a new image texture segmentation based on contourlet fractal features
TRANSCRIPT
Arab J Sci EngDOI 10.1007/s13369-013-0624-z
RESEARCH ARTICLE - ELECTRICAL ENGINEERING
A New Image Texture Segmentation Based on ContourletFractal Features
Katayoon Sarafrazi · Mehran Yazdi ·Mohammad Javad Abedini
Received: 27 September 2011 / Accepted: 3 July 2013© King Fahd University of Petroleum and Minerals 2013
Abstract Texture segmentation is one of the most difficultand important tasks in the remote sensing image process-ing. Through the years, several methods, based on statisticalfeatures, structural features and other image features, havebeen proposed for texture segmentation. This paper proposesa new feature based on Contourlet transform and Fractalanalysis for texture segmentation. The performance of theproposed feature is compared with that of fractal, statistical,fuzzy and power spectrum features in presence of noise andthe obtained results show superiority of the new proposedfeature.
Keywords Texture segmentation · Fractal theory ·Contourlet transform · Statistical features · Fuzzy features ·Power spectrum features
K. Sarafrazi (B) · M. YazdiDepartment of Communications and Electronics,Shiraz University, Shiraz, Irane-mail: [email protected]
M. Yazdie-mail: [email protected]
M. J. AbediniDepartment Civil and Environmental Engineering,Shiraz University, Shiraz, Irane-mail: [email protected]
1 Introduction
Texture segmentation is the process of grouping image pixelsinto regions with similar texture. Using traditional methodsin texture segmentation often results in an over segmentedimage because these methods rely solely on individual pixelvalues. Indeed, their overall goal is to segment the imageinto regions with homogenous pixel values while a texturedregion might consist of many small homogenous regions(texture elements). Therefore, an efficient texture analysismethod should consider using both the pixel under consider-ation and its neighbouring pixel values to represent texture.
Texture analysis methods can be divided into four groups[1]: statistical methods (e.g., co-occurrence matrices [2],autocorrelation features [3,4]), geometrical methods (e.g.,Voronoi tessellations [5]), Model base methods (e.g., randomfields, fractals) and signal processing methods (e.g., spatialdomain filtering, Fourier domain filtering and wavelet basedmethods).
In recent years, fractal geometry has been one of the mostpopular tools in texture analysis. This has been due to itsrobustness in image scaling and also because of the corre-lation between fractal dimension and human interpretation
123
Arab J Sci Eng
of roughness [6]. Fractal geometry has been used in applica-tions like texture segmentation [7–10], texture classification[11–14], and texture interpolation [15].
Introduced by Mandelbrot [16,17], fractal geometry hasbeen used to model many complex natural phenomena [18].One of the key parameters in fractal geometry is fractaldimension (FD). In Euclidean geometry, dimension (E) isan integer number (one for line, two for surface, etc.) whilein fractal geometry, dimension is a continuous parameter. Asan example, for a surface, the fractal dimension can be anyreal number between two and three. In general,
E ≤ FD < E + 1 (1)
Pentland [6] showed that there is a correlation between asurface’s roughness and its fractal dimension. In other words,the more complex the surface is, the higher will be its fractaldimension.
On the other hand, studies showed that a single fractaldimension cannot be used to distinguish different textures [8,13,17], so using other fractal parameters (e.g. lacunarity [11,13], fractal error [19], multifractals [7,9,10]) or combiningfractal analysis with other methods for texture segmentationand classification seems inevitable and useful.
Chaudhuri et al. [20] used fractal dimension of an originalimage along with the fractal dimension of five images derivedfrom the original image as texture descriptors. Betti et al.[21] and Marazzi et al. [22] combined wavelet transform andfractal analysis for texture segmentation. Charalampidis etal. [23] used fractal dimension of filtered versions of theoriginal image using directional Gabor filters as features forsegmentation.
Because Contourlet transform-based methods [24] use apyramidal directional filter bank to extract image details atdifferent scale and directions, in this paper, we use the fractaldimension of these detail images for texture segmentation.Simulation results show that a significant improvement hasbeen achieved by this method compared with other texturesegmentation methods.
2 Fractal Analysis
Natural objects rarely have simple shapes like circles, spheresand cones, and as a result, classic Euclidean geometry isunable to describe most of them. Mandelbrot introduced frac-tal geometry to describe such natural phenomena.
In 1968, Mandelbrot [25] showed that natural surfaces canbe modeled using the two-dimensional fractional Brownianmotion. Fractional Brownian motion (fBm) [26] is a gen-eralization of Brownian motion which is used to describemovements of a particle that is colliding to other particles inits surrounding environment. Pentland [6] argued that “anyprocess that acts locally to produce a permanent change in
shape will, after innumerable repetitions, result in a fractalsurface”. He also showed that under isotropic lightning con-ditions, intensity image of a fBm surface can also be modeledusing fBm.
Fractional Brownian motion is a zero-mean Gaussianincrement process which is described using Hurst parameter(0<H <1) as follows. Let BH(t) be a fBm signal then:
E {BH(t2) − BH(t1)} = 0 (2)
var {BH(t2) − BH(t1)} = σ 2|t2 − t1|2H (3)
For a signal in higher dimension (like an image), eq. (3)can be rewritten as (X is a vector containing E independentvariables):
var {BH(X2) − BH(X1)} = σ 2|X2 − X1|2Ha (4)
⇒ log(var {BH(X2) − BH(X1)})= k + 2H log(|X2 − X1|) (5)
So if we plot log(var {BH(X2) − BH H(X1)}) versuslog(|X2 − X1|) and fit a least square error regression linethrough data, slope of the line will be an estimation of 2H.Fractal dimension can be calculated via the following equa-tion:
FD = E + 1 − H (6)
The afore-mentioned method which directly estimates FDhas a high computational cost and, therefore, other methodsfor FD estimation have been proposed. In the next session, wedescribe the differential box counting (DBC) and triangularprism method which will be used in this paper to estimateFD.
2.1 Differential Box Counting (DBC)[27]
As proposed by Sarkar and Chaudhuri [28], DBC is one ofthe most popular fractal dimension estimation methods. Thismethod is accurate and also computationally efficient.
Let I (x, y) be an M×M image which is portioned into as×s grid, s be an integer which satisfies 1 < s ≤ M/2, and rbe a ratio defined as r = s/M . If G is the grey level range ofthe image, s′ = rG. We also partition grey level range of animage into intervals of length s′ and assign an integer numberto each interval. considered as an integer corresponding tointerval associated with the maximum grey level value of(i, j)th grid. Subsequently, ymin is an integer associated withthe minimum grey level value of (i, j)th grid. Hence,
nr (i, j) = ymax − ymin + 1 (7)
123
Arab J Sci Eng
Fig. 1 Triangular Prism method [27]
Taking all grids into account, we have:
Nr =∑
i, j
nr (i, j) (8)
For different values of r, Nr is calculated and a least squareerror line is fit to log(Nr )versus log(1/r) and if slope of thisline is considered as β, then FD = −β.
2.2 Triangular Prism
Triangular prism was first proposed by Clarke [29] andimproved by Lam [30]. In this method, an image is parti-tioned into δ × δ squares (see Fig. 1) and for (i,j)th square,we take the position of corners (a, b, c, d) and calculate thecentre point (e). Si, j is defined as the top area of the prismwhich is formed by raising corner and centre points to theirgrey level value (Si, j = A + B + C + D in Fig. 1). Totalarea is: Sδ = ∑
i, j Si, j .If we plot log(Sδ) versus log(δ) and fit a least square line
with slope β through the data, then FD = 2 − β.Both DBC and triangular prism methods are based on a
log-log plot and on fitting a least square error line throughthe data. Chenoweth et al. [19] proposed using mean squareerror of this fitting line (fractal error) as a feature for texturesegmentation.
3 Contourlet Transform
Very recently, wavelet transform has been widely used insignal processing. The reason for its success is that it gives
a good nonlinear approximation for piecewise smooth func-tions in one dimension [31]. However, as two-dimensionalwavelet is tensor product of one-dimensional wavelet, itcan only model zero-dimension discontinuities (points). Onthe other hand, in an image, edges are considered to beone-dimensional discontinuities and thus cannot be modeledusing wavelets.
As developed by Do and Vetterli [24], Contourlet trans-form is a “true two-dimensional representation of imageswhich can capture the intrinsic geometrical structure inher-ent in visual information” [32].
In this transform, firstly a multiscale filter bank like Lapla-cian is used to perform the pyramid decomposition. On eachdecomposition level, the prediction residual image is passedthrough a directional filter bank to calculate details in differ-ent directions.
Figure 2 shows the system for Laplacian pyramid decom-position. On each decomposition level, firstly the image ispassed through a low pass filter and the output is scaled downto generate a low-pass approximation of the input image.Then, the approximation image is scaled up and a band passimage is created by subtracting the up sampled approxima-tion image from the original image.
In Contourlet transform, at each level, firstly the Laplaciandecomposition is performed and then the residual image ispassed through a directional filter bank which extracts detailsin different directions. The approximation image becomesthe input for the next level (see Fig. 3).
In our work, ladder filters developed by Phoong et al. [33]are used for both Laplacian decomposition and directionalfiltering.
4 Materials and Methods
To evaluate the impact of using different features on textureimages, we create a supervised classifier using differentfeatures to segment several texture mosaics from Brodatzalbum [34] with different shapes (see Fig. 5) and aerial pho-tos of Shiraz city with 25 cm/pixel resolution (see Fig. 6).Sizes of all images are 512 × 512 pixels. A three-layer
Fig. 2 Laplaciandecomposition
123
Arab J Sci Eng
Fig. 3 Contourlet transform
Fig. 4 Hyperbolic tangent sigmoid transfer function
artificial neural network with 30 neurons in hidden layer wasused as our classifier (see Fig. 7). Hyperbolic tangent sig-moid transfer function was used for hidden and output layers(see Fig. 4).
Segmentation was also performed in the presence ofGaussian and speckle noises with different variances.
Nine different feature sets based on fractal analysis,Contourlet transform, power spectrum, fuzzy features andstatistical features were tested and compared. The follow-ing sections give a detailed explanation on each featureset.
4.1 Fractal Features
Fractal geometry is one of the most popular tools in textureanalysis, although studies have shown that a single fractaldimension is not enough for discriminating among differ-ent textures, so we use fractal dimension and fractal error oforiginal image (I1), high-grey valued image (I2), low-greyvalued image (I3), horizontally smoothed image (I4), verti-cally smoothed image (I5) and inverse image (I6) [22] usingboth triangular prism and DBC method in our work. Equa-
Fig. 5 Sample texture mosaics from Brodatz album
Fig. 6 Sample aerial photos of Shiraz city, Iran
Fig. 7 Neural networkarchitecture
123
Arab J Sci Eng
tions (8) through (12) show how these images are derivedfrom original image.
I2 (i, j) = I1 (i, j) − L1 if I (i, j) > L1
= 0 otherwiseL1 = gmin + gav/2
(9)
I3(i, j) = I1(i, j) − L2 if I (i, j) > L2
= I1(i, j) otherwiseL2 = gmax − gav/2
(10)
I4(i, j) = 1
2w + 1
w∑
k=−w
I (i, j + k) (11)
I5(i, j) = 1
2w + 1
w∑
k=−w
I (i + k, j) (12)
I6 = 255 − I (13)
Feature set 1 contains fractal dimension and fractal errorof images I1 through I6 using triangular prism while featureset 2 consists of fractal dimension and fractal error of imagesusing DBC method. Feature set 3 is a combination of featuresets 1 and 2.
4.2 Statistical Features
To extract statistical features for each pixel, a moving windowaround that pixel is used. Feature set 4 contains minimum,maximum, average, standard deviation, entropy, energy,Moran’s I [3] and Geary’s C [4] of pixel values inside thewindow. Moran’s I and Geary’s C are two autocorrelationfeatures which are calculated using Eqs. (14) and (15).
MI (d) = n∑
i∑
j wi j zi z j
W∑
i z2i
(14)
GC(d) = (n − 1)∑
i∑
j wi j (Ii − I j )2
2W∑
i z2i
, (15)
where n is the number of pixels inside the window, Ii is theith pixel value, Iav is the average of pixel values inside thewindow, zi = Ii − Iav, wi j = 1 if both pixels i and j areinside the window and wi j = 0 otherwise W = ∑
i j wi j .
4.3 Fuzzy Features
Hanmandlu et al. [35] proposed a set of fuzzy features fortexture segmentation and claimed it to be comparable to frac-tal features in images with few textures. We use a modifiedversion of these features in our study.
To extract fuzzy features for each pixel, a moving windowof size ws×ws is considered around the pixel. Then, for eachpixel inside the window a membership function is defined:
μ j (i) = exp
{−
(Ii − I j
ws
)2}
, (16)
where Ii is the grey level of the current pixel and I j is thegrey level of a neighbouring pixel inside the window. Next,the cumulative response of the current pixel is calculated as:
yi =∑
j
μ j (i).I j∑j μ j (i)
(17)
This process is repeated for all pixels in the window. D isdefined as the ws×ws image containing yi values. Han-mandlu et al. [35] used maximum entropy and uniformityof elements of D as useful features. In our work, we also useminimum, average and energy of elements of D. The addi-tional statistical parameters enhance overall performance ofthe segmentation algorithm.
4.4 Power Spectrum Features
To calculate power spectrum features, firstly we calculateFourier power spectrum using |F(u, v)|2 where F(u, v) isFourier transform of ws×ws subimage centred on the currentpixel. Next, we divide power spectrum space into 11 regionsas shown in Fig. 8.
The energy computed in each shaded band is a texturefeature indicating coarseness/fineness (for a coarse textureenergy is mostly concentrated in higher frequencies), and theenergy computed in each edge is a texture feature indicatingdirectionality. Energy of these regions is used as the featureset 6.
Fig. 8 Power spectrum space division for feature extraction a energy ofeach band indicates coarseness/fitness, b power spectrum space divisionfor extracting texture directionality
123
Arab J Sci Eng
Fig. 9 System forContourlet-fractal featureextraction
Table 1 Feature sets
Features Number offeatures
Set 1 Fractal dimension and fractal error of I1. . .I6 using triangular prism 12
Set 2 Fractal dimension and fractal error of I1. . .I6 using DBC 12
Set 3 Fractal dimension and fractal error of I1. . .I6 using triangular prism and DBC 24
Set 4 minimum, maximum, average, standard deviation, entropy, energy, Moran’s I andGeary’s C of pixel values
8
Set 5 minimum, maximum, average, standard deviation, entropy and energy of D 6
Set 6 Energy of 11 power spectrum features 11
Set 7 Contourlet decomposition result images 33
Set 8 Fractal dimension of Contourlet decomposition results 33
Table 2 Segmentation accuracy
Set 1 Set 2 Set 3 Set 4 Set 5 Set 6 Set 7 Set 8 Pixel value
Texture mosaics 90.74 % 94.22 % 94.01 % 88.31 % 87.60 % 88.07 % 69.54 % 98.89 % 90.16 %
Aerial photos 87.13 % 90.58 % 92.50 % 87.89 % 85.28 % 89.21 % 75.42 % 94.90 % 89.04 %
4.5 Contourlet Features
To extract these features, we perform a one level Contourletdecomposition in five directions. This will result in oneapproximation and 32 detail images containing Contourletcoefficients. Next, we resize all of these images to originalimage size using nearest neighbour method. Pixel value ofthe detail images along with approximation image is used asContourlet features.
4.6 Contourlet-Fractal Features
To extract Contourlet-fractal features, one level Contourletdecomposition using five directional filters is performedwhich will produce one approximation and 32 detail imagescontaining Contourlet coefficients. Next, using nearest neigh-bour method, all images are resized to original image size.Then, using a moving window of size ws×ws, local fractaldimension of each of these images is calculated by apply-ing DBC. The local fractal dimensions are considered as thefeature set 8 (see Fig. 9).
5 Experimental Results and Discussion
Supervised segmentation of 15 texture mosaics from Bro-datz album and five high resolution remote sensing images
using eight different feature sets (see Table 1) were tested.Segmentation was performed using the grey level value ofimages.
To evaluate different features, we performed a supervisedsegmentation in the presence of Gaussian and speckle noiseswith different variances. Window size of 35 was used forall methods. This window size was selected based on testresults obtained from trying different sizes. Table 2 shows theaverage segmentation accuracy (i.e., percentage of correctlyclassified pixels) for each feature set. Figures 10 through 13show the average segmentation accuracy for each feature setin the presence of speckle and Gaussian noises with differ-ent variances. Figures 14, 16, 18 and 20 show segmentationresults for three sample images. Figures 15, 17, 19 and 21show receiver operating characteristic (ROC) curves for seg-mentation results of Figs. 14, 16, 18 and 20.
As we expected, the image segmentation based on pixelvalues leads to poor results and this is due to the presence oftexture in the images and pixel value’s inability to representthe texture. Fractal and statistical features lead to accept-able results, but Contourlet-fractal features give rise to bettersegmentation results than other tested methods. This is dueto Contourlet’s ability to extract image details in differentdirections and fractal dimension’s correlation with texturecomplexity.
Segmentation based on statistical features and pixel valueseems to be very vulnerable to noise while fractal and
123
Arab J Sci Eng
0 0.1 0.2 0.3 0.4 0.50.6
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.6
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.6
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.6
0.7
0.8
0.9
1
Noise varianceS
egm
enta
tion
Acc
urac
y0 0.1 0.2 0.3 0.4 0.5
0.84
0.86
0.88
0.9
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.6
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.6
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.50.75
0.8
0.85
0.9
0.95
Noise variance
Seg
men
tatio
n A
ccur
acy
Set 4 Set 5 Set 6
Set 7 Set 8 Set 9
Set 1 Set 2 Set 3
Fig. 10 Average segmentation accuracy versus Gaussian noise variance for texture mosaic images
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
Set 4 Set 5 Set 6
Set 7 Set 8 Set 9
Set 1 Set 2 Set 3
Fig. 11 Average segmentation accuracy versus speckle noise variance for texture mosaic images
123
Arab J Sci Eng
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise varianceS
egm
enta
tion
Acc
urac
y0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
Set 4 Set 5 Set 6
Set 7 Set 8 Set 9
Set 1 Set 2 Set 3
Fig. 12 Average segmentation accuracy versus Gaussian noise variance for aerial photos
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
0 0.1 0.2 0.3 0.4 0.5
0.7
0.8
0.9
1
Noise variance
Seg
men
tatio
n A
ccur
acy
Set 4 Set 5 Set 6
Set 7 Set 8 Set 9
Set 1 Set 2 Set 3
Fig. 13 Average segmentation accuracy versus speckle noise variance for aerial photos
123
Arab J Sci Eng
Fig. 14 Segmentation resultsfor a sample mosaic image usingdifferent methods: a originalimage, b hand segmentedimage, c triangular prism result,d DBC result, e fractal result,f statistical features result,g fuzzy features result, h powerspectrum result, i Contourletresult, j Contourlet-fractalresult, k pixel value result
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k)
Fig. 15 ROC curves forsegmentation results of Fig. 12
123
Arab J Sci Eng
Fig. 16 Segmentation resultsfor a sample mosaic image usingdifferent methods: a originalimage, b hand segmentedimage, c triangular prism result,d DBC result, e fractal result,f statistical features result, gfuzzy features result, h powerspectrum result, i Contourletresult, j Contourlet-fractalresult, k pixel value result
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k)
Fig. 17 ROC curves forsegmentation results of Fig. 14
123
Arab J Sci Eng
Fig. 18 Segmentation resultsfor a sample aerial photo usingdifferent methods: a originalimage, b hand segmentedimage, c triangular prism result,d DBC result, e fractal result,f statistical features result,g fuzzy features result, h powerspectrum result, i Contourletresult, j Contourlet-fractalresult, k pixel value result
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k)
Fig. 19 ROC curves forsegmentation results of Fig. 16
123
Arab J Sci Eng
Fig. 20 Segmentation resultsfor a sample aerial photo usingdifferent methods: a originalimage, b hand segmentedimage, c triangular prism result,d DBC result, e fractal result,f statistical features result, gfuzzy features result, h powerspectrum result, i Contourletresult, j Contourlet-fractalresult, k pixel value result
(a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k)
Fig. 21 ROC curves forsegmentation results of Fig. 18
123
Arab J Sci Eng
Contourlet fractal features are much more robust and resis-tive to noise.
6 Conclusions
In this paper, we have proposed a set of Contourlet-fractalfeatures for texture image segmentation. The performance ofthe proposed features on image segmentation has been com-pared and contrasted with other features used in earlier tex-ture segmentation methods. Results have demonstrated thesuperior performance of new features in efficient segmenta-tion of texture images.
References
1. Tuceryan, M.; Jain, A.K.: Texture analysis. In: Chen, C.H.; Pau,L.F.; Wang, P.S.P. (eds.) The Handbook of Pattern Recognitionand Computer Vision, 2nd edn, Chap. 2.1, pp. 207–248. WorldScientific, Singapore (1998)
2. Haralick, R.M.; Shanmugam, K.; Dinstein, I.: Textural Features forImage Classification. IEEE Trans. Sys. Man Cybern. 3, 610–621(1973)
3. Moran, P.A.P.: Notes on Continuous Stochastic Phenomena. Bio-metrika 37, 17–23 (1950)
4. Geary, R.C.: The Contiguity Ratio and Statistical Mapping. Incorp.Statistician 3, 115–145 (1954)
5. Ahuja, N.: Dot Pattern Processing Using Voronoi Neighborhoods.IEEE Trans. Pattern Anal. Mach. Intell. 4, 336–343 (1982)
6. Pentland, A.P.: Fractal-based descriptions of natural scenes. IEEETrans. Pattern Anal. Mach. Intell. 5, 661–674 (1984)
7. Du, G.; Yeo, T.S.: A Novel Multifractal Estimation Method and ItsApplication to Remote Image Segmentation. IEEE Trans. Geosci.Remote Sens. 40, 980–982 (2002)
8. Myint, S.W.: Fractal approaches in texture analysis and classifica-tion of remotely sensed data: comparisons with spatial autocorre-lation techniques and simple descriptive statistics. Int. J. RemoteSens. 24, 1925–1947 (2003)
9. Abadi, M.; Grandchamp, E.: Texture Features and SegmentationBased on Multifractal Approach. In: Progress in Pattern Recogni-tion, Image Analysis and Applications (CIARP). Cancun Mexico,pp. 297–305 (2006)
10. Xia, Y.; Feng, D.; Zhao R.: Morphology-Based Multifractal Esti-mation for Texture Segmentation. IEEE Trans. Image Process. 14,614–623 (2006)
11. Myint, S.W.; Lam, N.: A study of lacunarity-based texture analysisapproaches to improve urban image classification. Comput. Envi-ron. Urban Sys. 29, 501–523 (2005)
12. Sankar, D.; Thomas, T.: Fractal Features based on Differential BoxCounting Method for the Categorization of Digital Mammograms.Int. J. Comput. Info. Sys. Ind. Manag. Appl. bf 2, 11–19 (2010)
13. Dong, P.: Lacunartity for Spatial Heterogeneity Measurement inGIS. Geograph. Info. Sci. 5, 20–26 (2000)
14. Dekker, R.J.: Texture Analysis and Classification of ERS SARImages for Map Updating of Urban Areas in The Netherlands.IEEE Trans. Geosci. Remote Sens. 41, 1950–1958 (2003)
15. Barnsley, M.: Fractal functions and interpolation. Constr. Approx.2, 303–329 (1986)
16. Mandelbrot, B.B.: Fractals: Form, Chance and Dimension. W.H.Freeman and Company, San Fransisco (1977)
17. Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H. Freemanand Company, New York (1982)
18. Abedini, M.J.; Shaghaghian, M.R.: Exploring scaling laws in sur-face topography Chaos. Solitons Fractals 42, 2373–2383 (2009)
19. Chenoweth, D.L.; Cooper, B.E.; Selvage, J.E.: Aerial Image Analy-sis Using Fractal-Based Models. In: Proceedings of the AerospaceApplications Conference, pp. 277–285 (1995)
20. Chaudhuri, B.B.; Sarkar, B.: Texture Segmentation Using Frac-tal Dimension. IEEE Trans. Pattern Anal. Mach. Intell. 16, 72–77(1995)
21. Betti, A.; Barni, M. Mecocci, A.: Using a Wavelet—Based FractalFeature to Improve Texture Discrimination on SAR Images. In:Proceedings of the International conference on image processing,pp: 251–254 (1997)
22. Marazzi, A.; Gamba, P.; Mecocci, A.; Costamagna, E.: A MixedFractal Wavelet Based Approach For Characterization of TexturedRemote Sensing Images. Geosci. Remote Sens. 2, 655–657 (1997)
23. Charalampidis, D.; Kasparis, T.; Rolland, J.P.: Segmentation oftextured images based on multiple fractal feature combinations.In: Park, S.K.; Juday, R.D. (eds.) Visual information processingVII, Proceedings of SPIE, vol. 3387, pp. 25–35 (1998)
24. Minh, N.D.: Directional Multiresolution Image Representations.PhD Thesis, Department of Communication Systems, Swiss Fed-eral Institute of Technology (2001)
25. Mandelbrot, B.B.: Stochastic Models for the Earth’s Relief, theShape and the Fractal Dimension of the Coastlines, and theNumber-Area Rule for Islands. In: Proceedings oj’tlze NationalAcademy of Sciences, USA, vol. 72. pp. 3825–3828 (1975)
26. Mandelbrot, B.B.; Van Ness, J.W.: Fractional Brownian Motions,Fractional Noises and Applications. SIAM Rev. 9, 422–437 (1968)
27. Sun, W.; Xu, G.; Gong, P. Liang, S.: Fractal analysis of remotelysensed images: A review of methods and applications. Int. J.Remote Sens. 27, 4963–4990 (2006)
28. Chaudhuri, B.B.; Sarkar, B.; Kundu, P. Improved fractal geometry-based texture segmentation technique. IEEE Proceedings E 140,233–241 (1993)
29. Clarke, K.C.: Computation of the fractal dimension of topographicsurfaces using the triangular prism surface area method. Comput.Geosci. 11, 713–722 (1986)
30. Lam, N.S.; Qiu, H.L.; Quattrochi, D.A.; Emerson, C.W. An eval-uation of fractal methods for characterizing image complexity.Cartogr. Geogr. Info. Sci. 29, 25–35 (2002)
31. Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Acad-emic Press, London (1999)
32. Do, M.N.; Vetterli, M.: The Contourlet Transform: An EfficientDirectional Multiresolution Image Representation. IEEE Trans.Image Process. 13, 2091–2106 (2005)
33. Phoong, S.M.; Kim, C.W.; Vaidyanathan, P.P.; Ansari R.: A newclass of two-channel biorthogonal filter banks and wavelet bases.IEEE Trans. Signal Process. 43, 649–665 (1995)
34. Brodatz, P.: Textures: A Photographic Album for Artists andDesigners. Dover, New York, 1966
35. Hanmandlu, M.; Madasu, V.K.; Vasikarla, S.: A Fuzzy Approachto Texture Segmentation. In: Proceedings of the InternationalConference on Information Technology: Coding and Computing,pp. 636–642 (2004)
123