Pergamon Computers Elect. Engng Vol. 21, No. I, pp. 21-32, 1995

0045-7!M6(!94)00012-3 Copyright 0 1994 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0045-7906/94 $7.00 + 0.00

TEXTURE DESCRIPTION USING FRACTAL AND ENERGY FEATURES

T. KASPARIS’, N. S. TZANNES’, M. BA~~IOUNI* and Q. CHEN’ ‘Department of Electrical and Computer Engineering and 2Department of Computer Science,

University of Central Florida, Orlando, FL 32816, U.S.A.

(Received for publication 23 June 1994)

Abstract-The fractal dimension of a texture has been used in the past as a segmentation feature, but since it cannot sufficiently describe enough textural characteristics, additional features are needed. In this paper we demonstrate that by combining the fractal dimension with a simple textural energy measure, a significant performance improvement is achieved compared to using each feature alone. The fractal dimension is computed using an efficient method that is also more accurate than most other popular methods, and the textural energy is easily computed using convolutional masks. Segmentation and classification of natural textures based on these two features is presented and the effect of additive noise is considered.

Kqv words: Texture segmentation, texture classification, fractals, textural energy.

1. INTRODUCTION

Textural features are important pattern elements in both human and machine vision [l]. The fractal model introduced by Mandelbrot [2] has been recently used in an effort to obtain a unified image model. Since some synthesized fractal sets have striking resemblance to natural objects (clouds, trees, mountains, etc.), the idea that fractals might also be successfully applied to image analysis was motivated. As a result, the fractal model received considerable attention in describing natural surfaces and in discriminating textures. A basic parameter of a fractal set is the fractal dimension D, which from an intuitive point of view it corresponds to the concept of roughness. The larger the D-value, the rougher the appearance of the set. For a plane D is 2 (equal to the topological dimension), while for highly irregular (space filling) surfaces D approaches 3.

2. FRACTAL TEXTURE ANALYSIS

Early fractal-based algorithms computed D using various techniques, and used it as the only feature in texture analysis [3-71. However, it was later noticed that fractal sets that look different may share the same D-value [8-lo] and as a result the performance of early algorithms was limited. Recent fractal based algorithms use more features, both fractal and non-fractal, including fractal matrices [l 11, higher order fractals [12], and multifractals [13]. Other techniques combine the D parameter with features that may have been previously used in other applications. Examples include the pixel variance [lo], human visual system properties [14], and lacunarity [9]. Lacunarity is a measure introduced by Mandelbrot to describe characteristics of fractals of the same dimension but different appearance. Most fractal-based techniques deal with segmentation where others [ 11,121 classify homogeneous textures assuming that they have already been segmented. The effect of additive noise is generally not accounted.

In this work we demonstrate that by combining the local fractal dimension with a textural energy measure which is simple to extract, we can obtain results much better than using each feature alone. Other features could have been combined with the fractal dimension (such as co-occurrence features, wavelet expansions, etc.), but we have chosen textural energy because it can be efficiently computed using convolutional masks, and because this feature pair exhibits good noise tolerance. In our simulations the feature combination performed very well for most of the natural textures we have tested, even at the presence of substantial amounts of noise.

21

22 T. KASPARIS ul (I/.

3. FRACTAL DIMENSION ESTIMATION

A basic relation which is a consequence of the self-similarity property of a fractal set is [2]:

A(ci) =ku -”

where A is the measured surface area, a is the elementary ruler area, k is a scaling constant, and D is the fractal dimension. Many of the approaches to compute D are essentially based on equation (1). The Brownian model has been popular in estimating D [3-71, but it is computationally expensive and not very accurate. Reliable estimation of the D parameter is essential in segmentation since (for surfaces) D takes values from 2 to 3 with usually small separability between different surfaces. One basic source of errors in estimating D is due to sampling. As D approaches the value of 3. a fractal set becomes highly irregular. Sampling however reduces this irregularity to finite levels. One proposed approach uses linear interpolation in an attempt to reduce this effect [9],

In our work, we have used a “variation” method which we found to be significantly more accurate and computationally cheaper than other techniques we have used. For 2-D curves embedded on a N x A4 grid, the computational complexity of this method is 0 (N) whereas for other standard algorithms such as box counting or Minkowski-Bouligand dimension the comput- ational complexity is of 0 (NM) [15]. The variation technique uses the notion of the s-variation to measure the peak of the function in an &-neighborhood, and the order of growth of the integral of the &-variation as E approaches zero is directly related to the D parameter [I S]. For a surface S (x, y ), this is expressed as:

where P’, is the &-variation of S in a neighborhood E around point (.u,J’). l.c..

ve,y. VI = max(S(s, t)} - min{S(s, f)j where maxi 1.~ -s 1, 1~. -. / ; <c. (3)

In applying equations (2) and (3) within texture blocks W,,. we directly computed I’, from (3) and approximated the integral in equation (2) [ 161. The fractal dimension was computed from the slope of the least mean square line passing through the points:

(4)

The variation I’, of a surface S can be efficiently computed from 2-D protile data by using a fast algorithm described in Ref. [17].

4. TEXTURAL ENERGY FEATURES

A class of texture figures is based on average degrees of match of pixel neighborhoods with a set of standard masks [18]. Experiments concluded that the most useful 5 x 5 masks are the four

Table I. Four 5 x 5 a zero-sum mask\

-I -2 0 2 I -1 0 2 0 I

4 -8 (I 8 4 4 0 8 (1 4

-6 -12 0 I? 6 6 0 I? 0 6

-4 -8 0 8 4 -4 0 x 0 ~4

-I -2 0 2 I I 0 2 0 I

E5E5 R5R5

-I 0 2 0 -I I -4 6 -4 I

-2 0 4 0 -2 4 I6 -24 16 4

0 0 0 0 0 6 -24 36 24 6

2 0 -4 0 2 -4 I6 -24 I6 4

I 0 ~ 2 0 I I -4 6 -4 I

Texture description using fractal and energy features 23

zero-sum matrices shown in Table 1 [l&19]. When used in texture analysis, a block W,, from texture S(x, y) is convolved with a mask, i.e.:

L (m, n) = i i M(k, l)S(m - k, n - I), (m, n) E W, (5) kc-2 ,=-I

where A4 (k, 1) is a mask from Table 1. The statistics of the convolutions L (m, n) are used as textural properties. One commonly used

parameter is the “energy” of block WV, de&red as the sum of the squares of all the convolutions. Since the dimensions of the block WV may vary, in our work we used normalized energy, defined as:

where C is a normalization parameter proportional to the dimensions of block Wu. Our experimentation with the masks of Table 1 indicated that the highest performance was achieved with the L5S5 mask, which is obtained from the convolution of a weighted average and a “spot” detector 1 -D operators [ 18,191. Computing the energy of a block W, with dimensions N x N using the L5S5 mask from equations (5) and (6) requires approximately 50N2 arithmetic operations (multiplications or squaring and additions) and one division. These operations can be significantly reduced to approximately 6N2 + 5N by using the separability of the L5S5 mask. Further reduction of the computation is possible by exploring the fact that the masks of Table 1 are generated from 1-D vectors with 3 elements each 118,191, and by utilizing the presence of zeros. In addition, the absolute value is sometimes used in equation (6) instead of the square, in order to further reduce the computation.

5. SEGMENTATION EXPERIMENTS

In the first part of our experiments we performed segmentation of 512 x 512 texture mosaics consisting of six textures taken from a photo album by Brodatz [20]. Each mosaic was first histogram equalized to impose a uniform gray level frequency distribution. The fractal dimension and the L5S5 features were extracted from blocks W,, of dimensions 64 x 64 pixels overlapped by 32 pixels in each direction and were assigned to the center of the block. A simple unsupervised K-means clustering algorithm which is based on the minimization of the sum of the squared distances from all the points in a cluster to the cluster center [21] was used for the final image partition.

Figure 1 presents a segmentation example. Figure la is the input texture mosaic and Figs 1 b and c are the segmentation results based either on the fractal dimension or the L5S5 energy measure alone. Figure Id is a much better segmentation result based on the combination of both features. Most errors occur around the boundaries, as expected.

6. NOISE EFFECTS

Since the noise will increase the texture irregularity, it is expected that the value of D will move closer to the upper bound of 3. Also, the high-pass characteristics of the L5S5 mask will produce higher convolution results at the presence of noise, and therefore the energy will increase. Normalization, however, will reduce the energy increase by a factor of C (see equation 6) and as a result the effect of the noise will be reduced. The feature histograms of several noise-free and noisy textures corrupted by uniform or Gaussian noise were compared. Figure 2 displays the histograms of D and L5S5 energy of two noise free texture samples (cork and sand) of Fig. 2a, and Fig. 3 displays the same histograms after zero mean Gaussian noise and variance of 50 (SNR z 10 dB) was added to the textures. Comparison of Figs 2 and 3 shows that the histograms of the noisy textures have shifted to higher values. Similar observations were made with other texture samples. Since both features increase, their separation will be less affected by the noise, and we can expect that segmentation will exhibit some noise immunity. However, excessive noise will “saturate” the features and the segmentation performance will rapidly drop. In our experiments the fractal dimension saturated first (since it is upper bounded by the value of 3) and the amount of noise for saturation varied among various textures.

Texture description using fractal and energy features 25

Fig. I. Segmentation of a noise-free texture mosaic. (a) Input mosaic. From top-left clockwise: leather, cork, sand, water, grass, raffia. (b) Segmentation using only the fractal dimension. (c) Segmentation using only the L5S5 energy measure. (d) Segmentation using both the fractal dimension and the L5S5 measure.

26 T. KASPARIS er ul.

I (a)

0.175.

0 . 2.. 0.15.

t; 0. 125,. 0.15,.

G c 0.1,.

3 $ 0.1. Go.075.

E g

0.05,.

0. 05 II\ gj Iii 0.025.

2.2 2.4 2.6 2.8 3 2.2 2.4 2.6 2.8 3

D. Cork D, Sand

0. 12.. (b) 0.1757

0 . 1.. 0.15,.

I

iY 0.08..

c 0. 125..

0.1. G 0. 06.. z 2 g 0.075..

r$ 0.04.. E 0.05. &

0.02. 0. 025..

f: 20 40 60 80 100 120 140 ’ 20 40 60 80 100 120 140

E. Cork E, Smd

Fig. 2. Feature histograms for cork and sand. (a) Fractal dimension. (b) L5S5 energy measure

Figure 4 presents segmentation of a noisy mosaic same as in Fig. 1. The combination of the two features produced at the presence of noise the same segmentation (Fig. 4d) as that obtained in the noise-free case (Fig. Id). This segmentation performance compares favorably with techniques that combine fractal dimension either with variance [lo] or with four lacunarity features [9], extracted from noise-free samples. The variance of the pixels is directly affected by the presence of noise, and therefore we would expect that the feature vector used in Ref. [lo] would be very sensitive to noise.

D. Cork (b)

0.14.

0. 12..

0.1

0.08.

0.06..

0.04

0.02.

’ 20 40 6.0 80 100 120 140

E. Cork E, Saud

Fig. 3. Feature histograms for noisy cork and sand. (a) Fractal dimension. (b) L5S5 energy measure.

Texture description using fractal and energy features 27

Fig. 4(a) and (b). Continued overleaf.

t 1p 4. Segmentation 01 d noi\> lcklurc mo\ak (.t) The mosaic of Fig. 2a with noise added. (h) Segmentation uw~g on14 the fractal dimenslc~n (0 Segmentation using only the LSS5 energy measure.

(d) Segmentation u,ing both the tracl,tl dimension and the L5SS measure.

Texture description using fractal and energy features

C. 85 -

- 80.

; 75-

: 70-

& 65. w g 60-

8 55.

50.

45 -

40 1 0 20 40 60 80 100 120 140 1

NOISE VARIANCE

Fig. 5. Segmentation performance vs additive noise variance.

29

0

As a comparison with one of the many available non-fractal texture analysis methods, one may consider the method of Ref. [22] which is based on gradient histograms. The segmentation results presented in this reference are generally good, however, the presence of noise may degrade the performance since the predominantly high-pass characteristics of gradient operators have a tendency to greatly enhance the presence of noise.

Figure 5 presents segmentation performance vs noise variance, evaluated by the number of correctly segmented blocks compared to the total number of blocks used in the segmentation (see Figs 1 and 2) and averaged for different mosaics. The performance remains fairly stable for g2 < 60.

7. CLASSIFICATION EXPERIMENTS

Classification is normally a two phase process. It requires an initial training phase during which the classifier is trained to recognize a class of reference feature vectors, and a testing phase during which unknown vectors are classified in the reference according to a best match criterion. During training, we computed the fractal dimension and texture energy histograms of (noise-free) texture samples under consideration. Reference features were placed in a scattering diagram in the feature space, and circular boundaries were established. In the testing phase classification took place after partitioning an unknown input noisy mosaic. For each partition a representative feature vector was computed by averaging the feature vectors of the blocks forming that partition, and classification was accomplished in the feature space by finding the location of the average vector in the pre-defined boundaries. In the event of overlapping boundaries the decision was random. The classification performance was evaluated by repeated experiments with random noisy mosaics, and the confusion results of 54 experiments with eight texture samples are summarized in Table 2. For four textures the classification was perfect, but there was some confusion between cork and weave.

Table 2. Confusion results with eight textures

Leather Cork Sand Raffia Grass Water Weave Pigskin

Leather 9 Cork 5 1 Sand I Raffia 6 I Grass I I 6 Water 9 Weave 4 Pigskin 4

10 T. KASPARIS er cd

The average classification accuracy from Table 2 is around 93%. computed by dividing the number of correct classifications with the total number of experiments. This result is quite satisfactory considering that the method uses only two features that can be easily computed. As a comparison, the technique of using fractal matrices proposed in Ref. [ 1 I] achieves about the same recognition rate with far more extensive computation and with features extracted globally (from the entire image rather than within smaller blocks) from noise-free samples. The higher-order fractal technique proposed in Ref. [ 121 uses three features which also require substantial computation to be globall> extracted. but no recognition rate is given (only the performance on discriminating a pair of nolhc-free textures is given in Ref. [I 21). The non-fractal technique of Ref. [22] reports perfect classitication among three rather dissimilar (and noise-free) textures from Brodatz’s album. These inconslstencies make a direct comparison of various methods difficult without implementing each and ever! algorithm.

x. (‘ON(‘LL!SION

We ha\e presented some texture segmentation and classification results based on a feature pair consisting of the fractal dimension and a textural energy measure. The main advantages of this approach are simple feature extraction and good noise tolerance. Also, the same local features extracted for the segmentation phase, are also used for the classification. In estimating the fractal dimension we used an efficient and accurate method that contributed to the overall performance. The segmentation performance under noisy conditions compares to results of other fractal-based techniques that use more than two features. obtained with noise-free texture mosaics. Classification experiments provided a recognition rate similar to techniques based on global features extracted also under ideal conditions. In our experiments. processing a 512 x 512 mosaic of six texture samples required approximately 1 h on a NeXT computer running Fortran. The computation time. hotvever, depends on several other factors such as programming efficiency, language used and most impor~ntly on the speed of the computer itself. Finally. the overall performance of the algorithm can be further improved by introducing additional features, at the expense of additional computarion.

-1~ /\rl~~~~ /c~/,~o~~~(,~~r \ Thus work has supported by a grant from the Florida High-Technology and Industry Council, with matching fund\ from Martin-Marietta Electronic Systems

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Texture description using fractal and energy features 31

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AUTHOR BIOGRAPHIES

T&Is Kasparis received the Diploma of Electrical Engineering from the National Technical University of Athens-Greece in 1980, and the MEEE and Ph.D. degrees in Electrical Engineering from the City College of New York in 1982 and 1988. From 1985 until 1989 he was an electronics consultant. In 1989 he joined the Electrical Engineering Department of the University of Central Florida, Orlando, where he is presently an assistant professor. His research interests are in non-linear signal and image processing, pattern recognition and computer vision.

Nicolaos Tzannes is presently Professor and Chairman of the Electrical Engineering Department of the University of Central Florida, Orlando. He holds a BSEE degree from the University of Minnesota, an MSEE degree from Syracuse University, and a Ph.D. degree from the Johns Hopkins University. He is the author, co-author or editor of several books. the most recent being Communication and Radar Systems (Prentice-Hall, 1985). He is the author or co-author of more than 70 papers on communication systems, digital signal processing, mathematical linguistics. etc., published in various journals or conference proceedings.

Mostafa Bassiouni received his B.Sc. and M.Sc. degrees in Computer Science from Alexandria University, Egypt, in 1974 and 1977, and the Ph.D. degree in Computer Science from Pennsylvania State University in 1982. He is currently an Associate Professor of Computer Science at the University of Central Florida, Orlando. He has been actively involved in research on data compression, computer networks, distributed systems, concurrency control protocols, and relational databases. He is the author or co-author of over 85 papers published in leading computer journals, book chapters, and conference proceedings. Dr Bassiouni is a member of the Institute of Electrical and Electronics Engineers (IEEE), and IEEE Computer and Communications Societies, the Association for Computing Machinery, and the American Society for Information Science.

32 T. KASPARIS er ul.

Qing Chen received the B.Sc. degree from Harbin Shipbuilding Engineering Institute, China in 1986, and the M.Sc. m Electrical Engineering from the University of Central Florida, Orlando, in 1992. Her research interests are in digital image processing and computer vision