Journal of Computational Methods in Sciences and Engineering 14 (2014) 219–225 DOI 10.3233/JCM-140498 IOS Press 219 Gabor transform applied to segmentation and skeletonization of digital images Ronald Pereza,d,∗ , William Lassob,d , Carlos Jimenezc , Lorenzo Mattosd and Cesar O. Torresd a Guajira University, Riohacha, Colombia University of Cesar, Cesar, Colombia c Applied Mathematics Group (GIMA), Guajira University, Riohacha, Colombia d Optics and Informatics Group (LOI), Popular University of Cesar, Cesar, Colombia b Popular Abstract. This paper implements a digital algorithm that allows the segmentation and skeletonization of digital images, this method is particularly important in the processing of retinographies where fundus analysis enables early detection of diseases such as diabetic retinopathy. The algorithm implemented using the Fourier transforms of two-dimensional Gabor and with the aid of conventional filters. The performance of the algorithm implemented under the Matlab mathematical platform has been compared with conventional methods resulting in obtaining filtered images with better contrast and higher resolution. Keywords: Gabor transform, segmentation, skeletonizer 1. Introduction The Gabor function has been used extensively in the texture segmentation [1] and defect detection [2, 3], as it can be tuned to a specific frequency and orientation, and achieve both localization in the time domain and frequency domain; two-dimensional (2-D) Gabor filters are multi-channel filters with a multiresolution decomposition process, which is similar to wavelet analysis. Time-frequency analysis, such as the Gabor transform, plays an important role in many signal processing applications. In harmonic analysis the algorithms of Fourier transformation are often applied [4,5]. However, this method gives significantly better results for stationary systems where frequency composition does not change during the investigation. Gabor filters have proved to be particularly useful in analyzing texture images containing highly specific frequency or orientation characteristics [6,7]. Research work on image extraction of fabrics has been extensive in the past few years. Its major application has been mainly in the area of automatic inspection. A fair amount of research papers have been published in literature in the field of the image processing based on the Gabor filter [8,9]. Besides face recognition, Gabor filters are successfully used in many other image processing and analysis domains, such as: image smoothing, image coding, texture analysis, shape analysis, edge detection, fingerprint and iris recognition. The Gabor function has been used extensively in defect detection [4] and texture segmentation [7], as it can be tuned to a specific frequency and orientation, and achieve both localization in the time domain and frequency domain. Thus it is a good tool to extract the slubs which can be regarded as the oriented texture in the slub yarn. For ∗ Corresponding author: Ronald Perez, Postgraduate Physics Student, Guajira University, Riohacha, Colombia. E-mail: [email protected]. c 2014 – IOS Press and the authors. All rights reserved 1472-7978/14/$27.50 220 R. Perez et al. / Gabor transform applied to segmentation and skeletonization of digital images realizing the automation of slub extraction. In this research, we have implemented a hybrid method of segmentation and skeletonization of images that have multiply object oriented, the procedure is based on the use of conventional filters, then using Gabor filter and finally the skeletonization of the image; This leads to the digital implementation of an automatic method of image segmentation, demonstrating the applicability of these filters in pattern recognition. Finally, the contribution of this work is the development of a digital algorithm and Matlab mathematical platform which becomes a powerful tool to aid in the process of segmented signal skeletonization. 2. Gabor wavelet transform Gabor transform was first defined by Gabor [10–12] and later extended to 2-D by Daugman [3]. In the spatial domain, the Gabor function is a Gaussian curve with aspect ratio modulated by a complex sinusoid; the Gabor filter was originally introduced by Dennis Gabor (Gabor, 1946). The one-dimensional Gabor filter is defined as the multiplication of a cosine/sine (even/odd) wave with a Gaussian windows as follows: ge (x) = √ x2 1 e− 2σ2 cos (2πω 0 x) 2πσ (1) x2 1 e− 2σ2 sin(2πω 0 x) (2) 2πσ Where ω0 defines the centre frequency (i.e., the frequency in which the filter yields the greatest response) and σ the spread of the Gaussian window. Daugman extended the Gabor filter to two dimensions as follows: go (x) = √ ge (x, y) = 1 − e 2πσ x σy go (x, y) = 1 − e 2πσ x σy 2 x2 σx + σy x2 σx + σy y 2 y cos (2πω x0 x + 2πω y0 y) (3) sin (2πω x0 x + 2πω y0 y) (4) Where (ωx0 , ωy0 ) defines the central frequency and (σx , σy ) the (potentially asymmetric) spread of the Gaussian window. 2 x−x0 0 −π ( x−x ) b =e ; the 2-D Gabor Transform; can be expressed as: Taking account that: Gaus b √ √ v u πx0 , πy0 ; √ , √ = Gf π π √ √ x − x0 y − y0 −iπ(ux0 +vy 0 ) Gaus e (5) f πx, πy πF Gaus 2 2 √ √ The Gabor transform can, of course, be considered as an inner product of the signal f ( πx, πy) and a shifted and modulated version of the analysis window. Note, however, that the Gabor y−ycan x−x0 transform also be considered as a sampled version of the windowed Fourier transform {Gaus 2 Gaus 2 0 √ √ e−iπ(ux0 +vy 0 ) } of the signal f ( πx, πy). This is a Fourier transform of a product of two functions, furthermore, in accord with the scaling factors by a properly scaled Fourier transform and taking account R. Perez et al. / Gabor transform applied to segmentation and skeletonization of digital images 221 Fig. 1. Bidimentional Gabor filter a specific frequency and orientation. Fig. 2. Algorithm’s block diagram. of scaling factor x → xa ; x0 → xa0 ; y → yb ; y0 → yb0 and coordinate inversions u → au; v → av , from the scaling factors present in the equation above we find that: √ x0 √ y0 au bv Gf π , π ;√ ,√ = a b π π √ x √ y x − x0 y − y0 −iπ(ux0 +vy 0 ) Gaus e (6) f π , π πF Gaus 2a 2b a b Which describes how such entities behave under scaling of coordinates. The Gabor filter (Gabor Wavelet) represents a band-pass linear filter whose impulse response is defined by a harmonic function multiplied by a Gaussian function, see Fig. 1. Thus, a bidimentional Gabor filter constitutes a complex sinusoidal plane of particular frequency and orientation modulated by a Gaussian envelope [13]. It achieves an optimal resolution in both spatial and frequency domains. 222 R. Perez et al. / Gabor transform applied to segmentation and skeletonization of digital images Fig. 3. Original image converted in gray-scale form. Fig. 4. Digital Simulation results using Gabor filter. 3. Methods and materials In algorithms based on time-frequency transforms such as the Gabor representation, there is often a tradeoff between performance and computational complexity, which can be controlled by adjusting the redundancy of the transform. Inspired by the multichannel operation of the Human Visual System (HVS), the proposed algorithm for the detection of frequency hopping signals, perform processing of the signal segment in four stages: In the first stage, the image and the Gabor transform are defined; in the second stage the Fourier transform is computed for the image and Gabor surface; in the next stage, the convolution of the Fourier transforms is developed, and finally in the next stage the Gabor transform of the image is allowed and the skeletonization of image is finally obtain. Block diagram of the algorithm is presented in Fig. 2. In the Gabor Transform, the Fast Fourier transform (FFT) is used for computation of the biorthogonal function and its multiplication complexity is of O(Nlog2N). Using the FFT, we find that the overall complexity of the Gabor Transform is O(NlogN). In this paper we propose an efficient Gabor expansion algorithm which replaces the FFT with the Arithmetic Fourier Transform (AFT). The algorithm reduces the overall multiplication complexity to O(N). 4. Image pre-processing In the first stage, some image pre-processing operations may be necessary [14]. First, the original images have to be converted to the grayscale form (see Fig. 3). Then, some contrast and illumination ad- R. Perez et al. / Gabor transform applied to segmentation and skeletonization of digital images 223 Fig. 5. Enhanced vessels in Gabor response compared with the conventional filters. justment operations are performed with the help of histograms techniques. All original images must be processed with the same illumination and contrast. Therefore, some histogram equalization operations are performed on these images, to obtain a satisfactory contrast [15,16]. Using Matlab platform was implemented mathematical computational algorithm that enables image segmentation and skeletonization, the numerical algorithm shown in Fig. 2. 5. Digital results Figure 4 shows the results of digital simulation, showing that Gabor filter effectively becomes a powerful tool for digital image processing particularly if it wants to develop filter edges and contours. 6. Fast retinal vessel tree extraction Early ocular disease diagnosis is an important field in medical research, manual diagnosis is usually performed by analyzing the images from a patient, as not all images show signs of diabetic retinopathy, it increases the time and decreases the efficiency of the analysis realized by ophthalmologists. From the image processing point of view, many strategies and algorithms have been developed to deal with the extraction optimal of the retinal vessel tree. In clinical ophthalmology with the help of the color images of the retinal, using the digital fundus camera are widely used for detection and diagnosis of diseases like diabetic retinopathy, hyper-tension and various vascular disorders. Therefore, an automatic segmentation of the vasculature could save workload of the ophthalmologists and may assist in characterizing the detected lesions and to identify false positives. The Gabor filters are widely applied to image processing and computer vision; since, the vessels in the retinal image are connected and piecewise linear, for their segmentation Gabor filters are better suited as they are capable of detecting oriented features and can be fine tuned to specific frequencies, see Fig. 5. As vessels detection is a fundamental step in Gabor response, it is necessary to point out the true vessels to get the best results from the matching process. In Fig. 5 we observed some advantages of 224 R. Perez et al. / Gabor transform applied to segmentation and skeletonization of digital images vessels detection using the Gabor filter: simplicity, detection of vessels and their orientations, having fixed characteristics in all directions, finding the correct places of vessels, testing wider area around the pixel. Gabor vessels detection algorithm is computationally less expensive compared to horizontal and vertical edges, laplacian, relief, sharpen, sobel, and prewitt operator. The Gabor filter detection algorithm performs better than all these operators under almost all scenarios, and finally better vessels detection specially in comparison with conventional filters 7. Conclusions In this paper, we presented, a detection algorithm using Gabor filter bank, and one efficient method for automatic segmentation of retinal blood vessels has been presented; the method choose the best Gabor filter in the time domain and the frequency domain, respectively, to filter the testing image to obtain the filtrated image. Images from three different datasets are used to evaluate the robustness and accuracy of the method, demonstrating that it may be useful in a wide range of retinal images. This method can be used in computer analyses of retinal images, e.g., in automated screening for diabetic retinopathy. The system is based on extraction of image ridges, which coincide approximately with vessel centerlines. Based on a brief comparison with some other classical vessel segmentation algorithms, we can conclude that the Gabor filter provides a better segmentation. References [1] Hoover, V. Kouznetsova and M. 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