Fractal Volume Compression

Wayne O. Cochran John C. Hart Patrick J. Flynn

School of Electrical Engineering and Computer Science

Washington State University

Pullman, WA 99164-2752

fwcochran,hart, ynng@eecs.wsu.edu

October 18, 1995

Abstract

This research is the �rst application of fractal compression to volumetric data. The

various components of the fractal image compression method extend simply and di-

rectly to the volumetric case. However, the additional dimension increases the already

high time complexity of the fractal technique, requiring the application of sophisticated

volumetric block classi�cation and search schemes to operate at a feasible rate. Nu-

merous experiments over the many parameters of fractal volume compression show it

to perform competitively against other volume compression methods, surpassing vector

quantization and approaching the discrete cosine transform.

Keywords: Data compression, fractal, iterated function system, volume visualization.

1 Introduction

The problem of managing extremely large data sets often arises in applications employing

volumetric data. This has prompted research in new techniques for economizing both storage

1

space and processing time. Data compression techniques reduce the size of volumetric data,

converting the array into a representation which is stored more e�ciently.

Fractal techniques based on iterated function systems [11, 2] have been successfully ap-

plied to the compression of one-dimensional signals [3, 31] and two dimensional images [13, 8],

by �nding a fractal representation that models the original signal as closely as possible, and

storing the model instead of the original data. Fractal volume compression uses analogous

techniques for the encoding of three-dimensional volumetric data.

This research in fractal volume compression is part of our recurrent modeling project,

which focuses on the abilities of the recurrent iterated function system [3] as a geometric

representation. Fractal volume compression provides a �rst step toward the use of linear

fractals to model arbitrary 3-D shapes.

Section 2 reviews the fractal image compression technique and summarizes four items of

previous work relating to fractal volume comrpession. Section 3 extends the components of

the fractal image compression method to volumetric data. Section 4 discusses optimization

of the search used by fractal compression methods, allowing fractal volume compression to

perform its search in a feasible amount of time. Section 5 brie y describes decompression al-

gorithms. Section 6 lists the results of various experiments and compares them with previous

results. Section 7 concludes and o�ers directions for further research.

2 Previous Research

This section begins by reviewing the essentials of the fractal image compression method,

mentioning elements that directly contributes to the fractal volume compression method.

Compression researchers have only recently applied their methods to the task of com-

pressing volumetric data. There appear to be only two previous extensions of sophisticated

image compression techniques (vector quantization and the discrete cosine transform) to vol-

umetric data. In addition, there is previous work in fractal compression of 3-D image data,

but only in the form of animations and multi-view imagery. The rest of this section summa-

2

rizes these previous techniques whereas their results appear in Sec 6.7 for better comparison

to fractal volume compression.

2.1 Fractal Image Compression

Fractal image compression partitions an image into contiguous, non-overlapping square range

blocks, and imposes a second, coarser partition on the image, separating it into larger

possibly-overlapping square domain blocks [13]. The collection of domain blocks forms the

domain pool D:

Next, a class of contractive block transformations, the transformation pool T is de�ned.

Each transformation consists of a value1 component, which alters the scalar values of voxels

within the block, and a geometry component, which shu�es (permutes) the positions of

voxels within a block. An image block transformation is contractive if and only if it brings

every two pixels in a block both nearer spatially (by re-sampling it at a lower resolution)

and closer in value (by reducing its contrast).

For each range block R; the encoding process searches for the (D;T ) 2 D � T such that

T (D) best matches R: The quality of this match is determined by a distortion measure.

The L2 distortion measure sums the square of the di�erence between corresponding pixels

between the two w � h blocks X and Y is

L2(X;Y ) =wX

x=0

hXy=0

(X(x; y)� Y (x; y))2 (1)

If the search fails to �nd a satisfactory match, the range block can subdivided and some or

all of its children encoded. Storage of the range block is replaced by storage of the indices of

the domain block and transformation. This technique yielded high-�delity image encoding

at compression rates ranging from 9 : 1 to 12 : 1 [13].

Numerous variations of the basic fractal image compression have appeared [28]. Tech-

niques to reduce the large search space of transformed domain blocks are of particular interest

1This component was called massic in [13], presumably due to fractal image compression's roots in iterated

function systems and measure theory.

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to the volumetric case due to its increased time and space complexity. A block classi�cation

scheme [26] trims the search space by a constant factor [13], segregating searches within

simple edge, mixed edge, midrange or shade block classes. The most promising method

reduces the time complexity of the search from linear to logarithmic by adapting existing

sophisticated nearest-neighbor techniques [27].

2.2 Vector Quantization of Volumetric Data

Vector quantization (VQ) is a popular technique for compressing images as well as many

other forms of data [10]. It attains high compression rates with little coding error by �nding

the representative blocks of pixels, and representing the input in terms of these blocks.

When applied to images, vector quantization partitions the input into contiguous, non-

overlapping blocks. Based on the distributions of the block contents and the desired com-

pression rate, a codebook is constructed containing a small but diverse subset of these rep-

resentative blocks. Each of the original blocks is then coded as an index to its nearest

representative in the codebook. The decoding algorithm simply performs a table lookup for

each stored index.

Vector quantization extends directly to volume data [23]. Encoding the gradients as

well as the scalar values of volume data accelerates the rendering of VQ compressed volume

data, directly from the compressed version [24]. Shading the codebook as a preprocesses

reduced total shading time by a factor of 1000. Rendering the codebook as a preprocess

and compositing the resulting pixmaps reduced orthographic-projection rendering time by a

factor of 12.

High-speed VQ rendering requires a \space-�lling" block arrangement, such that each

block contain enough information to interpolate the values of the cells. For example, their

1283 dataset consists of (128=2)3 = 262; 144 \voxel-spanning" 23-voxel blocks, but consists

of (127=1)3 = 2; 048; 383 space-�lling blocks. Moreover, high-speed VQ requires the storage

of not only the scalar value at each voxel, but also its three normal components and a color.

Hence, their example \compression" for high-speed rendering of a CT dataset using 23-voxel

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blocks �rst expands the data by a factor of �ve with shading information, then compresses

the result by a factor of only �ve due to the space-�lling arrangement of blocks, resulting in

no compression, just fast low-quality rendering.

2.3 DCT Volume Compression

The standard JPEG [25] image compression scheme, based on the discrete cosine transform

(DCT), also extends to volumetric data [32]. The scheme partitions the input volume into

83-voxel blocks and the DCT converts these 512 spatial values into 512 frequency values,

which quantize and entropy-encode better than the original spatial data.

Discrete cosine transform algorithms have been studied exhaustively, and existing opti-

mized algorithms for computing DCT coe�cients have yielded very fast compression times

(e.g. 132.1 seconds for a 12-bit 2563 CT head dataset).

An overlapping (space-�lling) arrangement of the blocks, as in the previous example, al-

lowed DCT-compressed volumes to be rendered directly from the compressed version. Non-

overlapping (voxel-spanning) blocks of 83 voxels were collected into overlapping (space-�lling)

\macro-blocks" of 323 voxels. Moreover, the overlap was increased from one voxel to three to

support gradient computation, but the redundancy was reclaimed through careful compres-

sion of macro-block boundaries. Macro-blocks were accessed and decompressed on demand,

one at a time, during the rendering process. This scheme saved time by avoiding the un-

necessary decompression of blocks that were completely occluded, and saved space, allowing

compressed volumes to be rendered on workstations that lack the necessary memory to store

the entire uncompressed volume.

2.4 Fractal Encoding of Moving Pictures

The fractal encoding of moving picture sequences was demonstrated in [5] as an extension

of one and two dimensional signal encoding. Using the temporal data as a third dimension,

43-voxel range blocks and 123-voxel domain blocks were de�ned over a series of 12 partial

video sequences (termed slabs) for 3-D fractal compression. Treating time simply as a third

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dimension caused annoying artifacts in the decompressed frames. Accurately-reproduced

high-contrast edges tended to visually mask errors that occurred in the lower frequency

background. Once an edge passed from the scene, these previously-masked errors became

more obvious, causing a visual incongruity between edge and non-edge frames. Increasing

the frame rate reduced this problem, but e�ectively reduced the system to a still image

coder.

As a result, this form of volume compression was abandoned and a new system was

developed that compressed the �rst frame using a fractal technique whereas successive frames

simply drew upon 2-D domain blocks from the previous frame. Reasonable quality was

reported at 80Kb=s for 352 � 288 monochrome frames at 10 frames per second.

2.5 Fractal Compression of a Multi-View Image

A fractal compression technique was devised for another form of three-dimensional image

data called a multi-view image, a collection of images obtained from viewing an object

from a discrete range of positions. Given enough images the object can be re-observed in

3-D with smooth transitions between continuously changing viewpoints as motion parallax

places the object in relief. The tremendous size of these 3-D datasets necessitates compression

techniques that exploit the view-to-view coherence between neighboring images.

The encoding technique used in [22] for multi-view images is an application of the algo-

rithm presented in [13] with very few extensions to take advantage of the extra dimension.

Their 3-D dataset contained color values along the x; y; and v (view) axis which was

partitioned into 82 � 5-voxel range blocks. Neighboring range blocks along the view axis

overlapped slightly, such that when decoded, these overlapping regions averaged together to

produce smoother transitions between reconstructed range blocks. Domain blocks consisted

of eight range blocks, and were searched in an outward spiraling path in the xy-plane,

extending from the goal range block, with the expectation that matching domain blocks are

spatially near the range block [13]. Only four isometry transformations were used.

As in [13], range and domain blocks were classi�ed, though in this case based solely on

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the variance of their brightness. Range blocks were also subdivided when necessary. After

entropy encoding, the parameters describing the fractal transformations, a 17-view color

multi-view image of a toy dog was coded at a bit rate of 0.1095 bits per pixel with a PSNR

of 37.52 dB.

Fractal coding is appreciated for its resolution independence. Since fractal transforma-

tions simply describe a contraction from one region of the dataset to another they can be

used at any resolution. Arti�cial data was interpolated from the compressed representation

by decoding at a higher resolution than the original dataset. This feature supported the

approximation of a 51-view 3-D image with a 17-view image, which e�ectively improved the

bit rate from 0.1095 to 0.0365 bits per pixel [22].

3 Fractal Volume Compression

Just as fractal image compression encodes images with block-restricted contractive self-

transforms, fractal volume compression likewise encodes volumes. The design issues for

creating a fractal block coding system for volumes are direct extensions of those used in

fractal image compression. The process partitions the volume into both �ne range blocks

and coarse domain blocks, and then �nds for each range block the transformed domain block

that best matches it.

This section begins with the characterization and notation of volumetric data, followed

by descriptions of the volumetric range, domain and transformation pools.

3.1 Volume Datasets

A volumetric dataset is a collection of voxels which represent some measurable property of

an object sampled on an integer 3-D grid. The algorithm presented here is constrained

to input volumes with scalar voxels. Volumes with vector samples may be divided into

separate datasets containing only scalar voxels and each resulting new dataset is encoded

7

independently2.

The functional notation V (x; y; z) 2 R denotes the voxel located in the input volume at

the grid point (x; y; z) 2 Z3: Each voxel outside the volume's region of support is de�ned

to be zero (i.e., a volume with dimensions W � H �D implies V (x; y; z) � 0 if (x; y; z) 62

[0;W � 1]� [0;H � 1]� [0;D � 1]).

Each voxel is normally quantized to b bits by mapping it to the integer range 0 : : : 2b � 1

thus requiring W �H �D � b bits to store the entire volume directly.

Volumetric data typically arises from the spatial measurement of real-world data, but also

from simulated sources. Medical applications use computed-tomography (CT) or magnetic-

resonance-imaging (MRI) scans. The study of aerodynamics depends on the results of wind

tunnel data and computational uid dynamics simulations. Computer graphics has found

situations in which a volumetric representation performs better than a surface description

[16, 15]. Volume compression makes the management of such massive amounts of volumetric

data feasible for general use in existing facilities.

The distortion metric

L2(X;Y ) =w�1Xx=0

h�1Xy=0

d�1Xz=0

(X(x; y; z)� Y (x; y; z))2 (2)

measures the similarity or \distance" between two w� h� d blocks of voxels X and Y: The

notation L2(X;Y ) will be used to represent this value.

3.2 Volumetric Range Partitioning

As with many compression algorithms, the dataset is partitioned into small spatial regions

and each region is encoded separately. The simple scheme used here dices the volume into

w � h � d non-overlapping cuboid cells (termed range blocks) and encodes each cell (i.e.,

subvolume) individually. A range block is uniquely located within the volume by a corner

grid point (xr; yr; zr) taken from the set

R = f(iw; jh; kd) j (i; j; k) 2 [0; dW=we � 1]� [0; dH=he � 1]� [0; dD=de � 1]g (3)

2In image compression, color images are analogously divided into independently-compressed color planes.

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which de�nes all the range blocks in this partitioning set. The voxels that comprise any

range block can be enumerated by the function

R(x; y; z) = V (xr + x; yr + y; zr + z); (x; y; z) 2 [0; w � 1]� [0; h� 1]� [0; d� 1] (4)

which conveniently allows us to reference voxels within a range block without regard to their

global positions within the volume.

Adaptive partitioning subdivides range blocks that fail to encode satisfactorily. Larger

range blocks yield a higher compression rate, though typically at a lower �delity. The

encoder �rst attempts to �nd maps for large range blocks. Each range block is subdivided

into eight children. If the coding of the range block results in a distortion above a speci�ed

threshold tmse for any child, then that child block is coded separately (and possibly subdivided

itself). If seven or eight children require separate coding, then a short stub code replaces the

range block's code, and all eight children are encoded separately. The overhead associated

with tracking this hierarchical coding requires each (non-stub) parent code contain child

con�guration information.

3.3 Volumetric Domain Pool

A set of nw � nh � nd (n = 2; 3; : : :) subvolumes called domain blocks are now de�ned by

the set of corner lattice points

D =

((i�x; j�y; k�z)

����� (i; j; k) 2�0;�W � nw

�x

���

"0;

$H � nh

�y

%#�

"0;

$D � nd

�z

%#)(5)

which are spread evenly throughout the volume by the integer domain spacing parameters

(�x; �y; �z): A domain block located by the grid point (xd; yd; zd) 2 D uses the function

D(x; y; z) = V (xd + x; yd + y; zd + z); (x; y; z) 2 [0; nw � 1]� [0; nh� 1]� [0; nd � 1]: (6)

to reference local constituent voxels. Note that the lattice support of these domain blocks

may overlap for large domain pools (i.e., �'s are small) or may be widely spread apart (i.e.,

�'s are large).

9

The domain block search is a minimization problem, but often the true minimum is

discarded in practice when other domain blocks yield higher compression with su�cient

�delity. While large domain pools are e�ective for providing accurate maps they can decrease

compression performance by forcing large indices to be transmitted to the decoder. Often a

small localized search can provide su�ciently accurate domain maps for many range blocks.

A common scheme for locating source domain blocks (see [22]) is a outward spiral search

emanating from the target range block. Transmitted indices along this spiral path will

usually provide low bit rates when entropy encoded [4].

Fractal volume compression utilizes a two pass system that �rst checks a small number

(e.g., 64) of spatially close domain blocks (using only the identity transformation). For

those range blocks that encode poorly during this �rst pass a larger (global) domain pool

is interrogated during a second pass. The �rst pass is very rapid (and signi�cantly reduces

the number of searches needed during the slower second pass). The global domain pool is

extremely large and Section 4 addresses its search.

3.4 Volumetric Transformation Pool

A set of transformations T are used to map (source) domain blocks to (target) range blocks.

Each transformation is composed of several components. The �rst component re-samples

domains blocks at a coarser resolution to obtain geometric contractivity. The second removes

the mean value (i.e., the DC component) from each of the domain block's voxels. Another set

of transformations alter the values of voxels within a block by scaling each voxel by a constant

and adding an o�set to the result. Constraining the magnitude of the scaling constant to be

strictly less than one guarantees contractivity, but the introduction of the orthogonalization

operator appears to removes any restrictions on the size of these scaling coe�cients [12]. The

diversity of the domain pool is increased by the last class of transformations which permute

voxels within the block.

The spatial contraction operator Cn decimates a nw � nh � nd domain block so it is

10

re-sampled with the same spatial resolution as a w � h � d range block:

Cn �D(x; y; z) =1

w � h � d

n�1Xi=0

n�1Xj=0

n�1Xk=0

D(nx + i; ny + j; nz + k): (7)

This e�ectively applies an n3 averaging �lter to the domain block followed by a simple

subsampling operation. Here we have forced the dimensions of the domain block to be

integral multiples of the range block dimensions, but the following equivalent decimation

operator C 0n can be used to contract w0 � h0 � d0 domain blocks without this restriction:

C 0n �D(x; y; z) =

1

w0 � h0 � d0

w0�1Xi=0

h0�1Xj=0

d0�1Xk=0

D

�x+ i

w

�;�y + j

h

�;

$z + k

d

%!: (8)

An orthogonalization transformation O is now applied which simply removes the contracted

domain block's DC component so that it is has zero-mean. Given the domain block mean

�d =1

w � h � d

w�1Xx=0

h�1Xy=0

d�1Xz=0

Cn �D(x; y; z); (9)

this operation is described by

O � Cn �D(x; y; z) = Cn �D(x; y; z)� �d: (10)

Next, the a�ne voxel value transformation G�;� scales each voxel in a block B(�) by � and

adds � to the result

G�;� �B(x; y; z) = �B(x; y; z) + �: (11)

When � = 0 this transformation simply de�nes a block with a single uniform value of �: In

place of the eight 2-D block isometries used in [13], we now have 48 3-D block isometries for

cubic partitions (i.e., w = h = d). Exactly half of them are listed in Tables 1 and 2, and

represent the rigid body rotations about thirteen axes (three face axes, six edge axes and

four vertex axes). These are doubled to include all re ections by complementing all three

coordinates in each case.

Non-cubic partitions only consider eight isometries: the identity, the three 180� rotations

about the principle axes, the re ections of the principle axes, and total re ection, shown in

Table 3.

11

x y z Identity

�x �y z Rotation of �

x �y �z about the three

�x y �z face-face axes.

�y x z Rotation of ��=2

y �x z about the three

x z �y face-face axes.

x �z y

�z y x

z y �x

Table 1: Ten rigid body cubic rotations, shown in terms of their result on the input vec-

tor (x; y; z):

The transformation pool T is the set of all possible transformations of the form

T = Ik �G�;� �O � Cn (12)

which can be parameterized by an isometry index 0 � k < 8; a \contrast scale" � and a

\luminance shift" �: For e�ective quantization the value of � is chosen from a �nite set

f�0; : : : ; �n�1g (determined a priori) and � is mapped to the nearest integer. We will use

the notation T (D) to denote the net e�ect of the transformation T 2 T on the domain block

D 2 D:

4 Searching

For each range block R 2 R a transformation T 2 T and domain block D 2 D are sought

after that yield a su�ciently small distortion measure L2(R;T (D)): This search dominates

the fractal compression process, and its extension to volumetric data causes this search time

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y x �z Rotation of �

z �y x about the six

�x z y edge-edge axes.

�y �x �z

�z �y �x

�x �z �y

z x y Rotation of �(2�=3)

�y z �x about the four

�z x �y vertex-vertex axes.

y �z �x

y z x

z �x �y

�y �z x

�z �x y

Table 2: Fourteen more rigid body cubic rotations shown in terms of their result on the

input vector (x; y; z):

13

x y z Identity

�x �y z Rotations

�x y �z

x �y �z

�x y z Re ections

x �y z

x y �z

�x �y �z

Table 3: Eight non-cubic transformations.

to greatly lengthen.

Designing a fast encoder hinges on the e�ciency of the search for matching transformed

domain blocks. This problem is approached by introducing heuristics to guide the encoder's

search towards regions of the dataset that share common characteristics and are therefore

more likely to provide self-similar mappings. A classi�cation scheme [26] provides such

guiding [13]. This scheme used an edge-oriented classi�cation system designed to alleviate

both the time complexity and edge degradation problems inherent to block image coders.

This system does not easily extend to the realm of 3-D block classi�cation as a consequence of

the complexity added by the extra dimension. One previous volumetric classi�cation scheme

[22] thresholded the sample variance of voxels within a block. This has the undesirable

e�ect of avoiding rapidly convergent transformations that map high contrast regions to low

contrast range blocks.

Even with classi�cation, this search for self-a�ne maps still remains sequential. Instead

of segregating blocks into a set of prede�ned classes, associating a real-valued key for each

block replace the linear global block search with a logarithmic multi-dimensional nearest-

neighbor search [27]. This solution is readily generalized to 3-D block encoding and allows

14

for much larger search spaces which improve coding �delity but also require larger domain

indices.

4.1 Brute Force Search

A brute force algorithm is e�ective for the �rst pass search of the local domain pool but is

too time consuming for the global domain pool.

For each range block R 2 R; the compression algorithm searches the domain and trans-

formation pools for the domain block D 2 D and transformation T 2 T that minimizes the

distortion L2(T (D); R) where T (D) is the result of applying the transform T to the domain

block D: The resulting set of \self-tiling" maps produce the \fractal" code that replaces the

input volume dataset. A complete search of the virtual codebook D � T involves examining

all possible parameterizations Tk;�i;� and their e�ect on every domain block in the domain

pool. The algorithm in Table 4 outlines this \brute force" solution. Every range block in

the partitioning set R is examined for which the entire domain pool D and transformation

pool T is scanned for the optimal transformation.

One can prune the exhaustive search of D � T by solving for the optimal gray-value

transformation coe�cients. If we represent range blocks and contracted domain blocks with

the m� 1 (m = w � h � d) column vectors [r0 : : : rm�1]T and [d0 : : : dm�1]T ; respectively, then

the optimal choice for � and � provides the least squares �t to an over-determined system

of the form Ax = b: 2666666664

d0 1

d1 1...

...

dm�1 1

3777777775

264 �

�

375 =

2666666664

r0

r1...

rm�1

3777777775: (13)

If we consider uniform domain blocks (i.e., d0 = d1 = � � � = dm�1) non-admissible then

the columns of A are linearly independent, the matrix ATA is invertible and the unique

least-squares solution �x = [�� ��]T is

�x = (ATA)�1ATb: (14)

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for R 2 R do begin

dist 1;

for D 2 D do

for T 2 T do begin

D0 T (D);

if L2(R;D0) < dist then begin

code fT;Dg;

dist L2(R;D0);

end

end;

replace R with quantized code;

end;

Table 4: Exhaustive search over D � T for each range block R 2 R:

16

Let �d = E(di) =1

m

Pm�1i=0 di be the �rst moment or mean of di (similarly for �r = E(ri)),

and let �2d = E(d2i ) � (E(di))2 be the second central moment or variance of di; and let

�rd = E(ridi)�E(ri)E(di) be the second central moment or covariance of ri and di; then

�x =

264 ��

��

375 =

264 �rd=�

2

d

�r � ���d

375 : (15)

The orthogonalization transformation O yields �d = 0: The o�set value simply becomes

the DC component �r of the range block. The value �i closest to �� from the set fa0; : : : an�1g

is selected as our contrast scaling coe�cient. � is simply mapped to the nearest integer.

It is well known from transform coding that much of the energy in an image represented in

the frequency domain resides in its DC term. Therefore, it is critical that the decoder can

faithfully reproduce � from whatever quantization scheme is used.

4.2 Volumetric Block Classi�cation

Block classi�cation was used in [13] not only for the encoding speedup that was obtained

by segregating the search, but for also determining the complexity of each block. This

allowed for more expressive transformations to be used for highly detailed blocks and simpler

transformations for less interesting blocks. Devoting a higher bit rate to more complex areas

of the image is worthwhile for maximum performance. Extension of the block classi�cation

techniques borrowed from classi�ed VQ [26] into 3-D is non-trivial and would no doubt be

computationally expensive. The �nal volume coder used in our experiments used a simple

variance threshold value to separate blocks into one of two classes: boring and active. Less

bits were devoted to contrast scaling coe�cients and no isometries (other than the identity)

were attempted for blocks belonging to the boring class.

In [6] principal component analysis (PCA) classi�ed blocks based on their gradients. PCA

is a well known technique for �nding a set of ordered orthogonal basis vectors that express the

directions of progressively smaller variance in a given dataset [21, 14]. The dimensionality of

highly correlated datasets can be reduced in this manner by expressing their values in terms

of these new basis vectors. This is often used for �nding normals to planes that best �t a set

17

of scattered data points fxig: The technique of weighted PCA assigns a relative importance

or weight wi to each point xi in the set. A set of principal vectors were extracted from

subvolume blocks in this manner by using the block's supporting lattice for each xi and the

corresponding voxel values for the weight wi: While this reduced encoding time and bit rate

for the volume coder, some complex blocks were misclassi�ed as not containing signi�cant

gradients thus leaving the possibility of critical blocks producing poor collage maps. Three-

dimensional block classi�cation remains a fertile area of study that could prove bene�cial to

many areas in volume visualization.

4.3 Nearest Neighbor Search

Even with classi�cation, we are still faced with a demanding search through N admissible

domain blocks (under the set of allowed isometries), computing least squares approximation

along the way, to �nd the optimal encoding for each range block. This daunting task can be

reduced to a multi-dimensional nearest neighbor search which can be performed in expected

logarithmic time [27].

Using the notation of Section 4.1, the transformed domain block A�x is the projection

of r onto the column space of A which is spanned by the orthonormal basis vectors e =

1pm[1 : : :1]T and �(d) where

�(x) =x� hx; eie

kx� hx; eiek: (16)

Thus the projection A�x is equivalent to

ProjA(r) = hr; eie+ hr; �(d)i�(d) (17)

for a given domain block d: Note that �(x) simply removes the DC term of x which is

accomplished by the operator in Equation 10. Since e and �(x) are orthogonal the coe�cients

� and � of the gray level transformation G from Equation 11 are not correlated.

In [27], the search for the domain block d that yieldsminimal coding error L2(r; P rojA(r))

is shown to be equivalent to the search for the nearest neighbor of �(r) in the set of 2N vectors

f��(di)gNi=1: The problem is now reduced to �nding nearest neighbors in Euclidean space for

18

which there are well known algorithms [9, 30]. Given a set of m-dimensional points [9] shows

how an optimal space dividing kd-tree can be constructed with O(mN logN) preprocessing

steps so that the search for the nearest neighbors of a given query point can be found in

expected O(logN) time. Since this technique su�ers when the dimension m becomes large

we down-�lter all blocks to 23(m = 8) vectors before processing. All domain blocks that are

close to uniform (e.g., �2d � 8) are discarded while the remaining domain block addresses are

stored in the kd-tree twice, once for each search key ��(d):

Since we are down-sampling r and d to compute �(r) and �(d) the kd-tree search will

not guarantee �nding the actual nearest neighbors. Fortunately, searching the kd-tree for

several (e.g., 10 or 20) neighbors can still be done in logarithmic time for a given query

point. For each neighbor the least squares method for �nding � and � are performed at the

range block's full resolution. Also, in order to reduce the memory requirements, we perform

the search for isometries of the range block fI�1k (r)g7k=0 instead of explicitly storing all of

the keys f�(Ik(d))g7k=0 for each domain block.

A substantial speedup is possible if we relax the constraint of �nding the true nearest

neighbors to simply �nding good matches. When searching for the n nearest neighbors, the

algorithm described in [9] keeps track of the dissimilarity measure r (i.e., radius) of the nth

best match found so far. It is this value r that informs us whether we need to traverse

another branch of the space partitioning tree. If we scale r by some number less than 1

(e.g., 0.2) when making this decision we can avoid inspecting large portions of the tree. This

modi�cation was added in our encoder without signi�cantly a�ecting the quality of the maps

found.

5 Fractal Volume Decompression

Complete decompression begins with an initial volume, typically initialized with zeroes. The

next volume in the sequence is partitioned into range blocks, and the previous volume into

domain blocks. Then the appropriate domain block in the previous volume is mapped to

19

each range block in the next volume. At each step of the iteration, the domain pool is

re�ned. As this process continues, it converges to the decompressed version.

In implementation, the process actually needs only one volume, partitioned into both

range and domain blocks, which is decompressed onto itself at each step of the iteration.

Each range block is overwritten with the transformed contents of the appropriate domain

block. If this overwrites a section of the volume later accessed as domain block, then some

or all of the domain block would be as it was in a later iteration. Some care is necessary for

hierarchical (e.g., octree) representations to make sure that the larger more lossy \parent"

range blocks do not overwrite previously decoded \child" blocks from a previous iteration.

Faster techniques exist for decompression fractal-coded images [20], and could be easily

extended to the volumetric case. However, these techniques are typically designed for anima-

tion playback, and volume visualization systems are not yet fast enough to render dynamic

volume datasets in real time.

The fractal volume compression algorithm can be adjusted to permit direct rendering of

compressed data, avoiding a complete decompression step. Separating a volumetric dataset

into a set of possibly overlapping \macro-blocks" supports on-demand decompression during

rendering [32]. Such macro-blocks may be incorporated into fractal volume compression by

selecting domain blocks within the macro-block containing the range block. In this fashion,

each macro-block of the compressed volume may be decompressed independently.

6 Results

The fractal volume compression algorithm was tested on a variety of popular, publicly-

available datasets, and its performance measured using a variety of metrics, to promote

better comparison with other existing and future volume compression methods.

20

6.1 Measurement

Several quantitative methods exist for indicating the �delity of a compression algorithm. In

the following, V is the original volumewith dimensionsW�H�D; and ~V is the decompressed

volume. The function "(x) represents the error between V (x) and ~V (x) :

"(x) = V (x)� ~V (x): (18)

The mean-square error (mse) is one numerical measure for determining the accuracy of

a compressed volume, and is de�ned

mse =

Px "

2(x)

W �H �D: (19)

Another way to express the di�erence or \noise" between V and ~V is the signal-to-noise

ratio (SNR). Several versions of SNR exist, which can cause confusion when comparing

results. The signal-to-noise ratio used in [24], denoted SNRf ; measures the ratio of the

signal variance to the error variance:

SNRf = 10 log10

var ~V (x)

var "(x)= 10 log

10

E( ~V 2(x))� E2( ~V (x))

E("2(x))� E2("(x)): (20)

If "(x) and V (x) each have a mean of zero then SNRf is equivalent to the mean squared

signal-to-noise ratio

SNRms = 10 log10

Px~V (x)2

L2(V; ~V ): (21)

Even for datasets that are not zero-mean (as is the case here) this is often used to measure

the quality of the reconstructed signal.

The peak-to-peak signal-to-noise ratio, PSNR, is de�ned

PSNR = 10 log10P 2

var "(x); (22)

where P 2 is the square of the dynamic range or peak signal value, the distance between the

largest and smallest voxel values, and the denominator is the variance of "(x) which is often

simply approximated by the mse.

21

6.2 CT Medical Data

A 2562 � 113 12-bit CT head dataset (containing voxel values ranging from �1117 to 2248)

was use to test the fractal volume compression algorithm over several di�erent range block

sizes w�h� d and octree subdivision down to an edge size no less than two voxels. An mse

distortion threshold value of tmse (the maximum allowed average mse per voxel) determined

which child blocks were recursively encoded. The domain pool blocks were spaced �x = 4;

�y = 4; and �z = 2 apart, yielding approximately 230; 000 domain block records for each

octree partition level. Encoding times were measured on a dual-processor HP-9000/J200

with 256 MB of RAM, and do not include �le I/O time.

w � h� d tmse comp. SNRms PSNR time

23 4; 608 10:99 : 1 27.62 42.57 134 s

43 1; 024 14:17 : 1 30.15 45.09 973 s

43 2; 048 18:41 : 1 28.74 43.68 711 s

43 4; 608 26:18 : 1 26.68 41.63 519 s

83 1; 024 15:93 : 1 29.70 44.64 1,607 s

83 2; 048 22:00 : 1 27.97 42.91 1,285 s

83 4; 608 34:37 : 1 25.50 40.43 1,081 s

32 � 2 4; 608 20:83 : 1 25.14 40.09 249 s

62 � 4 4; 608 39:21 : 1 21.08 38.06 616 s

Table 5: Encoding results for a 12-bit 2562 � 113 CT head using an mse octree partitioning

threshold of 4608.

To demonstrate the fast convergence of the decoder, several reconstructed 12-bit CT

heads were created using one to six iterations of the stored transformations. The �delity of

these reconstructed volumes (using the maps corresponding to the seventh row of Table 5)

are given in Table 6. One advantage to introducing the orthogonalization operator O (see

22

Equation 10) is a quick and guaranteed convergence to the attractor in a �xed number of

iterations [12]. In this example, only �ve iterations of the simple method outlined in Section 5

were required to decode the data. Each iteration took approximately 15 seconds.

iterations SNRms PSNR

1 13.24 28.08

2 21.09 36.03

3 24.99 39.34

4 25.47 40.40

5 25.50 40.43

6 25.50 40.43

Table 6: SNR for decoded 12 bit CT head using 1 to 6 iterations of the stored transformations.

This same dataset was reduced to 8 bit voxels by mapping each original voxel v to the

range f0 : : : 255g by round(255 � (v + 1117)=3365): The 8-bit volume was compressed using

various mse child threshold values tmse to determine when octree subdivision would occur.

Whereas images are generally compressed in their rendered state, volume data requires

further processing, such as classi�cation, shading and rendering [18, 7], before being displayed

as an image. Hence errors introduced through compression yield di�erent artifacts in the

volumetric case than in the image case. An isovalue surface rendering, such as Figure 2,

puts volume compression techniques to an extreme test. Moreover, faces are typically used

to analyze compression �delity since humans have developed visual skills speci�cally for

interpreting subtle changes in facial expression.

6.3 MRI Medical Data

Fractal volume compression was also tested on an 8-bit 2562 � 109 MRI dataset of a human

head. The noise in MRI datasets challenges compression algorithms more than CT datasets.

23

w � h� d tmse comp. SNRf PSNR time

23 18 11.02:1 25.90 41.49 166 s

32 � 2 18 13.92:1 24.53 40.10 241 s

42 � 2 9 24.61:1 22.04 37.63 448 s

42 � 2 18 29.07:1 21.52 37.12 298 s

43 18 18.02:1 25.30 40.87 607 s

62 � 4 9 20.78:1 22.81 38.39 830 s

62 � 4 18 27.51:1 21.74 37.32 640 s

82 � 4 18 55.35:1 14.41 30.13 592 s

83 18 22.27:1 24.18 39.75 1,003 s

122 � 8 9 24.08:1 20.69 36.29 1,362 s

Table 7: Encoding results for an 8-bit 2562�113 CT head using an using varying mse octree

partitioning thresholds

24

Figure 1: Slice #56 of the CT head volume dataset: Original (upper left), 18:1 43-voxel

range blocks (upper right), 22:1 6 � 42-voxel range blocks (lower left), 22:1 83-voxel range

blocks (lower right).

25

Figure 2: Isovalue surface rendering of the skin (isovalue = 50) of the CT head volume

dataset: Original (upper left), 18:1 43-voxel range blocks (upper right), 22:1 6 � 42-voxel

range blocks (lower left), 22:1 83-voxel range blocks (lower right).

26

For example, a simple run-length encoding of the CT head dataset reduces it to 68% of its

original size whereas the RLE method failed to compress the MRI head dataset.

tmse comp. mse SNRf PSNR

36 20.23:1 11.116 18.332 37.672

64 29.82:1 17.939 16.281 35.692

100 42.72:1 25.813 14.520 34.014

144 60.80:1 36.058 12.977 32.563

169 85.30:1 48.355 11.611 31.289

256 118.23:1 61.501 10.480 30.246

324 160.06:1 75.232 9.527 29.372

400 211.97:1 89.530 8.696 28.617

900 728.85:1 152.33 5.949 26.305

Table 8: Coding resulting for an 8-bit 2562 � 109 MRI of a human head with varying octree

partitioning threshold values tmse: Range block size varied dynamically from 163 to 23: A

domain spacing of �x;y;z = 4 was used along each axis.

Figure 3 demonstrates the �delity of the fractal volume compression on a slice of the MRI

dataset. Even at the extremely high 729:1 rate, the skin and bone edges are reproduced, but

the textured detail is obscured.

6.4 CT Engine Data

Table 12 contains results from an 8-bit 2562�110 CT scan of an engine block. The engine

contains many sharp edges, which fractal volume compression performs particularly well on.

A side-by-side comparison (Figure 4 shows the compressed version to be indistinguishable

from the original.

27

Figure 3: Slice #54 of the MRI head volume dataset: Original (upper left), 20:1 (upper

center), 25:1 (upper right), 30:1 (lower left), 43:1 (lower center), 729:1 (lower right).

28

block size comp. SNRf (dB) PSNR (dB)

23 8.59:1 21.31 40.60

22 � 3 11.09:1 20.24 39.53

32 � 2 15.35:1 18.99 38.30

43 12.14:1 20.98 40.26

42 � 6 16.29:1 19.08 38.39

83 13.59:1 20.72 39.99

163 13.91:1 20.47 39.76

323 13.96:1 20.00 39.30

Table 9: Comparison of compression rate and �delity versus range block sizes and hierarchy

depth of the MRI head dataset, using an mse octree partitioning threshold tmse = 18 and

domain spacing �x;y;z = 4 along each axis.

partition total local search admissible global

range size range codes codes found domain blocks

163 1,792 1,109 47,703

83 4,302 144 42,992

43 26,724 1,037 36,445

23 104,521 75,606 29,218

Total codes transmitted = 137,339.

Table 10: Coding statistics for an 8-bit 2562 � 109 MRI of a human head at various levels

in the octree using a partitioning threshold value tmse = 36 and domain spacing �x;y;z = 4

along each axis.

29

partition total local search admissible global

range size range codes codes found domain blocks

163 1,792 1,443 47,703

83 744 96 42,992

43 879 356 36,445

23 274 273 29,218

Total codes transmitted = 3,689.

Table 11: Coding statistics for an 8-bit 2562 � 109 MRI of a human head at various levels

in the octree using a partitioning threshold value tmse = 900 and domain spacing �x;y;z = 4

along each axis.

direct storage 7,208,960 bytes

RLE compressed 5,413,292 bytes

FVC compressed 216,460 bytes

compression time 552 s

compression rate 33.3:1

compression rate (over RLE) 25.0:1

mse 4.17

PSNR 41.93

SNRms 29.10

SNRf 28.13

Table 12: Results from compressing an 8-bit 2562�110 CT scan of an engine block. 83-voxel

range blocks (octree partitioned down to 23 voxels) were used with a partitioning treshold

of tmse = 18: The domain spacing �x;y;z = 4 was used along all three axes.

30

Figure 4: Original (left) and compressed (right) rendering an engine block MRI.

6.5 Slicing: Comparison with Fractal Image Compression

Volumetric encoding performs better than slice-by-slice encoding and also o�ers other ben-

e�ts such as direct rendering from the compressed version.

A fractal image coder was used to compress each cross section of some of the datasets.

Independent compression of the individual slices of the 8-bit CT head resulted in a volume

compressed by 18.06:1 with a PSNR of 39.21 dB, but Table 7 shows that similar �delity

(39.75 dB) resulted from fractal volume compression at a better rate of 22.27:1. Slice-by-

slice compression of the 8-bit MRI resulted in a 30.74:1 rate at a PSNR of 33.717 dB whereas

Table 9 shows the volumetric version compressed at a better �delity (34.014 dB) at a higher

ratio of 42.72:1. As the compression rate increases, the bene�ts of volumetric compression

of volumetric data appears to grow.

31

6.6 Dicing: Integration of Macro-Blocks

Fractal volume compression supports the rendering of a volume directly from the compressed

version by �rst dicing the volume into macro-blocks [32] (groups of blocks), and then com-

pressing each macro-block independently. Each macro-block may then be decompressed inde-

pendently. Hence large datasets may be rendered on workstations lacking su�cient memory

to contain the entire data, and only visible sections of the volume need be decompressed.

The 8-bit CT head was also diced into 323-voxel macro-blocks, which were compressed

individually, to measure any loss of �delity due to the limited domain pools. Since the

macro-blocks were signi�cantly smaller than the entire volume, a tighter domain spacing

was allowed, and resulted in an impressive compressed volume �delity PSNR of 41.28 dB

(SNRms of 27.50 dB). The compression rate for such a volume remains under investigation,

but will surely beat the 11 : 1 ratio in Table 7.

6.7 Comparison with Previous Results

The CT head dataset results compare with other volume compression algorithms based on

vector quantization and the discrete cosine transformation. The VQ algorithm was tested on

an 8-bit 1283 version of the dataset [24] whereas the DCT algorithm was tested on a 12-bit

2563 version [32]. As the resolution of the original dataset is 2562 � 113; both results are

guilty of some form of interpolation or padding, and such added redundancy can in ate the

compression ratio by as much as a factor of two.

Using voxel-spanning blocks, the VQ algorithm was capable of a compressing the 1283

volume by 40:1 at a SNRf of 17.6 dB [24]. At about 40:1, fractal volume compression yields

a measurably-better volume of 21.08 dB.

The performance of this fractal volume compression implementation falls short of the

mark set by the DCT volume compression algorithm [32]. Some have produced fractal

image compression algorithms that outperform the DCT [4], and one would expect similar

performance from fractal volume compression once fractal techniques receive the same level

of attention and optimization as the DCT currently has.

32

7 Conclusion

Fractal image compression extends simply and directly to three dimensions. As in the

image case, the technique still matches domain blocks to range blocks, but the additional

dimension produces many more isometries yielding a richer transformed-domain pool, higher

compression rate and better �delity.

The increased search time caused by the additional dimension is overcome by several

sophisticated classi�cation schemes. In particular, the nearest-neighbor method [27], which

is the only one to reduce the time complexity of the search, performed best for fractal volume

compression.

The performance of fractal volume compression beats VQ, and rivals DCT volume com-

pression. There is evidence that fractal image compression can beat DCT image compression

[4] and there are many common cases where fractal compression is the preferred technique.

The fractal volume compression system used for these results is available at:

ftp.eecs.wsu.edu:/pub/hart/fvc.tar.gz

7.1 Applications

Contrary to its name, fractal image compression performs better on sharp edges and worse

in textured areas. Hence, fractal volume compression performs better on clearly delineated

regions, such as bone and skin, but worse on tissue or �nely detailed areas. The DCT volume

compression method blurs both edges and �ne detail [32]. Hence, fractal volume compression

appears well suited for the compression of classi�ed, �ltered datasets, and also of synthesized

data sets.

Fractal compression techniques tend to perform better at high compression rates com-

pared to other methods. For medical volume compression applications, any data loss due

to compression is likely unacceptable. Hence, volume compression in general may �nd its

most suitable application in medicine for the high-rate low-�delity indexing, previewing and

presentation of volume datasets, for which fractal compression techniques are the best choice.

33

7.2 Future Research

Fast rendering algorithms appear available based on the fractal representation. As [23] pre-

ceded [24], further research on fractal volume compression will likely produce a fast rendering

algorithm.

Fractal compression research is much newer than other compression techniques such as

the DCT or VQ, and the technique is still not well understood nor fully developed. As new

enhancements appear for fractal image compression, they will likely improve fractal volume

compression as well.

The Bath fractal transform [19] operates without a domain search, and could be easily

extended the 3-D and applied to volumetric data. This technique compensates for the loss

of block diversity by augmenting the a�ne block transformation with linear, quadratic and

even cubic functions.

Compression of volumes sampled on an irregular grid would require a more sophisticated

partitioning and block-transformation scheme.

7.3 Acknowledgments

Several volume visualization systems were used during the development of fractal volume

compression [1, 17, 29]. Data for the CT head was obtained from the University of North

Carolina-Chapel Hill volume rendering test data set.

This research is part of the recurrent modeling project, which is supported in part by a

gift from Intel Corp. The research was performed using the facilities of the Imaging Research

Laboratory, which is supported in part under grants #CDA-9121675 and #CDA-9422044.

The second author is supported in part by the NSF Research Initiation Award #CCR-

9309210. The third author is supported in part by the NSF under grants #IRI-9209212 and

#IRI-9506414.

34

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