Fractal representation of data

A method is provided for the fractal representation of data in which a representative continuous n-dimensional surface is defined as a first approximation to the original data. Residual data is formed from the date and the n-dimensional surface, and is then divided into a first number of regions. For each of the first number of regions, the following steps are performed: A number of domain regions are determined from the data; for a current one of each of the domain regions, a second representative n-dimensional surface is formed from the current domain; a (domain residual is formed from the current domain region; and the second representative n-dimensional surface, and a closest distance or error measure is determined. After processing each of the regions, the representation of data is stored.

FIELD OF THE INVENTION 
The present invention relates to methods for the fractal representation of 
data and in particular to the use of such representations in the 
compression of data. 
BACKGROUND ART 
The representation and/or the compression of data, such as images or sound, 
can generally be divided into two classes. The first class, commonly 
called "lossless" allows for an exact reconstruction of the original data 
from the compressed or represented data in the reconstruction or 
decompression process. Lossless compression normally proceeds by removal 
of real redundancy in the data and includes such techniques as run length 
encoding, Hufftnan coding and LZW coding which are techniques well known 
to those skilled in the art of compression methods. 
A second class of techniques, known as "lossy" techniques represent or 
compress their data such that, upon decompression or reconstruction, the 
decompressed or reconstructed data may vary in comparison to the original 
data. Lossy techniques are able to be used where it is not necessary to 
exactly reconstruct the original data, with a close approximation to it, 
in most cases, being suitable. Of course, it will be readily apparent to 
those skilled in the art that, after initial compression via a lossy 
technique, the resulting compressed data can be further compressed 
utilising a lossless technique. 
One form of well known lossy image compression utilises the Discrete Cosine 
Transform (DCT) in order to compress an image as part of a lossy 
compression system. This system, as defined by the Joint Photographic 
Experts Group (JPEG) standard has become increasingly popular for the 
compression of still images. A further standard, known as the MPEG 
standard developed by Motion Pictures Expert Group, has also become a 
popular form of video image compression. The MPEG standard also relies on 
the Discrete Cosine Transform for the compression of images. 
An alternative compression or representation system is one based on 
"fractal" coding. The fractal coding technique relies on the creation of a 
series of transformations which, when applied to a set of data, result in 
that data approaching a "fixed point" which approximates the original data 
which is desired to be displayed. Therefore, in order to regenerate an 
approximation of the data utilising fractal techniques, it is only 
necessary to know the nature and content of the transformations as, given 
some initial arbitrary data, the transformations can be applied to that 
data in an iterative process such that the arbitrary data is transformed 
into a close approximation of the original data. 
As, the data can be represented by its transformations, fractal coding 
techniques offer the opportunity to produce substantial compression of 
data and, in particular, offer the opportunity to produce a better 
compression system than the JPEG standard. Examples of fractal compression 
or representation techniques can be seen from PCT Publication No. WO 
93/17519 entitled "Fractal Coding of Data" by Monro and Dudbridge, in 
addition to "Image Coding Based on a Fractal Theory of Iterated 
Contractive Image Transformations" by A. E. Jacquin published in IEEE 
Transactions on Image Processing, 1, pages 18-30, (1990). 
The iterated nature of producing a final image through the continuous 
utilisation of transformations has led to fractal coding techniques being 
described as an "Iterated Function System (IFS)". The transformations 
utilised in the IFS are commonly of a contractive nature in that the 
application of a transform results in the data "contracting" to a final 
set of data in accordance with a "difference measure" which can be a Root 
Mean Square (RMS) measure or a Hausdorff distance measure, known to those 
skilled in the art. 
Although the discussion will hereinafter be continued in relation to 
images, it will be readily apparent to those skilled in the art that the 
present invention is not limited to images and is readily applicable to 
any form of data for which a fractal representation or compression is 
desired. Furthermore, although 2-D data is referred to in the following 
description, lower and higher dimensional data sets can easily be 
accommodated with analogous methods by those skilled in the art. 
Referring now to FIGS. 1 and 2, the prior art iterated functions systems 
normally operate by partitioning an initial image 1 into a number of range 
blocks 2 which completely tile or cover the initial image 1. 
In the system set out in the aforementioned article by Jacquin, the pixels 
in each range block 2 are approximated by a linear combination of a first 
element which is derived from a range base function which is defined by 
Jacquin to be of uniform intensity. The second portion of each pixel of a 
range block 2 is made up of pixels derived from a transformation of pixels 
belonging to some other part of the image. These other portions of the 
image, which can be derived utilising a number of different methods, are 
known as "domain" blocks. These domain blocks are transformed utilising 
carefully selected affine transformation functions. The affine 
transformation functions and their selection corresponds to the "fractal" 
portion of the image and, in the Jacquin system, they are used to modulate 
each pixel of the range block 2. 
In PCT Publication No. WO 93/17519 entitled "Fractal Coding of Data", Monro 
and Dudbridge describe a system whereby the range base function consists 
of a planar facet. The range blocks are tiled in their entirety by four 
domain blocks. The planar facet is "corrected" or modulated by the 
application of a scaled version of the domain block which is one quarter 
the area of the original domain block. The major advantage of the Monro 
and Dudbridge technique is that an extensive search for the most suitable 
domain block is avoided. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide an improved form of 
fractal based coding of data. 
In accordance with a first aspect of the present invention, there is 
provided a method of coding original data comprising the steps of: 
(a) defining a representative continuous n-dimensional surface as a first 
approximation to said original data, 
(b) forming residual data from said original data and said representative 
continuous n-dimensional surface, 
(c) dividing the residual data into a first number of regions, 
(d) for each of said first number of regions: 
(e) determining a number of domain regions from said original data, said 
domain regions comprising transformed portions of said original data, 
(f) for a current one of each of said domain regions, forming a second 
representative n-dimensional surface from said current domain region, 
(g) forming a domain residual from said current domain region and said 
second representative n-dimensional surface, 
(h) determining a closest distance or error measure between each of said 
domain residual and said a current one of each of said first number of 
regions, and, 
(i) storing surface information defining said representative continuous 
n-dimensional surface and location and transformation data disclosing said 
domain region having a domain residual having a said closest distance or 
error measure, for each of said first number of regions. 
In accordance with another aspect of the present invention there is 
provided a method of decoding data which has been coded in accordance with 
preceding paragraphs, said method of decoding comprising the steps of: 
(a) defining said representative continuous n-dimensional surface from said 
surface information, 
(b) dividing said surface into said first number of regions, 
(c) for each of said first number of regions, applying the steps of: 
(d) determining a corresponding domain from said location and translation 
data for said region, 
(e) transforming said corresponding domain in accordance with said 
transformation data of said corresponding domain, 
(f) defining a second n-dimensional surface from said transformed 
corresponding domain, 
(g) calculating a second residue portion from said second n-dimensional 
surface and said transformed corresponding domain, 
(h) adding said second residue portion to the representative continuous 
n-dimensional surface of a current one of said first number of regions and 
updating current approximation of data, and 
(i) repeating steps (b)-(h) until said surface converges to an 
approximation to said decoded data. 
In accordance with still another aspect of the present invention there is 
provided a method of encoding data comprising the steps of: 
(a) deriving surface information, which defines an n-dimensional surface, 
as a first approximation to said data, 
(b) forming residual data from said data and said n-dimensional surface, 
(c) dividing said residual data into a first number of regions, 
(d) for a first one of said first number of regions, determining a 
corresponding series of domain regions in accordance with the following 
steps (e) to (j): 
(e) forming a domain region series indicator list, initially comprising an 
empty list, 
(f) determining a number of prospective domain regions from said residual 
data, 
(g) for each of said prospective domain regions, determining a residual 
prospective domain region from said prospective domain region and an 
approximation to said prospective domain region, and determining a 
candidate measure value being a measure of the difference between said 
residual prospective domain region and the first one of said first number 
of regions, 
(h) determining a best matching prospective domain region, being a domain 
region which is not a member of said domain region series indicator list 
and having the smallest of said candidate measure values, 
(i) appending an indicator corresponding to said best matching prospective 
domain region to a said domain region series list, and 
(j) subtracting the residual prospective domain region of said best 
matching prospective domain region from said residual data, 
(k) repeating steps (e) to (j) until said residual data is less than a 
predetermined amount, 
(l) repeating steps (d) to (k) for each region of said residual data, and 
(m) defining said coded data to comprise said surface information and a 
domain region series indicator list for each of said first number of 
regions. 
In accordance with still another aspect of the present invention there is 
provided a method of decoding data which has been encoded in accordance 
with preceding paragraphs, said method comprising the steps of: 
(a) deriving a current estimate of said data comprising an n-dimensional 
surface derived from said surface information, 
(b) dividing said current estimate into said first number of regions, 
(c) for a first one of said first number of regions, applying the steps (d) 
to (g): 
(d) for a first one of said indicators in said domain region series 
indicator list, applying the steps (e) to (f): 
(e) determining, from said current estimate of said data, a domain region 
corresponding to said indicator, 
(f) forming a domain region from said domain region and said n-dimensional 
surface, and 
(g) updating the current estimate of said current one of said first number 
of regions with new estimate of current domain regions, 
(h) repeating steps (d) to (g) for each of said indicators in said domain 
region series indicator list, 
(i) repeating steps (c) to (h) for each of said first number of regions, 
and 
(j) repeating steps (c) to (i) until said data converges.

DETAILED DESCRIPTION 
In the preferred embodiments a number of improvements are made over the 
prior art. In the preferred embodiment a representation of the original 
data, which for example may comprise image data, is constructed from two 
parts. The first part is a functional part which comprises a smooth 
continuous surface which approximates a representation of the data. In the 
case of image data, the surface will have coordinates in X and Y 
representing a particular pixel location and a third coordinate 
representing intensity. Of course, it will be readily apparent to those 
skilled in the art, that the present invention can be applied to other 
dimensional forms of data wherein the surface would become an "hyper 
surface". 
The second portion of approximation to the image comprises a fractal part 
which is designed to "modulate" or alter the smooth continuous surface 
part. It will therefore be apparent that, upon elimination of the fractal 
part, a smooth continuous surface will result and this surface will 
comprise the upper limit to the error in the representation of a given set 
of data or image. 
The use of a smooth continuous surface also eliminates discontinuities 
within block boundaries of an image when compression is extremely high. 
In the preferred embodiments, the smooth continuous surface is formed by an 
interpolation of node points of a spline. These node points are simply a 
subsampling of the image and are hence obtained for very little cost. The 
derivation of the fractal part need only return the best scaling parameter 
for each block. This is a simple operation which therefore permits a much 
more extensive search for the best matching block. 
In the preferred embodiments the mapping function of domain regions to 
range regions is not necessarily a fixed equation. This enables the node 
points in the interpolating spline to be set to be "fixed points" of the 
mapping equations and hence unaltered by the mapping. This occurs even 
when the large block has an arbitrary alignment with a small block. This 
flexibility is achieved by working from the small block and large block 
residuals with respect to an interpolation of the respective blocks on the 
node points for the small block and the points corresponding to the node 
points in the big block. 
Turning now to FIG. 1, there is shown a simple example of an image 1 of 
which is desired to form a fractal representation. In FIG. 2, the image 1 
of FIG. 1 is firstly divided into a number of small blocks 2. For the sake 
of simplicity, the small blocks form a uniform and regular array of 
squares. However, it will be apparent to those skilled in the art that 
other embodiments, not comprising square regions 2, could be utilised. 
Referring now to FIG. 3, there is shown a flow chart illustrating the 
encoding process of the first embodiment. The first step 60 in this 
process is the determination of the block size of each small block 2. The 
size of each small block 2 is determined to be an N.times.N pixel array 
with larger N allowing for greater compression. 
Turning to FIG. 4, there is shown an array of pixels 5 which corresponds to 
an arbitrary portion of the image of FIG. 2. Each pixel 6 has three 
components associated with it being the X and Y locations and an intensity 
or level magnitude. Turning now to FIG. 5, the array of pixels is divided 
into a number of small blocks 7, equivalent to small block 2 of FIG. 2. In 
the present example N will be equal to 4. 
After division of the image into small blocks 7 a "drop out" image or 
subsampled image is formed (step 61 of FIG. 3) from predetermined pixels 
e.g. 9, which are at the top left hand corners of each of the small blocks 
7. Additionally, pixels on the right of the image, e.g. pixel 10, and 
pixels on the bottom of the image, e.g. 11, also form part of the drop out 
image. If necessary, these pixels can be simply repetitions of the values 
in the right most column and bottom most row of pixels in the image. For 
example, pixel 10 can merely be a repetition of pixel 14. 
Upon determining the drop out image, a corresponding linear spline surface 
having node points corresponding to the points of the drop out image is 
formed (step 62 of FIG. 3). The formation of a linear spline surface from 
a set of node points is a process well known to those skilled in the art 
of computer graphics, see for example a standard textbook such as 
"Computer Graphics--Principles and Practice", second edition, by Foley, 
Van Darn, Feiner and Hughes, 1990, by Addison-Wesley Publishing Company 
Inc. 
The linear spline surface is then used in conjunction with all the pixels 5 
of the image to form a "residue" for each pixel, being the difference 
between the calculated surface and the pixel value (step 63 of FIG. 3). 
The set of residues values of each pixel for each small block is denoted 
Aj. Additionally, as will be seen hereinafter, the node points of the drop 
out image also form stationary or fixed points of the fractal mapping. 
The next step in the encoding process (step 64 of FIG. 3) is to form, for 
each N.times.N small block a corresponding (N+1).times.(N+1) grid block. 
Referring now to FIG. 6, a grid block denoted by the hatched line 16 
corresponds to the small block 7 of FIG., 5 and is bounded by the node 
points 9, 17-19, and has a boundary as represented by the hatched line 16. 
FIG. 7 illustrates the "tiling" of the pixel image of FIG. 4 with 
overlapping grid blocks 16. 
Referring now to FIG. 8, the next step in the encoding process (step 65 of 
FIG. 3) is to form, for each grid block, a series of big blocks 25 of size 
(2N+1).times.(2N+1) each having a centre 26 which falls on a pixel within 
a corresponding grid block. As there are (N+1).times.(N+1) pixels in a 
grid block, the number of possible big blocks 25 per grid block will be 
(N+1).sup.2. 
The big block 25 is then subsampled in the vertical and horizontal 
directions starting at the top left hand corner, with the subsampling 
occurring every second pixel in the big block 25 (step 66 of FIG. 3). The 
subsampling process gives a (N+1).times.(N+1) pixel block being of the 
same dimensions as the dimensions of the grid blocks. A linear 
interpolation of the subsampled big block is then calculated based on its 
four corner pixels. 
Each of these subsampled big blocks is then denoted .beta..sub.k, where k 
is an index indicating the pixel within the grid block which is overlayed 
by the centre pixel of the big block from which the subsampled big block 
.beta..sub.k has been formed. 
In addition, all possible horizontal, vertical and diagonal reflections of 
each subsampled big block are considered. As there are eight possible such 
reflections, all of which leave the centre point of the subsampled block 
unchanged, the eight reflections can be indicated by a further subscript 1 
in the range 0 to 7. This may include normalization of the big block 
(domain block) residual, described below. A residue block B.sub.kl is then 
formed by taking the top left N.times.N pixels of each reflected 
subsampled big block .beta..sub.kl and subtracting the linear 
interpolation of the corner points of the block. For simplicity, the 
indices k and l can be compounded into a single index j=8.times.k+1. 
For each residue block in the set A, a search is conducted of the residue 
block set B to determine the best matching residue block (step 68 of FIG. 
3). The matching of residue blocks can include a scaling factor .alpha.. 
The matching criterion between a residue block A and a corresponding 
residue block B is preferably a least squares criterion. This criterion 
can be formulated by considering two residue blocks A.sub.i and B.sub.j 
and considering them to be vectors comprising pixel components in some 
fixed order. In order to determine the closest Root Mean Square (RMS) 
match, it is necessity to determine which blocks minimise the RMS error 
which can be expressed as follows: 
EQU .epsilon..sub.ij =.parallel.A.sub.i -.alpha..sub.ij B.sub.j 
.parallel.Equation 1 
For a given i and j, the optimum scaling parameter .alpha..sub.ij can be 
calculated as follows: 
##EQU1## 
Referring now to FIGS. 9 and 10, there will now be explained an example of 
the matching process. In FIG. 9, there is shown a 5.times.5 grid block 30. 
Each of the pixels 31 includes indices to indicate that pixel's location 
within the grid block 30. The small block is indicated by the dark border 
32. 
Referring now to FIG. 10, there is shown various rotations and reflections 
of a subsampled big block, e.g. 35. These subsampled big blocks are formed 
from subsampling of corresponding big blocks (e.g. 25 of FIG. 8) in a 
manner as previously described. The portion 36 of the subsampled block 35 
forms the residue block B.sub.kl and is compared to a corresponding 
portion 32 of the grid block 30. Additionally, the corresponding residue 
block 37 of the horizontally reflected subsampled block 35 is also 
compared with portion 32. This process is also carried out for vertically 
reflected subsampled block 39, vertically and horizontally reflected 
subsampled block 40, diagonally reflected subsampled block 41, diagonally 
and horizontally reflected subsampled block 42, diagonally and vertically 
reflected subsampled block 43 and a diagonally, vertically and 
horizontally reflected subsampled block 44. The above matching process is 
repeated for each of the possible big blocks whose centre lies over one of 
the pixels within the grid 30. In the present case, as there are 25 pixels 
per grid block 30 and 8 subsampled blocks formed per pixel, a total of 200 
possible big blocks are tried for each small block 32. 
Upon completion of this process, the representation of the image comprises 
the "drop out" image, and for each small block, the relative address j of 
the closest matching big block centre of a given rotation and the scaling 
factor .alpha..sub.ij encode the entire image (step 69 of FIG. 3). 
The Decoding Process 
Referring now to FIG. 11, there is shown a flow chart of the decoding 
process 50. The first step in the decoding process is to receive 51 the 
drop out image and the index j and scaling factor data for each small 
block of the image. 
The drop out image forms the node points for a linearly interpolated base 
image whose size can be different from the initial image (step 52 of FIG. 
11). It will be assumed that the desired final image can be broken up into 
small blocks of size M.times.M pixels. For each M.times.M small block, the 
associated (2M+1).times.(2M+1) big block, as denoted by the index j, which 
includes the reflection component L, is formed from the linearly 
interpolated image (step 53 of FIG. 11). 
For each small block, the resulting big block is subsampled 55 at every 
second pixel location in X and Y starting at the top left hand corner of 
the big block so as to form to a subsampled block. The big blocks (domain 
blocks) are drawn from the current best estimate of the image. Initially, 
this is the interpolated set of grid points. After one iteration, it will 
have some big block information in it. Next, the appropriate X flip, Y 
flip or diagonal (D) flip operations are carried out 56 on the subsampled 
block. 
A linear interpolation of the subsampled block, based on the corner points 
is then calculated and subtracted from the subsampled block to form 57 a 
residue block. The residue block is then multiplied by the scaling factor 
for the block. The top left M.times.M pixels of the resultant scaled 
residue is added to the corresponding interpolated small block of 
interpolated image determined under step 52. 
The steps 55-58 are then repeated for each of the small blocks. 
Subsequently, the steps 53-58 are repeated until the image converges. 
Convergence is measured in the normal way whereby the differences between 
the data of two iterations of steps 55-58 is below a predetermined error 
tolerance. The number of iterations before convergence will be of the 
order of p where 2.sup.p-1 &lt;M&lt;2.sup.P. This completes the decoding 
process. 
For a given desired compression ratio, it may be the case that the loss of 
image resolution is higher than desired in certain areas of the image. In 
this situation, one approach could be to decrease the small block size 
uniformly across the image. However, this is likely to be inefficient if 
some blocks are already represented well by the current small block size. 
An alternative method is to divide those residue blocks of the image which 
are problematical in two separate regions and to match these regions 
individually to corresponding B subdivided residue regions. In accordance 
with the above process, this block division need not result in regular or 
even contiguous regions so long as a division is compactly representable. 
Because each region of a divided small block will have a small number of 
pixels to match, the match will be improved over that for the whole block. 
It is likely that this process will cause minimal decrease in compression 
ratios. Of course, some encoding of the division process is necessary. One 
obvious form of encoding is a quadtree encoding, known to those skilled in 
the art and requiring only one bit for each node in the quadtree to encode 
the quadtree structure. This allows a small block to be subdivided only 
where necessary thereby maintaining high compression ratios. 
It should be noted that for reasonably well encoded images, the error 
vector A.sub.i -.alpha..sub.ij B.sub.j is approximately orthogonal to 
A.sub.i. This means that the size of the error is relatively insensitive 
to the scale factor .alpha..sub.ij. It is therefore possible to coarsely 
quantise the scale factors .alpha..sub.ij without significant impact on 
accuracy. Further, for a given accuracy, the values of interpolation nodes 
need only to be encoded to the order of the accuracy required. Therefore, 
for an RMS error requirement of say 8/255, for an image having 256 
intensity levels for each pixel, it would only be necessary to utilise 
four bits for each of the grid points or interpolation nodes and the RMS 
error at these node points would still be of the order of the required 
accuracy. 
A second embodiment of the present invention which allows each range block 
to draw on multiple domain blocks will now be illustrated. In this 
alternative embodiment, a form of interpolation is used which is able to 
fit an irregular set of points. One form of suitable interpolation, well 
known to those skilled in the art of numerical methods, is that of 
triangular linear patches. 
1. The alternative embodiment divides an image up in the same manner as the 
first embodiment. This includes a series of grid blocks 16 (FIG. 5), and 
grid points 9 and grid blocks 16 (FIG. 6). 
2. An interpolated image 61 is determined based on the grid points e.g. 9 
using the chosen interpolation method. This interpolated image forms a 
current approximation to the range block. 
3. The residual of each pixel of each range block is determined with 
respect to the interpolation. This residual is the difference between the 
actual pixel values and the interpolated pixel values. This residual forms 
a current range block residual 62 for each range block. 
4. A big block 25 (FIG. 12) of size (2N+1).times.(2N+1) is then formed, 
with the big block 25 having a centre pixel 26 which falls on one of the 
pixels of a corresponding grid block. This big block is then subsampled in 
the same manner as the first embodiment to form a subsampled big block 67 
having the same dimensions as the corresponding grid block 16 (FIG. 13). 
5. Each grid block 16, will have, at its corners, one of four grid points 
9,17-19 from which the interpolation of the range block was found in step 
2. A corresponding interpolation of the (N+1).times.(N+1) sized subsampled 
big block is obtained 68 using the corner pixels of the subsampled big 
block 67. 
6. A residual domain block 69, being the difference between the subsampled 
big block 67 and the interpolated subsampled big block 68 is obtained. 
This difference is then normalised 79, with the preferred method of 
normalisation being a division by the square root of the sum of the square 
of the pixel differences. 
7. The normalised residual domain block 79 is then scaled to form scaled 
residual domain block 80, the scaling being such that the pixel 81 is of 
equivalent magnitude to pixel 82 of the residual grid block 62 of FIG. 13. 
8. The steps 4-7 are repeated for all possible big blocks whose centre 
falls on one of a corresponding grid block pixel. The scaled residual 
domain block 80 which best matches a current residual grid block 62 in a 
root mean square sense is then chosen as the best scaled domain block. 
9. The scaled residual domain block 80 is then added to the current 
approximation to the range block 61 to form a new approximation to the 
range block 84 (FIG. 13). 
10. A new residual grid block 85 is then calculated, being the difference 
between grid block 16 and the new approximation to the range block 84. 
This new residual grid block will have zero values at the points 87-91 as 
they have been fixed by the scaling process. 
11. Next, an alternative big block 25a (FIG. 14) is chosen, the alternative 
big block being one which has not, as yet, been used in the encoding 
process. 
12. This big block is subsampled to form subsampled big block 67a. The 
subsampled big block is then interpolated at equivalent points 93-97 to 
all points which have thus far been fixed by the encoding process. 
13. A residual 69a is then determined, being the difference between 
subsampled big block 67a and interpolated subsample big block 68a. This 
residual is then subsequently normalised 79A. 
14. The normalised residual domain block 79a is then scaled to form a 
scaled domain block 80a such that the scaled residual at the next point to 
be fixed e.g. 81a, for the second domain block, has the value of the new 
residual grid block 85 (FIG. 13) at an equivalent point 93. 
15. The steps 11-14 are repeated for each of the remaining candidate domain 
blocks (big blocks 26A) and the domain block which best matches the 
current range block residual in a root mean square sense is chosen. 
16. The steps 9-14 are then repeated, for a predetermined sequence of fixed 
points, until the RMS error in the range block approximation is reduced 
below a predetermined level. 
17. The steps 1-16 can then be repeated for each of the range blocks within 
the image. 
The encoding for each range block of an image will then consist of the 
value of the grid point e.g. 9 in the top left hand corner of each range 
block and the big block offset values (i.e. domain block offset values) 
for the series of chosen big blocks in addition to a scaling factor for 
each of the chosen big blocks. 
The Decoding Process of the Second Embodiment 
The decoding process can be carried out for a pixel array of any resolution 
and proceeds as follows: 
1. Each of the pixel grid points stored with each range block are set to 
their stored values. 
2. The pixel image can then be formed from an interpolation of the grid 
points. This interpolation process then forms a base image approximation. 
3. The base image approximation is copied to a temporary array called the 
"current image estimate". 
4. A copy of a "current range block" from the base image approximation is 
made and forms the "current range block estimate". 
5. For a current range block, the domain block, as determined by the big 
block offset, is extracted from the current image estimate and is 
subsampled so as to be of the same pixel dimensions as the range block. 
6. An interpolation of the subsampled domain block using the appropriate 
predetermined fixed pixels for that domain block is calculated. 
7. The normalised residual is in turn calculated from the interpolation of 
the subsampled domain block and the subsampled domain block itself. 
8. The normalised residual is scaled by the relevant stored scale factor 
for the current domain block. 
9. The scaled normalised residual from step 8 is added to the current range 
block estimate. 
10. The current range block estimate is copied to the appropriate range 
block in the current image approximation. 
11. The steps 5 to 10 are repeated for all the domain blocks stored for the 
current range block. 
12. The steps 4 to 11 are repeated for all the range blocks in the image. 
13. The steps 4 to 12 are iterated until the current image estimate 
converges, with convergence being measured in the normal sense. 
Further Aspects of the Invention 
Further embodiments of the invention will now be described with reference 
to FIGS. 15 to 18. 
A flow chart of a method of encoding original data 1502 is shown in FIG. 
15. In step 1504, a representative continuous n-dimensional surface is 
defined as a first approximation to the original data 1502. In step 1506, 
residual data is formed from the original data and the representative 
continuous n-dimensional surface. In step 1508, the residual data is 
divided into a first number of regions. 
In decision block 1510, a check is made to determine if each of the first 
number of regions has been processed. This is done to ensure that, for 
each of the first number of regions, steps 1512 to 1518 are performed. If 
decision block 1510 returns false (NO), processing continues at step 1512. 
In step 1512, a number of domain regions are determined from the original 
data in which the domain regions comprise transformed portions of the 
original data. In step 1514, for a current one of each of the domain 
regions, a second representative n-dimensional surface is formed from the 
current domain region. In step 1516, a domain residual is formed from the 
current domain region and the second representative n-dimensional surface. 
In step 1518, a closest distance, or error measure, between each of the 
domain residual and a current one of each of the first number of regions 
is determined. Processing continues at decision block 1510. 
Otherwise, if decision block 1510 returns true (YES), processing continues 
at step 1520. In step 1520, surface information defining the 
representative continuous n-dimensional surface and location and 
transformation data disclosing the domain region having a domain residual 
having a closest distance or error measure are stored, for each of the 
first number of regions. 
Preferably, the surface information defining the representative continuous 
n-dimensional surface includes data values at predetermined locations of 
the surface. Further, at least one data value may be located within each 
corresponding region of the first number of regions. Still further, the 
surface may be defined from one of linear interpolation or cubic spline 
interpolation of the data values. 
Preferably, each of the domain regions is derived from an area overlapping 
the current one of each of the first number of regions. Further, the 
transformed portion of the residual data may include sub-sampling the 
residual data. Still further, the transformation optionally may include 
one of scaling, rotation, horizontal reflection, vertical reflection or 
diagonal reflection of the portion of the residual data. 
Preferably, the domain regions include a domain region derived from a block 
of data having a centre corresponding to the location of a data value 
within the current one of each of the first number of regions and a 
separate domain region being defined for each of the locations of data 
values within the regions. 
Preferably, the number of dimensions is three. 
A flowchart of a method of decoding data 1602, which has been coded in 
accordance with FIG. 15, is shown in FIG. 16. In step 1604, the 
representative continuous n-dimensional surface is defined from the 
surface information. The n-dimensional surface is the basis for the new 
range block approximation. In step 1605, the current approximation of the 
data is initialised. The current approximation is where domain blocks are 
taken from. In step 1606, the surface is divided into the first number of 
regions. 
In decision block 1608, a check is made to determine if each of the first 
number of regions has been processed. This is done to ensure that, for 
each of the first number of regions, steps 1610 to 1618 are applied. If 
decision block 1608 returns false (NO), processing continues at step 1610. 
In step 1610, a corresponding domain from the location and translation data 
for the region are determined. In step 1612, the corresponding domain is 
transformed in accordance with the transformation data of the 
corresponding domain. In step 1614, a second n-dimensional surface is 
defined from the transformed corresponding domain. In step 1616, a second 
residue portion is calculated from the second n-dimensional surface and 
the transformed corresponding domain. In step 1618, the second residue 
portion is added to the representative continuous n-dimensional surface 
for the current one of the first number of regions and the result is used 
to update current approximation of data. Processing then continues at 
decision block 1608. If decision block 1608 returns true (YES), processing 
continues at decision block 1620. 
In decision block 1620, a check is made to determine if the surface has 
converged to an approximation to the decoded data. This is done to ensure 
that steps 1606 to 1618 are repeated until the surface converges to an 
approximation to the decoded data. If decision block 1620 returns false 
(NO), processing continues at step 1606. Otherwise, if decision block 1620 
returns true (YES), processing terminates. 
Preferably, the convergence comprises one of repeating steps 1606 to 1618 a 
predetermined number of times or the surface undergoing less than a 
predetermined amount of alteration between successive repetitions of steps 
1606 to 1618. 
A flowchart of a method of encoding data 1702 according to a fourth 
embodiment is shown in FIG. 17. In step 1704, surface information, which 
defines an n-dimensional surface, is derived as a first approximation to 
the data. In step 1706, residual data is formed from the data in the 
n-dimensional surface. In step 1708, the residual data is divided into a 
first number of regions. 
In decision block 1710, a check is made to determine if each region of the 
residual data has been processed. This is done to ensure that, for each of 
the first number of regions, a corresponding series of domain regions are 
determined in accordance with steps 1712 to 1722. If decision block 1710 
returns false (NO), processing continues at step 1712. 
In step 1712 a domain region series indicator list is formed, which 
initially comprises an empty list. In step 1714, a number of prospective 
domain regions are determined from the residual data. In step 1716, for 
each of the prospective domain regions, a residual prospective domain 
region is determined from the prospective domain region in an 
approximation to the prospective domain region, and a candidate measure 
value, being a measure of the difference between the residual prospective 
domain region and the first one of the first number of regions, is 
determined. In step 1718, a best matching prospective domain region, being 
a domain region which is not a member of the domain region series 
indicator list and having the smallest of the candidate measure values, is 
determined. In step 1720, an indicator corresponding to the best matching 
prospective domain region is appended to the domain region series list. In 
step 1722, the residual prospective domain region of the best matching 
prospective domain region is subtracted from the residual data. 
In decision block 1724, a check is made to determine if the residual data 
is less than a predetermined amount. This is done to ensure that steps 
1712 to 1722 are repeated until the residual data is less than a 
predetermined amount. If decision block 1724 returns false (NO), 
processing continues at step 1712. Otherwise, if decision block 1724 
returns true (YES), processing continues at decision block 1710. 
If decision block 1710 returns true (YES), processing continues at step 
1726. In step 1726, the coded data are defined to be the surface 
information and a domain region series indicator list for each of the 
first number of regions. 
Preferably, step 1716 comprises scaling the residual prospective domain 
region such that a predetermined portion of the residual prospective 
domain region is equal to a corresponding portion of the residual data and 
step 1726 includes defining the coded data to include a scale factor for 
each of the domain regions in the domain region indicator list. Further, 
the prospective domain regions may include a subsampling of the data. 
Optionally, the prospective regions may overlap corresponding ones of the 
first number of regions. 
Still further, optionally step 1716 may further comprise normalising the 
residual prospective domain region before determining the candidate 
measure value. 
A flowchart of a method of decoding data 1802, which has been encoded in 
accordance with FIG. 17, is shown in FIG. 18. In step 1804, a current 
estimate is derived of the data comprising an n-dimensional surface 
derived from the surface information. In step 1805, the current estimate 
is divided into the first number of regions. 
In decision block 1808, a check is made to determine if each of the first 
number of regions has been processed. This is done to ensure that, for 
each of the first number of regions, steps 1810 to 1814 are applied. If 
decision block 1808 returns false (NO), processing continues at decision 
block 1810. 
In decision block 1810, a check is made to determine if each of the 
indicators in the domain region series indicator list has been processed. 
This is done to ensure that, for each of the indicators in the domain 
series indicator list, steps 1812 and 1814 are applied. 
If decision block 1810 returns false (NO), processing continues at step 
1812. In step 1812, from the current estimate of the data, a domain region 
corresponding to the indicator is determined. In step 1814, the current 
one of the first number of regions is updated by subtracting the previous 
estimate of the current domain region from the current one of the first 
number of regions and adding the current estimate of the current domain 
region. If this is the first time this domain region has been derived then 
the subtraction step can be omitted. Processing continues at decision 
block 1810. If decision block 1810 returns true (YES), processing 
continues at decision block 1808. If decision block 1808 returns true 
(YES), processing continues at decision block 1816. 
In decision block 1816, a check is made to determine if the data has 
converged. This is done to ensure that steps 1808 to 1814 are repeated 
until the data converges. If decision block 1816 returns false (NO), 
processing continues at decision block 1808. Otherwise, if decision block 
1816 returns true (YES), processing terminates. 
Preferably, step 1812 comprises deriving a residue comprising the 
difference between the current estimate of the data and a surface 
approximation to the current estimate of the data. 
Preferably, step 1812 includes sub-sampling the current estimate of the 
data. 
Preferably, step 1812 comprises normalising the domain region and scaling 
by the corresponding scaling factor stored in the encoded data. 
Preferably, step 1814 comprises subtracting the previous estimate of the 
current domain region from the current one of the first number of regions 
and adding in the domain region derived in step 1812. 
The embodiments of the invention described above rely on matching pixel 
values contained in the image in a well-defined sequence. In such 
processes, the first pixel is matched by the lowest-level approximation to 
the range block. Subsequent pixels are matched by adding an appropriately 
scaled domain block residual in which the residual is formed with respect 
to an interpolation of the set of points in the domain block corresponding 
to the current set of fixed points in the range block. This in effect can 
be understood to be matching the terms in a non-orthogonal basis expansion 
in which the basis functions are the interpolation kernels defined by the 
interpolation process, as described above. 
In a further embodiment of the invention, terms are matched in another 
basis expansion where the terms have finite support. For example, this 
could include a wavelet expansion. As is well known in the art, the 
wavelet basis space is comprised of a space defined by translates of a 
scaling function with compact support which describes the lowest frequency 
part of the data. It is also comprises a space spanned by translates and 
dilates of "mother wavelets". 
A method of encoding data 1902 according to the further embodiment is 
illustrated in FIG. 19. For such a wavelet decomposition of an image, an 
ordering of the basis terms is defined such that each scaling function is 
associated with a range block. The rest of the basis functions are ordered 
so that they are associated with one, and only one, of the range blocks 
(the range block with which they are spatially most naturally associated) 
and are ordered on the basis of their scale. This is done so that the 
coarse scale terms have higher priority than the fine scale terms. 
In step 1904, the image is projected into a subspace defined by the scaling 
function terms. This produces the scaling function coefficients 1906. 
For each range block, a set of domain blocks is defined which can be 
considered in the formation of a fractal expansion. For example, these may 
be blocks of twice the linear size of the range block and that have 
centers within the range block. 
In step 1908, the domain block to best match the next basis term is found, 
and the domain block parameters 1909 are produced. This comprises the 
following steps: 
(i) For a chosen domain block candidate, the domain block is shrunk and it 
is translated to give a block with the same geometry as the range block. 
(ii) The domain block is projected onto the subspace defined by all the 
basis wavelets associated with the range block and which have not 
previously been matched in the representation of the range block. This 
includes initially all associated basis terms except the one translate of 
the scaling function which defines the range block. 
(iii) The resulting projection is normalised. 
(iv) The residual is calculated for the range block. This is the difference 
between the current representation of the range block and the sum of all 
the basis terms in the full image associated with this range block. 
(v) The projection from step (iii) is scaled so that, in its expansion, the 
coefficient of the next unmatched basis term in the sequence matches the 
corresponding coefficient in the residual obtained from step (iv). 
(vi) All candidate domain blocks are tried and the block for which the 
error in the resulting representation is minimised is that which is 
chosen. The error may be the root mean square error or the other chosen 
measure. 
In step 1910, a check is made to determine if the range block residual is 
less than a predetermined threshold value. This is done to ensure that 
step 1908, which comprises steps 1908(i) to 1908(vi), are repeated, adding 
domain block terms in sequence order, until the error (rms or other chosen 
measure) from step 1908(iv) falls below a predetermined threshold. If 
decision block 1910 returns false (NO), processing continues at step 1908. 
Otherwise, if decision block 1910 returns true (YES), processing continues 
at decision block 1912. 
In decision block 1912, a check is made to determine if all range blocks 
are finished. This is done to ensure that step 1908, including steps 
1908(i) to 1908(vi), is repeated for all range blocks. If decision block 
1912 returns false (NO), processing continues at step 1908. Otherwise, if 
decision block 1912 returns true (YES), processing terminates. 
A method of decoding data 2002, which is encoded in accordance with the 
method of FIG. 19, is illustrated in FIG. 20. In step 2002, the low 
frequency representation of the image is calculated from the scaling 
function coefficients. The calculated result from step 2002 is used to 
initialise the current approximation to the image. 
In step 2006, an approximation to the next domain term is calculated and 
the image is updated. This comprises the following steps: 
(i) For the current range block, the approximation to the current domain 
block is obtained from the current approximation to the image. 
(ii) The block is shrunk and shifted to adjust the geometry to match the 
range block. 
(iii) The result of step (ii) is projected onto the subspace spanned by the 
basis terms later in the defined sequence than and including the basis 
term associated with the current domain block. 
(iv) The resulting projection from step (iii) is normalised. 
(v) The result obtained from step (iv) is scaled by the stored scaling 
factor. 
(vi) The scaled result from step (v) is used to update the current 
representation of the range block. 
(vii) The current approximation to the image is updated to reflect the 
changes in this range block. 
In decision block 2008, a check is made to determine if all domain terms 
have been used. This is done to ensure that step 2006, comprising steps 
2006(i) to 2006(vii), is repeated to incorporate all domain blocks for the 
range block. If decision block 2008 returns false (NO), processing 
continues at step 2006. Otherwise, if decision block 2008 returns true 
(YES), processing continues at decision block 2010. 
In decision block 2010, a check is made to determine if all range blocks 
are done. This is done to ensure that steps 2006, comprising steps 2006(i) 
to 2006(vii), and 2008 are repeated for all range blocks. If decision 
block 2010 returns false (NO), processing continues at step 2006. 
Otherwise, if decision block 2010 returns true (YES), processing continues 
at decision block 2012. 
In decision block 2012, a check is made to determine if the image has 
converged. This is done to ensure that steps 2006, comprising steps 
2006(i) to 2006(vii), and 2010 are repeated until the result converges. If 
decision block 2012 returns false (NO), processing continues at step 2006. 
Otherwise, if decision block 2012 returns true (YES), processing ends. 
The foregoing describes a number of embodiments of the present invention 
with modifications. Further modifications, obvious to those skilled in the 
art, can be made thereto without departing from the scope of the 
invention.