texture description using fractal and energy features - CiteSeerX

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

Pergamon

0045-7!M6(!94)00012-3

TEXTURE

Copyright0 1994ElsevierScienceLtd Printedin GreatBritain.All rightsreserved 0045-7906/94 $7.00+ 0.00

DESCRIPTION USING FRACTAL ENERGY FEATURES

AND

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

T.

22

3. A basic relation

FRACTAL

KASPARIS

ul (I/.

DIMENSION

which is a consequence

ESTIMATION

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 computational 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

ve,y. VI= max(S(s,

t)} - min{S(s,

f)j

E around

point

(.u,J’). l.c.. 1, 1~.-. / ; lcklurc mo\ak (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-

& w g

65.

8

55.

29

60-

50. 45 40 1 0

20

40

60 80 100 NOISE VARIANCE

120

1 0

140

Fig. 5. Segmentation performance vs additive noise variance.

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 Leather Cork Sand Raffia

9

Grass

I

Water Weave Pigskin

Cork

Sand

Raffia

Grass

Water

Weave

Pigskin

1

5 I

I

6

I

6 9 4 4

T.

10

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. [ 1I] 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 matching fund\ from Martin-Marietta Electronic Systems

High-Technology

and Industry

Council,

with

REFERENCES Statistical and structural approaches to texture. fEEE Proc. 67(S), 786-804 (1979). ’ B. 13. Mandelbrot, The Frucrul Geometry of‘ Nurure. Freeman, New York (1977). ;’ A. I? Pentland, Fractal-based description of natural sciences. IEEE Truns. Pccttern Awl. Much. Intell. PAMI-6(6), 661 674 (1984). 4. M. C. Stem. Fractal image models and object detectlon. I’rsual Commun. Image Process. If Proc. SPIE 845, 293-300 (1987) S W. Ohle! and I. Lundham, Discrete 2-dimenslonal fractional Brownian motion as a model for medical images. Visual Commrrrr. Irnrcgc, Process. II Proc. SPIE 845, 221 ~232 (1987). 6. S. Dellepiane. S. B. Seprico. G. Vernazza and R. Viviani. Fractal-based image analysis in radiological applications. 1‘isuul C‘ommu~~.Image Procr.w. II. Proc. SPIE 845, 396 -403 (1987). 7. T Lundahi. W. Onley. S. Kay, H. White, D. Williams and A. Most, Texture analysis of diagnostic x-ray images by the use of fractals. Visual Commun. Image Process. Proc. SPIE 707, 23-30 (1986). X. G. G Medioni and Y. Yasumoto. A note on using the fractal dimension for segmentation. IEEE Computer Vision Work.vhvp, Annapolis Md.. 25~.30 (1984). 9. J. M. Keller. S. Chen and R. M. Crownover. Texture description and segmentation through fractal geometry. Computer I ‘kion. Gruphics Image Process. 45, I50- I66 (1989). IO. J. Garding, Properties of fractal intensity surfaces. Putrern Recognil. Lert. 8, 319-324 (1988). I I. F1.Kaneko. A generalized fractal dimension and its application to texture analysis. ICASSP Proc., I71 I-1714 (1989). 12. A. Ait-Kheddache and S. A. Rajala, Texture classification based on higher-order fractals. ICASSP Proc., 1112-II IS (198X) I.3 f’ .4rdulm, 5. Fioravanti and D. D. Giousto. Muitifractdls towards remote-sensing surface characterization. Proc. IEEE (‘WI/ /GA RSSOI. Espoo. Finland (1991). I4 J Jang and S. A. Rajala. Segmentation based image coding usmg fractals and the human visual system. KASSP Proc.. 1957 1960 (1990). Ii. I). Dubuc. C. R. Carmes. C. Trlco~ and S. W. Zucker. The variation method: a technique to estimate the fractal dimension of surfaces. b’i.yuul Commun. Imuge Process. II Proc. SPIE 845, 241-248 (1987). 16 Q (‘hen. 7i,.\-trrrc Segmentalion and Classifcarion Using Fracral and Energy Measures, M.S. Thesis, Univ. of Central FlorIda. School of Engng.. Orlando (1991)

1. K M. Harallck.

Texture

description

using fractal

and energy

features

17. B. Dubuc, M. Eng. Thesis, Dept of Electrical Engineering, McGill University (1987). 18. K. I. Laws, Texture energy measures. Proc. Image Understanding Workshop, 47-51 (1979). 19. M. Pietikainen, A. Rosenfeld and L. Davis, Experiments with texture classification using matches. IEEE Trans. Syst. Man Cybern. SMC-13, 421-426 (1983). 20. P. Brodatz, Textures: A Photographic Album for Artists and Designers. Dover, New York 21. J. Tou and R. Gonzalez, Pattern Recognition Principles. Addison-Wesley, Reading, Mass. 22. R. Liang, M. Shridhar and M. Ahmad, texture in images: algorithms for comparison and Elect. Engng. 16(2), 65-77 (1990).

AUTHOR

31

averages

of local pattern

(1966). (1974). segmentation.

Computers

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

Qing Chen

T. KASPARISer ul.

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