit 213. The outputs of the adder circuits 217 and 217' are connected respec- -tively to the output terminals 222 and 223. v It is clear that when ui 0 to ui 5 are applied to the terminals 200 to 205, the aample xi 2 appears at the terminals ~
222 and the sample xi 3 at the terminal 223. -Figure'3b shows a second calculator of x as a function , of u based on the application of equations (7') and (8'). Instead of being'applied to the terminals 200 to 205 in the order ui 0 to `'`
Ui 5~ the samples of the image transform are applied to these ' -terminals in the order ui,0~ Ui~ Ui,2' Ui,l' ui,4~ uij3.................. ;-' which is the natural order of genration of these coefficients~ The .
terminals 200 and 204 are, as in Figure 3a, connected to the sub- ' , tractor circuit 215 whose output is connected to the multiplier ' '~
by ~ circu~t 216. The output of the circuit 216 is connected to an adder circuit 218 for which the second input is connected to the '~
terminal 205. It is clear that the signal wi 2 is to'be found at the output of the cricuit 217. The signal is applied to the adder circuit 219 and to the subtractor circuit 219' for which the second inputs are'connected to the terminal 202. The output of the adder circuit 219 is connected to the output terminal 222 and '~

the output of the subtrac~or clrcuit 219' is conn~ct~ to the ~"
output terminal 223. ' -12- ;

~slao Figure 4 shows the coder and decoder of images by matrix transform in accordance with the invention for partial images of 2x2 points.
P Xi~2p~ Xi,2P+l~ f the partial image are applied to the input terminal 30 of a register 31 which delay8 the samples by the duration ~ between two successive samples. The input terminal 30 and the output of the register 31 are connected to an adder circuit 32 and to a subtractor circuit 23 which carry out the sum and difference of the two successive sample5 (xi 2p 1 xi 2p+1). The outputs of the adder circuit 32 and subtractor circuit 33 are connected respectively to registers 34 and 35.
having delays of 2 ~ ~nd 4 T. At the moment when the register 34 receives, at its input, the signal ui 2p+2' it provides at its output the signal ui 2p 2. These two signals are applies to the s.ubtractor circui~ 36 which provides (ui 2p+2 ~ Ui 2p 2) This latter difference signal.is applied to a multiplier circuit 37 which multiplies it by ~.
g (Xi,2p xi,2p+1) leaving the register 35 has been delayed by 2T to make it concomitant with the signal ~ ;
~(ui 2p+2 ~ Ui 2p 2) which leaves the register 34. These two signals are added in the adder register 38 to form ui 2p+1. Thes ~.
, ~ signals representing the coefficients ui 2p and ui 2p+1 are stored :~ in the memory 39 so.that the coeficients corresponding to a line . of the limage form a line in the memory 39. Then the coefficients ~:~ are rearranged in the memory 39 as will be explained.
~ The second stage 41 to 49 is identical to the first stage ~.
,~ 3~1 to 39, with the differénce that the circuit 45 has a delay of 2 lines and the circuit 44 a dealy of 4 lines. Instead of providing the coefficients u, it provides the coefficients U. The circuits ~30; at the two stages to which the reference numerals te~minate by the same unit digit are identical, except for what has just been sa.id -13- :
.
.:

10~91~0 regarding the delay.
As many coefficients U as there are points in the image are found in the memory 49. These coefficients U are grouped in squares in memory 49. If a suitable direction of subtraction is given to subtractor circuits 33, 36, 43, 46, in each square the coefficient Ui j at the top left is of the type of Figure l(A), i.e. that which results from the contribution of the four image points (equation (10)), the coefficient Ui j+l at the top right is of the type of Figure 1 (B) (equation (11)), i.e. that which results from a contribution of twelve image points, the coefficient .. ~ .
Ui+l j at bottom left is of the type of Figure l(C) (equation(12)), i.e. that which results from a contribution of twelve image points, and the coefficient Ui+l j+l at bottom right is of the type of Figure l(D) (equation (13)), which results from a contribution of ~ ;
thirty six image points.
The signals U are applied to the compressor-multiplexer 101; then they are transmitted through the transmission channel 100 to the demultiplexer 102. The compressor contained in the compressor-multiplexer 101 compresses differently the signals of types of Figures l(A), (B), (C) and (D). It can, for exampler, transmit with a certain number of bits the signals of the type of ~ -Figure l(A), with a certain lesser number of bits the signals of the type of Figures l(B) and l(C), and with an even lesser number `~
,: . ... ...
of bits the signals of the type of Figure l(D). These latter signals need not be transmitted at all. The compressor can use -. ~, . . ..
the compression processes known in the case of the conventional ~ Hadamard transformation. ~ `
! The coefficient ~ is determined experimentally. The applicant has found tha~ the best results are obtain~d by taking ~ between 0.1 and 0~2. The value 0.125 see~s particularly in~
teresting as much for its results as for its simplicity of digital -14- ~
: ;' ,~ , lowla~

coding.
The decoder comprises, like the coder, two identical delay stages of register, one relative to a decoding by lines and the other a decoding by columns. The second stage will be described.
The memory 59 of the first stage has been filled by lines and the coefficients have been rearranged as will be explained.
The signal ui 2p+2 is applied to the shift register 61 where it is delayed by ~ as well as to a subtractor circuit 63. The ~ ~
signal ui 2p+1 leaving the register 61 is applied to a register ~ ~ -62 for delaying it by ~ and to an adder circuit 65. The signal -~
Ui 2p leaving the register 62 iS applied to an adder circuit 66, to a subtractor circuit 68 and to a shift register having a delay 2T. The signal ui 2p 2 leaving the register 64 is applied to the subtractor circuit 63 whose output is connected to a multiplier by circuit 67. The output of the multiplier circuit 67 is connec-ted to the adder circuit 65 and the output of the latter is connec-.
ted to adder circuit 66 and subtractor circuit 68. Finally, these ~
two last mentioned circuits are connected to the memory 69 from ~ -~ , . , which the output 60 is the output of the decoder.
- In Figure 6(A), an image having a dozen points per line and a dozen lines has been shown. A rectangle T surrounding six samples moves along the line from a position To 0 in which it contains two zero points on the left outside the image to a position To 5 in which it contains two zero points on the right -~ outside the image. Each position of the rectangle T on a line of ; thé image gives rise to two coefficients of a line of the trans-form of Figure 6(B). These coefficients are of two different types. One shown in white is of the type ui 2p and the other shown in cross hatching is of the type ui 2p+1 These are samples which ; are written into the memory 39.
' I . ' -10~9~(i0 Before subjecting the line transform to the column trans-formation the coefficient u2r 2p+1 is exchanged in the matrix of Figure 6(B) with the coefficient u2r+l 2p so as to obtain the matrix of Figure 6(C). In other words, it can be said that the ~ ;
matrix of order 12x12 of Figure 6(B) is split into matrices of order 2x2 and that these latter matrices are transposed. This re-arrangement of the matrix 6(B), in order to obtain the matrix 6(C), allows a stage identical to the coding stage for the lines to be taken as the coding stage for the columns. Returning to the case of a 6x6 sample matrix of equation (9) the coding stage for thelines serves to multiply each sample line forming a line . .
matrix of order lx6 by a general multiplication matrix of order 6x2 thus to obtain a matrix of order lx2 for each sample line. -~
All these line matrices form a resulting matrix of order 6x2. But, when this 6x2 matrix has been obtained, i.e. when the middle matrix and the right hand matrix of equation (9) has been multiplied -therebetween, there remains the following multiplication of the ~;
resulting matrix by left hand matrix of equation (9):
(II) I

0 ~ ~ 0 ul 1ui+2 2p --ui+2~2p+l~ (17) ~ x .:, a . ~ ~:
+4,2p Ui+4,2p+1 ~ -Ui+5,2p Ui+5~2P+l , This multiplication i8 a multiplication of a matrix of order 2x6 by a matrix of order 6x2 which gives a resulting matrix of order 2x2. It is advantageous to convert the "u" coefficient , ~ ' 'I

losslao matrix of order 6x2 (matrix (II) in expression 17) into a 2x6 matrix and to convert the general multiplication matrix of order 2x6 (matrix (I) in expressionl7) into a 6x2 matrix which allows the same calculation algorithm as in the line transformation to be used for the column transformation.
Considering that the transpose of a matrix product is equal to the product of the transposes taken in the reverse order, -~
the product of the transposes taken in the reverse order, the ~
product (17) can be written as follows by replacing matrix (I) by ' matrix (I') and matrix (II) by matrix (II') and changing the mul-tiplication direction~
~ ~O ~ 1 ¦ui 2p Ui+1~2P ¦ ui+2,2p Ui+3,2P-¦Ui+4,2p Ui+4~2p ¦ 1 -1 ui 2p+1 Ui+l,2p+11, 1ui+2~2p+l Ui+3,2p~lli+4,2p+1 Ui+5,2p+~ 1 -l ~ '~
O + a ::;
:~ . . . ~+ a l . .
J . 18~
`;
! ~ - If each of the component matrices of order 2x2 constitut-ing the matrix for the "u coefficients" of the expressions 17 and ., - . . .
18 i6 considered, it can be seen that they are obtained relative t 20 ~ ~
'~ to each other by transposition. , ' ~ The transformation applied to the matrix of Fig. 6A is ~ again applied to the matrix of Fig. 6C, i.e. a rectangle R sur-, ii .
~ rounding six samples is displaced'along the line from a position '`'~` ~ Ro O surrounding two zero samples of the proceeding line, the --~
'~ samples uO O and ul O and the samples u2 0 and U3 0 to the position R5 O. In fact, the calculation device-is fed by the reading line `
~, ' by line of the matrix C, the data of the upper,and lower lines , being retained in the internal memories of the device (44 and 45 , 'I of Figure 4).
When the rectangle R is displaced along an even line ' , (white squares) of C, the coefficients of type Ui 2p' Ui~l 2p are . , . , . . . -: .:
~ . _ : . , 10~91~0 obtained by moving along an odd line (cross hatched) the coefficients Ui,2p+1 Ui+l,2p+1 are obtained.
The matrix D is filled line by line with coefficients of different types, the first shown ~ is o t yp 2r,2p second shown ~ is of the type U2r+1,2p' the t ~ is ~ :
of the type U2r 2p+1 and the fourth shown ~ is of the type U2r+l 2p+1. It should be noticed that in order to recover the normal matrix configuration, the elementary matrices 2x2 for the U coefficients must be transposed.
Figure 5 shows a line cader for the 4x4 dimension compose~
of two stages 71 and 72 of the type of the first stage 4 of Figure ~ ~
4. The first stage 71 operates at the rate f, receives the samples ~ ;
of the image and provides the coefficients ui and ui 2p+1 The coefficients ui 2p+1 ar.e outputted and the coefficients ui 2p :~
are applied to the second stage 72 identical to the first with the sole difference that it operates at the frequency f/2.
The second stage 72 provides vi 4p and vi 4p+1 The first St~.ge provides one time in two ui 2p+1 = Vi 4p+2 and the !
: 20 me ui, 2p+3 vi,4p+3. The coefficients vi 4 , vi Vi 4p+2 and vi 4p+3 are written in thememory 79. In order to carry ~ ~:
out the complete transform, a column coder having two stages is arrange behing the line coder 71-72. ~ -, Figure 5b shows a line decoder for the 4x4 dimehsion composed of two stages 81 and 82 of the type of the second stage ~ ~.
6 of Figure 4. The first stage 81 operates at the rate f/2, re-ceives from thememory 79 the coefficients of the transform vi 4p and vi 4p+3 and provides the coefficlents ui 2p to.the second stage 82.
.The second stage 82 operates at the rate f. The coeffi-. cients vi 4p+2 and vi 4p+3 are applied~ each on time in two, to .. ;~, I .

~ , 10~91~0 the stage 82. At the output of this stage, the samples of the reconstructed image xi 2p ~ Xi 2p+1~ are found. In order to carry out the complete transform, a column decoder having two stages is arranged in front of the line decoder 81-82. The putting in cascade of a line stages and b column stages allow us the transformation to be carried out on a sub-image having 2a 2b points.

.. :

' , "

:. :
, ~,',, , I , '' -19- ~ , ~ ~"'" ' ' _.. .

Claims (5)

The embodiments of the invention in which an exclusive property or privilege is claimed are defined as follows:

1. A matrix transform system for coding images with weighted contribution of several points of the image to each point of the coded image and for decoding coded images with weighted contribution of several points of the coded image to each point of the decoded image, said system comprising:
a) means for sampling the lines of an image to be coded and forming with said image samples a square matrix associated with the image;
b) means for splitting said square matrix asso-ciated with the image into a plurality of first input component matrices of order 3N x 3N having a central part of order N x N;
c) means for multiplying each of said first input component matrices by a first rectangular coefficient matrix of order N x 3N having a square central part in which the coeffi-cients are equal to positive or negative unity and two square lateral parts in which the coefficients are selectively equal to zero and a predetermined factor smaller than unity, and for forming first intermediate matrices of order N x 3N;
d) means for multiplying each of said first intermediate matrices by a second rectangular coefficient matrix of order 3N x N which is the transpose of said first rectangular coefficient matrix, and forming first output matrices of order N x N, each of said first output matrices being the transform of the central part of a first input component matrix;
e) means for forming with said first output matrices a square matrix associated with the coded image;
f) means for splitting said square matrix associated with the coded image into a plurality of second input component matrices of order 3N x 3N having a central part of order N x N;
g) means for multiplying each of said second input component matrices by a third rectangular coefficient matrix of order N x 3N having a square central part of order N x N in which the coefficients are equal to unity and two square lateral parts of order N x N in which the coefficients are selectively equal to zero and a predetermined factor smaller than unity and for forming second intermediate matrices of order N x 3N;
h) means for multiplying each of said second intermediate matrices by a fourth rectangular coefficient matrix of order 3N x N which is the transpose of said third rectangular coefficient matrix and for forming second output matrices of order N x N, each of said second output matrices being the transform of the central part of a second input component matrix;
and i) means for forming with said second output matrices a square matrix associated with the decoded image.

2. A matrix transform system for coding and decoding images according to claim 1 in which N is a power of two and the central part of the first, second, third and fourth rectangular coefficient matrices is a Hadamard matrix.

3. A matrix transform system for coding and decoding images according to claim 1 in which N is a power of two and the central part of the first, second, third and fourth rectangular coefficient matrices is a Haar matrix.

4. A matrix transform system for coding images according to claim 1 in which N = 2 , the means for multiplying each of the first input component matrices by a first rectangular coefficient matrix of order 2 x 6 having a square central part in which the coefficients are equal to positive or negative unity and two square lateral parts in which the coefficients are selectively equal to zero and a predetermined factor smaller than unity comprises:
a) means for splitting the samples of each line of a first input component matrix into a central group of two samples and two lateral groups of two samples;
b) a first adder and subtractor circuit for forming the sum and the difference of the samples of the central sample group;
c) a first multiplier and adder circuit for multi-plying the samples of the lateral sample groups by a predetermined factor smaller than unity and forming the sums of the multiplied samples of each lateral group;
d) a second adder and subtractor circuit for forming the difference of the sums of the multiplied samples of the two lateral groups and forming the sum of said difference and of the difference of the samples of the central group;
the signals provided by said first and second adder and subtractor circuits forming the coefficients of a first intermediate matrix;
e) means for splitting the coefficients of each line of a first intermediate matrix into a central group of two coefficients and two lateral groups of two coefficients;
f) a third adder and subtractor circuit for forming the sum and the difference of the intermediate coeffi-cients of the central intermediate coefficient group;
g) a second multiplier and adder circuit for multiplying the intermediate coefficients of the lateral intermediate coefficient groups by a predetermined factor smaller than unity and forming the sums of the multiplied intermediate coefficients of each lateral group;

h) a fourth adder and subtractor circuit for forming the difference of the sums of the multiplied intermediate coefficients of the two lateral groups and forming the sum of said difference and of the difference of the intermediate coefficients of the central group;
the signal provided by the said third and fourth adder and subtractor circuits forming the coded image samples.

5. A matrix transform system for decoding coded images according to claim 1 in which N = 2, the means for multiplying each of the second input component matrices by a third rectangular coefficient matrix of order 2 x 6 having a square central part in which the coefficients are equal to posi-tive or negative unity and two square lateral parts in which the coefficients are selectively equal to zero and a predetermined factor smaller than unity comprises:
a) means for splitting the samples of each line of a second input component matrix into a central group of two samples and two lateral groups of two samples;
b) a first adder and subtractor circuit for forming the sum and the difference of the samples of the central sample group;
c) a first multiplier and adder circuit for multi-plying one sample in each lateral sample group by a predetermined factor smaller than unity and forming the difference of said multiplied samples;
d) a second adder and subtractor circuit for forming the sum of the central group sample sum and of the lateral group multiplied sample difference and the difference of the central group sample difference and of the lateral group multiplied sample difference;

the signals provided by said second adder and subtractor circuits forming the coefficients of a second intermediate matrix;
e) means for splitting the coefficients of each line of a second intermediate matrix into a central group of two coefficients and two lateral groups of two coefficients;
f) a third adder and subtractor circuit for forming the sum and the difference of the intermediate coeffi-cients of the central intermediate coefficient group;
g) a second multiplier and adder circuit for multiplying one intermediate coefficient in each lateral inter-mediate coefficient group by a predetermined factor smaller than unity and forming the difference of said multiplied intermediate coefficients;
h) a fourth adder and subtractor circuit for forming the sum of the central group intermediate coefficient sum and of the lateral group multiplied intermediate coefficient difference and the difference of the central group intermediate coefficient difference and of the lateral group multiplied intermediate coefficient difference;
the signals provided by the said fourth adder and subtractor circuits forming the decoded image samples.