Volume 3, Number 3, Pages 492{509 · Key Words. Hexagonal structure, image scaling, Spiral Architecture, fractal image compression, image partitioning. 1. Introduction Computer Vision

$Page 1: Volume 3, Number 3, Pages 492{509 · Key Words. Hexagonal structure, image scaling, Spiral Architecture, fractal image compression, image partitioning. 1. Introduction Computer Vision$
INTERNATIONAL JOURNAL OF c© 2007 Institute for ScientificINFORMATION AND SYSTEMS SCIENCES Computing and InformationVolume 3, Number 3, Pages 492–509

UNIFORM IMAGE PARTITIONING FOR FRACTALCOMPRESSION ON VIRTUAL HEXAGONAL STRUCTURE

XIANGJIAN HE, HUAQING WANG, WENJING JIA, QIANG WU, NAMHO HUR,JINWOONG KIM, AND TOM HINTZ

Abstract. Hexagonal structure is different from the traditional square struc-

ture for image representation. The geometrical arrangement of pixels on hexag-

onal structure can be described in terms of a hexagonal grid. Uniformly sepa-

rating image into seven similar copies with a smaller scale has commonly been

used for parallel and accurate image processing including image compression on

hexagonal structure. However, all the existing hardware for capturing image

and for displaying image are produced based on square architecture. It has

become a serious problem affecting the advanced research based on hexagonal

structure. Furthermore, the current techniques used for uniform separation of

images on hexagonal structure do not coincide with the rectangular shape of

images. This has been an obstacle in the use of hexagonal structure for image

processing. In this paper, we briefly review a newly developed virtual hexagonal

structure that is scalable. Based on this virtual structure, algorithms for uni-

form image separation are presented. The virtual hexagonal structure retains

image resolution during the process of image separation, and does not intro-

duce distortion. Furthermore, images can be smoothly and easily transferred

between the traditional square structure and the hexagonal structure while the

image shape is kept in rectangle. As an application of image partitioning, we

present a Fractal Image Compression (FIC) method on the virtual image struc-

ture by adopting Fisher’s basic FIC method on the traditional square image

structure. The modification on the definition of range block and domain block

is implemented in order to utilize the enhanced image structure. The results

of the FIC approach applied to testing images are analyzed and show higher

fidelity.

Key Words. Hexagonal structure, image scaling, Spiral Architecture, fractal

image compression, image partitioning.

1. Introduction

Computer Vision and Image Processing is a computationally expensive field inwhich many operations require massive computations, especially when large datasets are involved such as those for stereo image matching and feature extraction.Parallel processing using a cluster consisting of multiple computers is a straightfor-ward method to speed up image processing. Cluster-based parallel computing hasthe advantages of low cost and high utility. On the other hand, it has the disadvan-tages of high communication latency and irregular load patterns on the computingnodes [1]. Its performance mainly depends on the amount of and the structure forcommunications between processing nodes. Therefore, as an application for parallel

Received by the editors October 1, 2006 and, in revised form, November 28, 2006.

492

FRACTAL COMPRESSION ON VIRTUAL HEXAGONAL STRUCTURE 493

image processing, image separation or partitioning for task dividing is critical in aparallel algorithm for image processing.

Many types of image partitioning have been proposed [2] on traditional squareimage structure such as Row Partition, Column Partition and Block Partition. Allof these partition methods do not intend to separate an image into similar copies(or sub-images). Hence, when the sub-images are assigned to various computernodes for parallel processing, the load-balancing is not guaranteed.

The method for image partitioning presented in [3] is based on a hexagonal imagestructure, called Spiral Architecture (SA) [4], which is inspired from anatomicalconsiderations of the primate’s vision. It partitions the input image uniformlyinto a number of sub-images. Each sub-image is a near (or similar) copy of theinput image with a smaller scale, while all of the sub-images are mutually exclusiveand the original image information is all kept in the sub-images. Every sub-imagecan be processed independently and in parallel by an individual node without dataexchange between the nodes. Furthermore, the workload on every node is almost thesame as that on any other node because of the uniform partition. Consequently, thecomputational complexity is greatly reduced and the processing time is significantlyshortened. The work shown in [3] has the following shortcomings. First of all, theshape of input images is hexagon-like and does not coincide with rectangular shapeof traditional images. Secondly, the image partition method is computationallyexpensive. Thirdly, images are represented on a mimic Spiral Architecture thatintroduces a loss on image resolution and creates image distortion when the sub-images have a rotated angle from the original image.

In order to take the advantages and suppress the disadvantages of SA, in thispaper, we present a novel method for uniformly separating image into four similarsub-images based on a newly defined virtual hexagonal image structure [5]. Thehexagonal structure can be smoothly converted from the square structure withoutchanging the image shape. In this virtual structure, hexagonal pixels can be easilylocated through simple computations. It avoids the necessity of using a computationexpensive method to compute pixel locations and hence avoids the need to build alarge table to record the location information. This virtual structure hardly changesthe image resolution and almost does not introduce image distortion.

As another application of uniform image partitioning, a Fractal Image Compres-sion (FIC) method is presented in this paper. FIC is an area of intensive researchin the previous decades [6][7]. During more than two decades of the development inFIC, a lot of researches have been done to improve the performance. The existentenhanced approaches can be classified into the following categories [8]:

• Partition methods imposed on the image to obtain range blocks;• Methods for selection of the pool of domain blocks;• Selection of transforms on the domain blocks;• Searching methods for locating suitable domain blocks;• Representation and quantization of the transform parameters.

The research work on FIC to be presented in this paper, however, is based on adifferent image structure, namely hexagonal structure. A hexagonal structure calledthe Spiral Architecture (SA) [4] was proposed by Sheridan in 1996, on which eachhexagonal pixel is identified by a designated positive seven-based integer. Thesenumbered hexagons tile the plane in a recursive modular manner along the spiraldirection (see Figure 1). Based on the spiral addressing system, there are twooperations defined on SA, i.e. spiral addition and spiral multiplication. These two

494 X. HE, H. WANG, W. JIA, Q. WU, N. HUR, J. KIM, AND T. HINTZ

operations were used in our previous work [9] for the definition of range blocks anddomain blocks when adopting FIC on SA.

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Figure 1. Spiral Architecture with spiral addressing.

The FIC method based on a hexagonal structure in this paper inherits the FICmethods on square image structure introduced by Fisher [6] and also inherits theideas proposed on SA for uniform image separation. All range blocks have thesame size. Although the performance of this algorithm in terms of compressionratio and fidelity can be further improved, the performance analysis well evaluatesthe capabilities of the proposed FIC on hexagonal structure in comparison to FICon square image structure. For a better comparison, the proposed method followsthe same contractive transformation applied to domain blocks and nearest neighborsearch used in locating suitable domain blocks on square structure. The definitionof range block and domain block, however, is slightly adjusted to take the geo-metrical advantages of hexagonal structure. It can be easily found as shown in theexperimental results that this FIC on hexagonal structure offers better fidelity withdifferent compression ratios for all test images.

The rest of this paper is organized as follows. In Section 2, we briefly review theSpiral Architecture. In Section 3, we introduce the construction of a new virtualhexagonal structure. A new technique for uniform image partition is presented inSection 4. Experimental results for uniform image partitioning are demonstratedin Section 5. The approach for FIC on the virtual structure following a uniformpartitioning algorithm is presented in Section 6. Section 7 demonstrates the exper-imental results using the FIC method. We conclude in Section 8.

2. Spiral Architecture

On Spiral Architecture, an image is represented as a collection of hexagonalpixels. Each pixel has only six neighbouring pixels with the same distance to it.Each pixel is identified by a number of base 7 called a spiral address. The numbered(or addressed) hexagons form the cluster of size 7n, where n is a positive integer.These hexagons starting from address 0 towards address 7n tile the plane in a


recursive modular manner along a spiral-like curve. As an example, a cluster withsize of 72 and the corresponding spiral addresses are shown in Figure 1. The imagespace formed on the Spiral Architecture always has a hexagon-like shape. Thisshortcoming restricts the applications of image processing on hexagonal structures.Our approach in this paper will overcome this disadvantage.

Two algebraic, useful and important operations have been defined on SpiralArchitecture based on spiral addresses. They are Spiral Addition and Spiral Mul-tiplication. These two operations correspond to two transformations on SpiralArchitecture, which are translation and rotation with a scaling. Spiral Multiplica-tion is also often applied to uniformly separate images on Spiral Architecture forparallel processing. In this paper, we perform an algorithm for image partition on ahexagonal structure without using computationally expensive Spiral Multiplication.Our algorithm will maintain the important property that none of image (intensity)information is lost during the separation process.

3. Virtual Hexagonal Structure (VHS)

In this section, we review the construction of a new virtual hexagonal struc-ture [5].

To construct hexagonal pixels, each square pixel is first separated into 7 × 7smaller pixels, called sub-pixels. To be simple, the light intensity for each of thesesub-pixels is set to be the same as that of the pixel from which the sub-pixels areseparated. Each virtual hexagonal pixel is formed by 56 sub-pixels arranged asshown in Figure 2. To be simple, the light intensity of each constructed hexagonalpixel is computed as the average of the intensities of the 56 sub-pixels forming thehexagonal pixel when necessary.

Figure 2. The structure of a single hexagonal pixel in the virtualhexagonal structure.

Figure 3 shows a collection of seven hexagonal pixels constructed with spiraladdresses from 0 to 6. From Figure 3, it is easy to see that the hexagonal pixelsconstructed in this way tile the whole plane without and spaces and overlaps.

From Figure 3, it can be easily computed that the distance from pixel 0 to pixel1 or pixel 4 is 8. The distance from pixel 0 to pixel 2, pixel 3, pixel 5 or pixel 6 is√

72 + 42 = 8.06

which is close to 8. Hence, the feature of equal distance is almost retained andhence this construction hardly introduces image distortion.


k

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Figure 3. A cluster of seven constructed virtual hexagonal pixels.

Through this paper, we assume that original images are represented on a squarestructure arranged as 16M rows and 16N columns, where M and N are two positiveintegers. This assumption guarantees that sub-images to be obtained in this paperhas exactly the same size and are of a complete rectangle shape. Let the centre ofthe virtual hexagonal structure be located at the middle of 8M -th row and (8M+1)-th row, and at 8N -th column. Let us denote the light intensity of the square pixelat row m and column n by Q(m,n), where 0 ≤ m ≤ 16M and 0 ≤ n ≤ 16N . Here,we use row 0 to represent the first row (i.e., top row) and column 0 to representthe first column (i.e., left column) and so forth.

Note that there are (7×16M =) 112M rows and (7×16N =) 112N columns in thevirtual square structure consisting of virtual sub-pixels obtained from the originalsquare pixels. Let us denote the light intensity of the square sub-pixel at row m andcolumn n by P (m,n), where 0 ≤ m ≤ 112M and 0 ≤ n ≤ 112N . Let us constructthe first hexagonal pixel using the 56 sub-pixels with centre located in the middleof rows 56M − 1 and 56M and at column 56N − 1 of the virtual square structure.This first pixel is called the central hexagonal pixel in the hexagonal structure.It is corresponding to the pixel with spiral address 0 in the Spiral Architecture.After the 56 sub-pixels for the first hexagonal pixel are allocated, all sub-pixelsfor all hexagonal pixels can be assigned. For our uniform partition algorithms, theassignment of sub-pixels to corresponding hexagonal pixels is not explicitly required.We do not need to compute the intensities for the virtual hexagonal pixels. Afterimage processing for uniform partition, the intensity of each square pixel can becomputed as the average of the light intensities of the 7 × 7 sub-pixels separatedfrom this square pixel.

4. Uniform Image Partition

In this section, we will perform algorithms for separating image represented onthe virtual hexagonal structure into four sub-images that look similar and haveexact the same image size. The idea is to define rows and columns as images repre-sented on the square structure. Then, each sub-image is a collection of hexagonalpixels from every second row and column. It is not as obvious as in the squarestructure to find the column and the row of each hexagonal pixel. It is not obvious


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C=1 C=0 C=-1 C=-2 C=-3 C=-4 C=4 C=3 C=2

Figure 4. Columns on a hexagonal structure.

either to know which pixels form the first row or the first column of a sub-image.We follow the following three steps for the uniform image separation. We firstdefine the rows and columns on the virtual hexagonal structure, and propose analgorithm for computing the row and column of the hexagonal pixel that a givensub-pixel belongs to. Then we present an algorithm for the extraction of the firstsub-image. In the third step, algorithms for construction of the second, the thirdand the fourth sub-images are proposed.

4.1. Pixel Row and Column. Assume a given sub-pixel is at row p and columnq. Let R and C represent the number of rows and number of columns needed tomove from the central hexagonal pixel to the hexagonal pixel containing the givensub-pixel taking into account the moving direction corresponding to the signs ofR and C. Here, pixels on the same column are on the same vertical line. Forexample, as shown in Figure 4, pixels with addresses 43, 42, 5, 6, 64, 60 and 61 areon the same column with C = 1. The row with R = 0 consists of the pixels on thehorizontal line passing the central pixel and on the columns with even C values,and the pixels on the horizontal line passing the pixel with address 3 and on thecolumns with odd C values. Other rows are formed in the same way. For example,pixels with addresses 21, 14, 2, 1, 6, 52, 50 and 56 are on the same row with R = 1.Figure 5 shows rows in a hexagonal structure consisting of 49 hexagons. Followingthe algorithm proposed in [10], the C and R corresponding to the given sub-pixelcan be computed from P1, P2, Q1 and Q2 defined below.

Q1 = sgn{q − 56N + 1}max{

int c|c ≤ |q − 56N + 1|7

},

Q2 = (q − 56N + 1) mod 7,

P1 = sgn{p− 56M + 1}max{

int c|c ≤ |p− 56M + 1|8

},

P2 = (p− 56M + 1)mod 8.


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Figure 5. Rows on a hexagonal structure.

4.2. Construct the First Sub-image. Let h be an arbitrary given hexagonalpixel on a hexagonal structure. Let us denote the values of C and R correspondingto h by Ch and Rh respectively. Then the first sub-image S1 is formed by thehexagonal pixels that satisfy the following conditions:

S1 = {h|Ch mod2 = 0, (Ch

2mod 2)− (Rh mod 2) = 0}.

From the above representation, we can see that any hexagonal pixel must be onthe column with even C value; and if the pixel falls onto column with even C/2value then R value for the pixel must be even as well, otherwise R must be odd.

The arrangement of the hexagonal pixels in S1 is made as follows. We keepthe location of the central pixel (with C = 0 and R = 0) unchanged, and moveany other hexagonal pixel towards the central pixel by its half distance from it.Note that the distance between two adjacent columns is 7 sub-pixels long, and thedistance between two adjacent rows is 8 sub-pixels long. Hence, any pixel belongingto S1 except the central pixel will move by |Ch|/2 × 7 sub-pixels leftwards (orrightwards when Ch is negative) then by |Rh|/2× 8 upwards (or downwards whenRh is negative).

After the above-mentioned procedure, the first sub-image is formed and it issitting in the middle of the original image area. In order to fit all four sub-imagesonto the same image area after the four sub-images are formed, in the next step, wemove S1 to the top left corner of the original image area such that S1 will exactlyoccupy the top 8M rows and left 8N columns of the original rectangular imagearea. Note that the new centre of S1 will be located in the middle of rows 28M − 1and 28M and at column 28N − 1 of the virtual square structure. Hence, to movethe S1 to the top left corner, we need only move all sub-pixels in S1 upwards by28M sub-pixels and leftwards by 28N sub-pixels.

4.3. Construct Other Sub-images. The second sub-image S2 consists of allhexagonal pixels that are one hexagonal above or below the pixels in S1. The thirdsub-image S3 consists of all hexagonal pixels that are next to the pixels in S1 butsitting at the left-bottom side. Similarly, the fourth sub-image S4 consists of all


hexagonal pixels that are next to the pixels in S1 but sitting at the right-bottomside. Let us now construct sub-images 2 trough 4 one after another in the following.

4.3.1. Construction of S2. To construct S2, we first translate all sub-pixelsdownwards by 8 sub-pixels. By doing so, the first hexagonal pixel (with spiraladdress 4 as shown in Figure 1) of S2 that is the one right above the central pixelwith spiral address 0 is moved to the central position with C and R both equal to 0.After the translation, the bottom eight rows in the virtual square structure will bemoved out from the original image area. In order not to lose any image informationafter the translation, we fill the top eight rows of the virtual square structure bythose sub-pixels at the bottom eight rows while performing the translation. If weuse M1 to denote 112×M , N1 to denote 112×N , and P [i][j] to denote the valueat the sub-pixel (i, j), the pseudo code for this movement is shown as follows.

for ( i = 0 ; i < M1 ; i ++ ){for ( j = 0 ; j < N1 ; j ++ ){

if ( 0 <= i && i < 8 )P[i][j] = P[M1-8+i][j] ;

elseP[i][j] = P[i-8][j] ;

}}

We can now use exactly the same algorithm as shown in Subsection 4.2 aboveto construct S2 and display it in the middle of the original image area. The nextstep then is to move S2 to the top right corner of the original image area suchthat S2 will exactly occupy the top 8M rows and right 8N columns of the originalrectangular image area.

4.3.2. Construction of S3. To construct S3, we first translate all sub-pixels right-wards by seven sub-pixels, and then upwards by four sub-pixels. By doing so, thefirst hexagonal pixel (with spiral address 2 as shown in Figure 1) of S3 is movedto the central position. Again, in order not to lose any image information afterthe translation, the sub-pixels that will be moved out from the image area mustbe filled into the pixels that will not be translated from any other sub-pixels. Thepseudo code for this movement is shown as follows.

for ( i = 0 ; i < M1 ; i ++ ){for ( j = 0 ; j < N1 ; j ++ ){

if ( M1 - 4 <= i && i < M1 ){if (j < 7)

P[i][j] = P[i-M1+4][N1-7+j] ;else

P[i][j] = P[i-M1+4][j-7] ;}else{

if ( j < 7 )P[i][j] = P[i+4][N1-7+j] ;

elseP[i][j] = P[i+4][j-7] ;

}}

}


We can now use exactly the same algorithm as shown in Subsection 4.2 aboveto construct S3 and display it in the middle of the original image area. The nextstep then is to move S3 to the left bottom corner of the original image area suchthat S3 will exactly occupy the bottom 8M rows and left 8N columns of the originalrectangular image area.

4.3.3. Construction of S4. The construction of S4 is almost the same as theconstruction of S3. We first translate all sub-pixels leftwards by seven sub-pixels,and then upwards by four sub-pixels. By doing so, the first hexagonal pixel (withspiral address 6 as shown in Figure 1) of S4 is moved to the central position.

We can then use exactly the same algorithm as shown in Subsection B above toconstruct S4 and display it in the middle of the original image area. The next stepthen is to move S4 to the right bottom corner of the original image area such thatS4 will exactly occupy the bottom 8M rows and right 8N columns of the originalrectangular image area.

4.3.4. Map Back to Square Structure. After the process for image separation,to be simple, the sub-images represented on the virtual hexagonal structure can bemapped back to the square structure. Note that each square pixel on a sub-imageis formed by 7 × 7 sub-pixels. We can simply compute the light intensity of eachsquare pixel as the average of the light intensities of the 7× 7 sub-pixels.

Note that, for better display result, a better method for assignments of intensityvalues of sub-pixels is required through a better interpolation technique. This is,however, beyond the discussion and the objectives of this paper.

5. Experimental Results of Image Partitioning

The above algorithms for uniform image separation on the newly developed vir-tual hexagonal structure are implemented using C++ programming language andtested on a desktop of Pentium IV, 2.8GHz CPU and 480MB memory. Experimen-tal results of the proposed image partitioning algorithms on grey-level images arepresented here.

A sample image, called “building” with size of 384 × 384 pixels is shown inFigure 6. An example of image separation into four sub-images on the virtualhexagonal structure is shown in Figure 7.

In Figure 7, it can be seen clearly that four sub-images have exactly the samesize, look identical and are still in the shape of a square. There are no gaps andno overlapping between the sub-images. All information of the original image ismaintained during the whole separation process based on the hexagonal structure.The resolution of the whole image represented on the virtual hexagonal imagestructure and shown in Figure 7 is almost the same as that for Figure 6.

As an illustration, the results of a further partitioning process are displayed inFigure 8 that contains 16 similar sub-images.

Note that the row and column numbers of the original image do not have to bethe same for image separation. Figure 10 shows four sub-image images partitionedfrom a 512× 384 image in Figure 9.

The total time to complete the computation for image separation for an imageis less than 1 second. Compared with the method introduced in [11] that takesminutes to complete a separation, a great improvement has been achieved usingthis new virtual structure.


Figure 6. A sample image, “building”

Figure 7. The “building” image is uniformly separated into foursub-images of equal size and represented on a virtual hexagonalstructure.

6. Principles of FIC on Virtual Hexagonal Structure

In our previous research, Spiral Addition and Spiral Multiplication were appliedto obtain the range blocks and domain blocks on SA. The addition and multipli-cation operations were defined based on Spiral Counting [4]. Spiral Counting canbe considered as a Spiral movement from a hexagonal pixel to another hexagonalpixel. Any hexagonal pixel in an image can be reached by Spiral counting from anyother given hexagon in the same image. It is important to note that each of theaddition and multiplication operations takes two spiral addresses and maps themto another spiral address. They can then be used to define two transformations onthe spiral address space: image translation (by spiral addition) and rotating imagepartitioning (by spiral multiplication). In our previous research on FIC, the rangeblocks can be easily traced by spiral counting. However, the definition of domain


Figure 8. 16 similar sub-images separated from the “building”.

Figure 9. A 512× 384 car image.

blocks and the construction of domain pool rely on the uniform image partitioningon SA.

Figure 11 shows an example of image partitioning using spiral multiplication.The original image has 1680710 (a number of base 10) hexagon pixels. The multi-plier used in spiral multiplication is 555557 (a number of base 7). The separatedsub-image areas are shown using different illuminations and labeled by differentarea numbers (see Figure 11(b)). Finally, the fragments of each sub-image are col-lected together to produce a complete partitioned sub-image. The addends usedin spiral addition for fragment collection are also shown on each sub-image (seeFigure 11(c)).The uniform image partitioning on SA applied here reveals the one-to-four relationship between the original image and its four sub-images. Therefore,for any given block in the SA space, it would be possible to find another corre-sponding area of four-time larger size by implementing the opposite operation of


Figure 10. Four similar sub-images separated from a 512 × 384a car image.

this uniform image partitioning. If we consider the smaller block as a range block,the larger block then would be one of the potential domain blocks for this range.

(a)

2

2 3

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(b)

1

3 4

55555addend = 66666addend =

2

44444addend =0addend =

(c)

Figure 11. Uniform image partitioning on SA. (a) Original im-age. (b) Four labeled sub-images after spiral multiplication. (c)Four complete sub-images after fragment collecting with spiral ad-dition.

The idea of uniformly separating an image into four sub-images is adopted in thispaper for the definition of range and domain blocks on a virtual hexagonal structure.There are the following differences between our previous approach and the work tobe presented in this paper. First of all, we will not explicitly use the time consumingSpiral Addition and Multiplications for image separation. Secondly, the shape ofimages to be compressed is no longer hexagon-like, and can be a rectangle. Thethird but not the last difference is that the conversion between image representedon square structure and the hexagonal structure is fast and easy. Figure 10 showsthe results of an image separation.

In this section, we first define the rows and columns on the hexagonal structureand then define the range and domain blocks for FIC in this paper. Note that therows and columns defined in this section has a different meaning from those definedin Section 4.


6.1. Construction of Hexagonal Images. Let the central virtual hexagonalpixel locate at the image centre. Let R and C, as defined in Section 4, represent thenumber of rows and number of columns needed to move from the central hexagonalpixel to the hexagonal pixel containing the given sub-pixel taking into account themoving direction corresponding to the signs of R and C. Again, we assume thatthe size of original images is 16M×16N , where M and N are two positive integers.Then, after converted to the virtual hexagonal structure, each image has 14N rowsand 16M columns. Let us use X and Y to denote the column and row of anarbitrary hexagonal pixel. Define

X ={

C + 8M, if − 8M < C < 8M,0, otherwise,

Y =

R + 7N, if C mod2 = 0 and − 7N < R < 7N,0, if C mod 2 = 0 and |R| = 7N,

R + 7N − 1, if C mod2 = 1.

Recall that the light intensity of the square pixel at column m and row n isdenoted by Q(m, n), where 0 ≤ m ≤ 16M and 0 ≤ n ≤ 16N and the intensity ofthe sub-pixel at column i and row j by P (i, j) for 0 ≤ i ≤ 112M and 0 ≤ j ≤ 112N .Let us also denote the light intensity of the hexagonal pixel at column x and rowy by O(x, y), where 0 ≤ x ≤ 16M and 0 ≤ y ≤ 14N . Then, the pseudo code tocompute the light intensities of hexagons from given intensities of square pixels isshows as follows.int O[16 * M][14 * N];int P[112 * M][112 * N];

for ( x = 0 ; x < 16 * M ; x ++ )for ( y = 0 ; y < 14 * N ; y ++ )

O[x][y] = 0 ;for ( i = 0 ; i < 112 * M ; i ++ ){

for ( j = 0 ; j < 112 * N , j ++ ){findXY(i,j) ; // get X&Y and return resultY = result.y ; // ‘‘R’’ - rowX = result.x ; // ‘‘C’’ - columnO[X][Y] = O[X][Y]+P[i][j] ;

}}for ( x = 0; x < 16 * M ; x ++ )

for ( y = 0 ; y < 14 * N ; y ++ )O[x][y] = O[x][y]/56;

Up to now, we have computed the intensity values of all virtual hexagonal pixels.For our FIC work in this paper, we use 16M × 16N square pixels to keep theinformation of all hexagonal pixels. An image of size 16M × 14N is then obtained.It looks similar to the original image.

6.2. FIC on Hexagonal Structure. After the hexagonal image represented us-ing square pixels is obtained as presented in the above Subsection, FIC proceduresbecome obvious by simply following Fisher’s basic FIC algorithm on square struc-ture. The size of range blocks is scalable and can be 2× 2, 3× 3, 4× 4, 6× 6, 8× 8and so forth. We can compare the results with those obtained from the originalusing the same scales.


After compression step, the compressed image based on the virtual structure canbe displayed on the original image area. The pseudo code for displaying compressedimages is written below.

for ( i = 0 ; i < 112 * M ; i ++ ){for ( j = 0 ; j < 112 * N , j ++ ){

findXY(i,j) ; //get X&Y and return resultY = result.y ; //‘‘R’’ - rowX = result.x ; //‘‘C’’ - columnP[i][j] = O[X][Y] ;

}}for ( m = 0 ; m < 16 * M ; m ++ ){

for ( n = 0 ; n < 16 * N ; n ++ ){Q[m][n] = 0 ;for ( i = 0 ; i < 7 ; i ++ )

for ( j = 0 ; j < 7 ; j ++ )Q[m][n] = Q[m][n]+P[7 * m+i][7 * n+j] ;

Q[m][n] = Q[m][n]/56 ;}

}

7. Experimental results of FIC

Four testing images are all 180× 180 one channel grey level images as shown inFigure 12. For both FIC on square structure and spiral architecture, each rangeblock is encoded by storing the following information: the index to the best domainblock (eight bits), the intensity coefficient (seven bits) and contrast coefficient (fivebits). The domain block pool is designed to be the spatially closest 256 domainblocks available to any given range block.

For every experiment image, the performance of FIC on the traditional squarestructure is compared with that on SA and the results are shown in the figuresfrom Figures 13(a) to (d). In these figures, the PSNR (dB) is a function of thecompression ratio. The PSNR vs. Compression Ratio curves are selected instead ofPSNR vs. Bit Rate because in our experiment result, it distributes more equablythan bit rate along X axis.

As seen in Figures 13(a) to (d), the curve representing FIC performance on SAis always above that for square structure. It is easy to see that the proposed FICon SA always performs better than that on square because there are higher PSNRvalues for different compression ratios for all tested images.

The improved performance is introduced by the enhanced image structure, i.e.SA. First of all, SA has higher degrees of symmetry that results in a considerablesaving in storage [12]. In our preliminary research on image compression on SA, weselect a block of certain size on an image and unwind the pixels into a one-dimensiondata set. It has been found that the data set on SA introduces less average variancethan on the traditional square image structure [13]. This less variance can also befound in the range block and domain block defined in our proposed method so thatit reduces the error between the fixed point and the original image and eventuallyincreases the fidelity in both encoding and decoding. In addition, SA has a greaterangular resolution, which also increases the accuracy in the search of the bestdomain block.


(a) Building (b) Boat

(c) House (d) View

Figure 12. The four testing images.

8. Conclusion and Discussion

In this paper, we have presented algorithms to uniformly separate an image rep-resented on hexagonal structure into four sub-images. The partitioned sub-imageslook similar. It is important to know that the arrangement of the pixels in thesub-images does not change their symmetry relationship in the original image. Forexample, if in the original image a pixel A was in one of the six neighbouring direc-tions as shown in Figure 4 or Figure 5 corresponding to another pixel B belongingto the same sub-image, pixel A is in exactly the same direction corresponding topixel B in their sub-image. Another example is that if three pixels C, D and Ebelong to the same sub-image pixel set and if the distance between C and D wasthe same as the distance between C and E in the original image, then in theirsub-image, these two distances are still the same. Furthermore, we do not split anyhexagonal pixel during the whole process, and hence all pixel information includingthe intensity values are maintained. The importance of uniform partition has beendemonstrated in many papers addressing parallel image processing on hexagonalimage structure. The maintenance of the symmetry property after image separa-tion is critical for image processing such as finding gradient and its magnitude thatrequest neighbouring information for the computations.


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Figure 13. The PSNR versus Compression Ratio for the fourtesting images.

One may argue that on the traditional square image structure, we can also sep-arate image uniformly into four parts by simply split every four pixels evenly intofour sub-images. However, the sub-images obtained in this way also split the vir-tual hexagonal pixels and hence no longer hold the features of hexagonal structure.Therefore, those algorithms and applications developed based on hexagonal struc-ture and proved to be more accurate or faster have lost their groundwork to beapplied to these sub-images.

In this paper, we have also improved our translation algorithms proposed in [10]by placing the pixels moved out from the image area onto the empty region of theimage area. This keeps all image information after image translation. The trans-lation process and the separation process are both reversible. The original imagecan be fully reconstructed from the separated sub-images using inverse algorithmsif needed.

Another contribution in this paper is a proposal to scale down an image to fourtimes smaller images based on the hexagonal structure. Scaling operation is withthe operations for image separation.

In our implementation, we implicitly adopt the ideas of two operations definedon Spiral Architecture, namely spiral addition and spiral multiplication, and usethem for image separation. However, we do not perform these two operationsto avoid a large amount of time requested for the complex computations on thevirtual structure. This is very different from any of previous approaches, and hassignificantly improved the performance in terms of speed and complexity.

Another advantage of using this new approach is that the image shape is con-sistent with the traditional shape of a rectangle. This make it easier and moreflexible for image processing based on hexagonal structure using images capturedand displayed on square structure.


As we do not compute the light intensities for virtual hexagons, image resolutionis maintained during the separation process, and we save the processing time andmemory storage.

As there are simple non-overlapping mappings between the sub-pixels and thesquare pixels, and the mappings between the sub-pixels and the hexagonal pixels,the results of image processing on the hexagonal structure can be easily mappedback to the square structure for display.

On Spiral Architecture, we have performed image separation into any givennumber of sub-images in our previous research work. This work can be extendedto the virtual hexagonal structure for future approaches.

For FIC on the virtual hexagonal structure, the investigation carried out by usand the performance analysis show the advantages of the new proposed FIC on SA.By inheriting the basic FIC method directly from square structure to SA, it givesmore accurate compression result for all testing images. The proposed FIC also hasflexible range block sizes so that the compression ratio is adjustable.

The analysis of experimental results indicates SA has great potential in FIC im-provement. For future work, other FIC methods on SA can be applied by followingsimilar ways that have been tried on square structure. Various image partitioningmethods (such as polygonal partitioning and edge-based partitioning), parallel pro-cessing enhanced by spiral multiplication and different definition of domain blockpool are of great research interest.

Acknowledgments

This work was partly supported by the Australian Research Council DiscoveryGrant (DP0451666). The authors appreciate the reviewers’ valuable comments onthe two papers appeared in ICPR2006 [14] and ICARCV2006 [15] on which thispaper is built.

References

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[2] C. H. Koelbel, D. B. Loveman, R. S. Schreiber, G. L. S. Jr. and M. E. Zosel, The HighPerformance Fortran Handbook, MIT Press, Cambridge, MA., 1994.

[3] Q. Wu, X. He, T. Hintz and Y. Ye, Complete Image Partitioning on Spiral Architecture,Lecture Notes in Computer Science, Vol. 2745, M.Guo and L.T.Yang (eds.), pp. 304–315,2003, Springer.

[4] P. Sheridan, T. Hintz and D. Alexander, Pseudo-invariant Image Transformations on a Hexag-onal Lattice, Image and Vision Computing, vol. 18, pp. 907–917, 2000.

[5] X. He, T. Hintz, Q. Wu, H. Wang and W. Jia, A new simulation of Spiral Architecture,Proceedings of International Conference on Image Processing, Computer Vision, and PatternRecognition, pp. 570–575, 2006.

[6] Y. Fisher, Fractal Image Encoding and Analysis. Berlin Heidelberg, Germany: Springer-Verlag, 1998.

[7] M. Barnsley and L. P. Hurd, Fractal Image Compression. New York, U.S.A: AK Peters. Ltd,1993.

[8] B. Wohlberg and G. d. Jager, A Review of the Fractal Image Coding Literature, IEEETransaction on Image Processing, vol. 8, 1999.

[9] H. Wang, Q. Wu, X. He and T. Hintz, A New Approach to Fractal Image Compression on aVirtual Spiral Architecture, Proceedings of the International Conference on Image Processing(ICIP2006), Atlanta, USA, pp. 3101–3104, October, 2006.

[10] X. He, W. Jia, Q. Wu, N. Hur, T. Hintz, H. Wang and J. Kim, Basic Transformationson Virtual Hexagonal Structure, Proceedings of the International Conference on ComputerGraphics, Imaging and Visualization, pp. 243–248, 2006.


[11] Q. Wu,, X. He and T. Hintz, Virtual Spiral Architecture, Proceedings of the InternationalConference on Parallel and Distributed Processing Techniques and Applications, vol.1, pp.399–405, 2004.

[12] I. Her, Geometric Transformations on the hexagonal Grid, IEEE Transaction on Image Pro-cessing, vol. 4, 1995.

[13] Q. Wu, X. He and T. Hintz, Preliminary Image Compression Research Using Uniform ImagePartitioning on Spiral Architecture, Proceedings of the Second International Conference onInformation Technology and Applications, pp. 216–221, 2004.

[14] H. Wang, X. He, Q. Wu and T. Hintz, A New Approach for Fractal Image Compression ona Virtual Hexagonal Structure, Proceedings of the 18th International Conference of PatternRecognition (ICPR2006), Vol.III, pp. 909–912, 2006.

[15] X. He, H. Wang, N. Hur, W. Jia, Q. Wu, J. Kim and T. Hintz, Uniformly Partitioning Imageson Virtual Hexagonal Structure, Proceedings of the 9th International Conference on Control,Automation, Robotics and Vision (ICARCV2006), Singapore, December 2006, to appear.

Authors, Xiangjian He, Huaqing Wang, Wenjing Jia, Qiang Wu and Tom Hintz, are with theDepartment of Computer Systems, Faculty of Information Technology at University of Technology,Sydney, Australia.

E-mail : {sean,huwang,wejia,wuq,hintz}@it.uts.edu.auAuthors, Namho Hur and Jinwoong Kim, are with the Broadcasting System Research Group,

Digital Broadcasting Research Division at Electronics Telecommunications Research Institute,Korea.

E-mail : {namho,jwkim}@etri.re.kr

Documents

Volume 3, Number 3, Pages 492{509 · Key Words. Hexagonal structure, image scaling, Spiral Architecture, fractal image compression, image partitioning. 1. Introduction Computer Vision