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Fast algorithm for hybrid region‐based active contours optimisation

2016; Institution of Engineering and Technology; Volume: 11; Issue: 3 Linguagem: Inglês

10.1049/iet-ipr.2016.0648

1751-9667

Yamina Boutiche, Abdelhamid Abdesselam,

Image Processing Techniques and Applications

IET Image ProcessingVolume 11, Issue 3 p. 200-209 Research ArticleFree Access Fast algorithm for hybrid region-based active contours optimisation Yamina Boutiche, Corresponding Author Yamina Boutiche bouticheyami@gmail.com Image and Signal Processing Laboratory, Research Center in Industrial Technologies CRTI, ex CSC, B. P. 64, Cheraga, Algiers, Algeria Département d'Electronique, Faculté de Technologie, Université Saad Dahlab de Blida, BP 120, Route de Soumaa, 09000 Blida, AlgérieSearch for more papers by this authorAbdelhamid Abdesselam, Abdelhamid Abdesselam Department Computer Science, Sultan Qaboos University, Muscat, OmanSearch for more papers by this author Yamina Boutiche, Corresponding Author Yamina Boutiche bouticheyami@gmail.com Image and Signal Processing Laboratory, Research Center in Industrial Technologies CRTI, ex CSC, B. P. 64, Cheraga, Algiers, Algeria Département d'Electronique, Faculté de Technologie, Université Saad Dahlab de Blida, BP 120, Route de Soumaa, 09000 Blida, AlgérieSearch for more papers by this authorAbdelhamid Abdesselam, Abdelhamid Abdesselam Department Computer Science, Sultan Qaboos University, Muscat, OmanSearch for more papers by this author First published: 04 January 2017 https://doi.org/10.1049/iet-ipr.2016.0648Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Active contours are usually based on the optimisation of energy functionals that are built to attract the curve towards the objects' boundaries. This study describes a hybrid region-based active contours technique that uses global means to define the global fitting energy and local means and variances to define the local fitting energy. The optimisation of the functional is performed by applying a sweeping-principle algorithm, which avoids solving any partial differential equation and removes the need for any stability conditions. Furthermore, sweeping-principle algorithm is not based on the computation of derivatives, which allows using a binary level set function during the minimisation process instead of the signed distance function, consequently this removes the need for the distance regularisation term, avoiding its subtle side effects and speeding up the optimisation process. Successful and accurate segmentation results are obtained on synthetic and real images with a significant gain in the CPU execution time when compared with the minimisation via the commonly used gradient descent method. 1 Introduction Image segmentation and restoration are two primordial tasks for computer vision applications. Image segmentation consists of partitioning an image into segments in order to make it more meaningful and easier to analyse. It is usually used to locate objects' boundaries and/or regions in the image. Image restoration has the goal to undo or at least reduce the degradations affecting the image, such as noise and blur. Implicit active contour techniques have been successfully used to perform these two tasks. The main common point of active contour techniques is the optimisation of an energy functional to get objects' contours and/or smooth version of the original image (cartoon image). The original and famous model for image segmentation is the one proposed by Mumford and Shah [1]. It is expressed by the following equation: (1)where are constants introduced to control the contribution of the different terms. is the image plane, is the given original image, u is an optimal approximation of and is the curve's length. The optimisation of the model expressed by (1) resolves two important problems [1]: segmentation of the image (since contours of the objects it contains are extracted) and its restoration (since a smooth version of is estimated). Several approximations of the result image u have been proposed during the last decade, they can be grouped in two classes. The first class is called global region-based active contours [2-7], it is governed by the famous Chan–Vese model [2, 5]; where authors approximated the image by a set of constants that represent the means of intensity inside and outside the active curve. Such approximation ignores totally the intensity variation inside image's regions, and therefore the model fails to segment inhomogeneous images. The second group is known as local region-based active contours [8-13]. It was initially introduced by Li et al. [8], where local statistic information has been considered. Such local propriety is usually obtained by using the kernel Gaussian function. This class of techniques improves considerably the segmentation results on inhomogeneous images but has strong dependence on the curve initialisation. A hybrid approach was introduced to take advantage of both approaches and overcome some of their drawbacks [14-18]. It is not sensitive to initial conditions (like local region-based models) and works properly with inhomogeneous images (unlike global region-based models). Roughly, the functionals of the hybrid approach are written as follows: (2)where and are parameters introduced to control the contribution of the corresponding terms. Here the fidelity to data term is ensured by global fitting energy and local fitting energy . The regularisation of the curve in evolution is guaranteed by , which is its length. is called distance regularisation term that ensures keeping the level set to a signed distance function (SDF) during the process of minimisation. All the three approaches mentioned above are based on energy optimisation which is performed by different techniques. The most used technique is the gradient descent (GD) method. It is based on introducing a virtual temporal variable in the corresponding Euler–Lagrange static energy (resolve partial differential equation) and evolves it iteratively until reaching the minimum. To insure the stable evolution, Courant–Friedrichs–Lewy condition is required, it consists of linking the spatial step with the temporal step. As the time step must be very small, the evolution is slow and the overall process is time consuming. Some other alternatives are proposed in literature to overcome the drawbacks of GD method. In [19], authors have used the split-Bregman technique to minimise the energy functional for image segmentation. The split-Bregman method transforms the model into solving Poisson equations by introducing an auxiliary variable. The same technique is used in [20] but in the multiphase level set framework. Another technique, less often used in image segmentation, is the projection approach (Chambolle's algorithm) which is used in image denoising, deblurring, inpainting and colourisation [21, 22]. In this paper and inspired from [23], we introduce a new algorithm for the optimisation of the hybrid region-based active contour technique for image segmentation. The algorithm is based on scanning the image and checking the energy variation for each pixel when this one is moved from the inside of the curve to the outside and vice versa. Unlike the techniques mentioned above, the proposed algorithm does not introduce any auxiliary variable and does not require any numerical stability conditions, which allows very fast convergence. The paper is structured as follows. In Section 2, we introduce the hybrid region-based active contour with a dynamic combination. The proposed fast algorithm to minimise the functional energy is described in Section 3. Section 4 presents and discusses the experiment results on synthetic and real images. Section 5 concludes the paper. 2 Adopted approach Hybrid region-based active contours approach is based on local and global statistical image information. It includes different models depending on the region descriptors used and the way to incorporate them. For example, in [17, 18] the energy functional is based on global and local means. Such models fail to extract regions that have similar mean but different variances. In [14], authors use global means and variances with local means and variances to define the energy fitting. However, the estimation of all those terms requires heavy computations. In this paper, we assume that globally, image intensities are not homogeneous but locally they are; which means that the local variances in Gaussian kernel windows Ωi have small values, contrary to the global variances which are large. As the introduction of global variance in the global fitting energy could mislead the curve in evolution and yield the convergence to be slower, we propose to incorporate the local means and variances inside and outside the curve, to build the local fitting energy; and global means for the global fitting energy. Furthermore, the constant that controls the weights of the local and global energies is replaced by a function that takes into account the local neighbourhood. In addition, in this work the distance regularisation term is completely ignored due to the way that the functional is minimised (see Section 3). The functional is therefore formulated as follows: (3)The local fitting energy incorporates the means and variances. They are computed in small windows via the Gaussian kernel function . It is formulated as follows [14]: (4)where is the original image, the level set function, is a regularised Heaviside function defined as (5)The global fitting energy corresponds to the piecewise constant Chan–Vese model [2] (6)The curve regularisation term is its length and it is computed via the well-known integral [8, 9] (7)Traditionally, is chosen as a constant , which adjusts the contribution of the local and global region fitting energies. In order to get good segmentation results, this constant has to be chosen very carefully, which is not an easy task. This constant is replaced by a function that reflects the intensity variation inside small windows of size . It is formulated as follows [16]: (8)where is a positif constant, and are the maximum and minimum of the intensity values within this local window, respectively, is the intensity level of the image, for grey scale images, it is usually equal to 255. Based on the calculation of variation, and keeping the level set fixed, we minimise (3) with respect to other variables [13] (9) (10) (11) (12) (13) (14)The Euler–Lagrange equation that minimises the functional of (3) is given by (15)where denotes the regularised version of Dirac function, and are the local terms and are given by (16)and the global terms and are formulated as follows: (17)More often the minimisation of (15) is ensured by GD method, where temporal discretisation is introduced (, F is the right hand of (15)). In the next section, we introduce an algorithm that optimises the functional in (3) without using the Euler–Lagrange equation. 3 Proposed fast algorithm The main idea of the proposed algorithm is inspired from the work described in [23]. Initially, the image domain is subdivided into two parts using simple closed curve. Let be the image domain, the inside of the curve and its outside. The evolution of the initial curve is obtained by scanning the image row by row and computing, at each pixel, the energy variation resulting from moving the pixel x from inside the curve to its outside and vice versa. When this variation is negative, the sign of is changed (i.e. ). The sweeping optimisation principle is used in [6, 18, 24]. Where in [6] this principle is applied to optimise the piecewise constant Chan–Vese model proposed in [2], such model deals only with homogenous images [9, 14]. In [18], authors apply this principle to optimise global and local fuzzy energies, derived from the global and the local means. In [24], this principle is used to minimise the fuzzy energy-based functional described by a Gaussian mixture model. In all those three works the energy alteration is calculated from a fuzzy membership function while in the proposed algorithm it is calculated directly from the level set function. It is used to optimise the hybrid functional that incorporates the local means and variances to build the local fitting energy; and global means for the global fitting energy as described in 2. In the proposed functional (3), the curve evolution is governed by the fidelity to the data term , from which we infer easily the energies inside and outside the curve as follows: (18)From (18), and in order to evaluate the energy variation of each pixel when this pixel is moved from the inside of the curve to its outside, we propose the following formulation: (19)Similarly, when the pixel is moved from outside to inside of the curve, the energy variation is calculated as follows: (20)where m and n represent the area (number of pixels) inside and outside the curve, respectively. Furthermore, the computation of curve's perimeter in (7) is replaced by the following approximation [23]: (21)The stop criterion of the proposed algorithm is based on the energy variation. It is given by the energy difference where corresponds to the energy in the iteration and to the energy in the iteration n. The convergence is achieved when the variation of energy is equal to zero (see Algorithm (Fig. 1)). Fig. 1Open in figure viewerPowerPoint Proposed algorithm This optimisation technique does not involve any derivative calculations, instead, it just changes the sign of whenever the energy variation is negative. Consequently, there is no need for F to be differentiable, and there is no need for the distance regularisation term. Due to those reasons, the convergence is very fast. As mentioned above, region-based active contours have the advantage of addressing simultaneously the segmentation and restoration problems. Thereby, at the end of the minimisation process the restored image u is directly deducted as follows: (22)where and approximate the image inside and outside the curve, respectively, according to the computed global and local means. They are formulated by (23)This work addresses a two-phase partitions problem (i.e. one level set function), thereby at the convergence, the level set map displays the two regions that are the foreground represented by and background by . 4 Experiment results Several experiments have been conducted to evaluate the effectiveness and efficiency of the proposed algorithm on both synthetic and real images. In all those experiments, the proposed implementation represents the contour implicitly by a binary level set (BLS) function, where the inside pixels are set to 1 and the outside pixels to −1. Besides, the parameter and the value of the Gaussian kernel . Furthermore, the two parameters, used to compute the dynamic function . Finally, the constant that controls the contribution of the length term is set to , unless mentioned otherwise, where the value of is equal to . All the codes are developed using MatlabR2014a software and run on a computer with core(TM) i7-2600, 3.40 GHz and 4 GB RAM. We would like also to indicate that in all the figures shown in this section, the dashed green lines represent the initial contour while the solid red lines represent the final contours (Please refer to the web version of this article for a better understanding of the colour figures). The first experiment demonstrates the effectiveness of the proposed algorithm on clean (with no noise) synthetic images. As the perimeter term is used to improve the processing of noisy images, it has been omitted () in this experiment. Perfect segmentation is obtained for both images as shown in Fig. 2. It is worth noting that the convergence was very fast, it is obtained in only one sweep on both images, which demonstrates the efficiency of the proposed algorithm. Fig. 2Open in figure viewerPowerPoint Segmentation results on two clean (without noise) synthetic images, where the length term is omitted . The results are obtained in one sweep of the images In the second experiment, relatively clean (without noise) and complex real image is used. Again, a perfect segmentation is obtained and the convergence was very fast, it is achieved in few scans (sweeps) of the image. Fig. 3 shows some steps of the curve evolution including the final segmentation. Fig. 3Open in figure viewerPowerPoint Steps of curve evolution on inhomogeneous image (a) Initialisation, (b), (c) Intermediate curves, (d) Convergence, (e) Restored image given by (23) In the third experiment, more complex real images are used and the execution time is recorded. Fig. 4 displays obtained segmentations and Table 1 shows recorded execution times. We can clearly see that in all the three images, the curve converges successfully toward the objects' boundaries and the execution time is very fast. Table 1. Iterations and CPU time of the experiments in Fig. 4 Fig. 4: 1st row Fig. 4: 2nd row Fig. 4: 3rd row size sweep 17 08 09 CPU, s 2.89 0.65 0.31 Fig. 4Open in figure viewerPowerPoint Obtained results with the proposed algorithm. First column: original images, second column: initial contour, third column: final contours, fourth column: the restored image Hybrid models are proposed to overcome the sensitivity of the local statistic measurements to contour initialisation. The fourth experiment is devoted to demonstrate that this property is well-preserved in the proposed fast hybrid model. Figs. 5a–d show that the proposed algorithm converges to the same location (objects' boundaries) for different initialisations of the contour (surrounding two objects, inside one of the object, crossing two objects and outside both objects). However, the experiment showed also that the initialisation affects the rapidity of convergence. The convergence is achieved in six sweeps for the case shown in Fig. 5a, in five sweeps for cases shown in Figs. 5b and c, and in seven sweeps for the case shown in Fig. 5d. Fig. 5e draws the four energy versus sweeps graphs resulting from those four initialisations. Fig. 5Open in figure viewerPowerPoint Robustness to contour initialisation. All initialisations converge to the same contour (a)–(d) Four initialisations, (e) Energy variation in each image sweep corresponding to the four initialisations The following two experiments are conducted to compare the accuracy of the segmentation produced by the proposed hybrid model optimised via sweeping algorithm (referred to as fast GLF or F_GLF) with the same model but minimised using GD method (referred to as classic GLF or C_GLF). We would like to mention that, C_GLF uses the SDF to represent the curve; the distance regularisation term is therefore indispensable in the functional to keep this property of the level set during the minimisation process. This term is known to have some undesirable side effects as indicated in [22]. On the other hand, the proposed algorithm uses BLS function and there is no need to include the distance regularisation term, as explained in Section 3. Fig. 6 shows the segmentation obtained on an inhomogeneous image with very concave regions. The first row shows the initial contour and the corresponding SDF and BLS. The second row displays the outcomes of C_GLF and the third row those of F_GLF. A zoom window is made on the concave zones to show that, in the case of F_GLF, the curve converges to the concave zones slightly better than in the case of the C_GLF. Fig. 6Open in figure viewerPowerPoint Results of segmentation of object with very concave shape (a) Contour initialisation, (b) Level set as SDF, (c) Level set as binary function BLS, (d) Segmentation results with the classic GLF, (e) Final SDF, (f) Segmentation results with the proposed algorithm, (g) Final BLS To provide a quantitative evaluation of the accuracy of the two algorithms, we selected ten images from the ‘single object segmentation evaluation dataset’ provided by the Weizmann Institute of Science [25]. The dataset contains for each sample, a colour and greyscale image, as well as segmentations done by three different individuals. The authors have suggested to take as ground truth segmentation the intersection of the three human segmentations. They have also suggested to use the F-score value as a metric for evaluating the performance of the segmentation. The F-score is defined as [26] (24)where P is the precision (i.e. the ratio of pixels correctly labelled as foreground to the total pixels labelled as background) and R is the recall (i.e. the ratio of pixels correctly labelled as foreground to the total pixels labelled as foreground in the ground truth image). The first column in Fig. 7 shows the greyscale image, the second, third and fourth columns, the human segmentations, the fifth and sixth columns, the level set obtained by the classic GD (C_GLF) and the proposed algorithms (F_GLF), respectively. We have also indicated the F-scores obtained by the two algorithms on top of the resulting level sets. Fig. 7Open in figure viewerPowerPoint Segmentation results obtained on ten greyscale images taken from the Weizmann Institute of Science dataset Obtained results confirm that the proposed algorithm performs slightly better for objects with strong concavities as indicated by the results obtained on 100_0497, bw4, img_1965 and leafpav images; but in general the two algorithms perform similarly well, as indicated by the high average F-scores (0.915 for C_GLF and 0.917 for F_GLF). Note: All the images in Fig. 7 use the rectangle surrounding the object as an initial contour, just like in Fig. 8a. Fig. 8Open in figure viewerPowerPoint Results for real brain MRI image with the hybrid region-based model (a) Original image and curve initialisation, (b) Minimisation via GD method, (c) Minimisation via proposed algorithm The main advantage of the proposed algorithm is its low computational cost which makes it suitable for real-time applications. The next experiment is conducted to demonstrate this property for the proposed algorithm. Fig. 8 shows the segmentation results obtained on a magnetic resonance imaging (MRI) of size (256 × 256). In the experiments in Figs. 7 and 8, the stop criterion for C_GLF is () and for the proposed algorithm it is set automatically to zero (). Although the two techniques obtain similar segmentation results, the proposed F_GLF converges much faster than the C_GLF, where a great CPU time gain is obtained as shown in Table 2, that summarises the results obtained on the images used in Figs. 3, 5 and 6 and those taken from the Weizmann institute dataset. Table 2. Comparison of the computation cost of the GD method and the proposed algorithm Image C_GLF F_GLF CPU time gain, % Iterations CPU, s Sweeps CPU, s Fig. 3 image (247 × 254) 380 11.85 7 0.22 98.14 Fig. 5 image (84 × 84) 347 1.96 7 0.05 97.44 Fig. 6 image (248 × 296) 393 11.97 7 0.23 98.07 100_0497 (300 × 400) 480 42.77 9 0.56 98.69 bbmf_lancaster_july_06 (300 × 225) 6048 344.68 13 0.51 99.85 bw4 (300 × 200) 1137 45.59 11 0.35 99.23 dsc04575 (300 × 225) 30,030 1678.5 28 1.08 99.93 Egret_face (300 × 200) 526 22.29 7 0.23 98.97 img_1965 (300 × 400) 16 1.50 6 0.38 74.66 Leafpav(300 × 203) 2675 119.04 9 0.32 99.73 nitpix_p1280114 (300 × 225) 1119 64.25 7 0.29 99.55 oscar2005_05_07 (300 × 225) 199 11.42 6 0.25 97.81 outside_guggenheim_walls (300 × 400) 96 8.78 5 0.31 96.47 Additional experiments have been conducted to study the impact of the local and global statistics on the performance of the segmentation. In Section 2, we have indicated that incorporating global variance in the global fitting energy term would make the model slower and more sensitive to noise. This has been confirmed by the experiment that uses a noisy synthetic image (Fig. 9). Fig. 9b shows the resulting segmentation when global variance is used and Fig. 9c shows the segmentation resulting from running the proposed model (that does not include the global variance in the global fitting energy). It is clear that the proposed algorithm produces cleaner segmentation (less noisy). Besides, when global variance is used, the convergence is obtained in 16 sweeps while the proposed algorithm achieves the convergence in only five sweeps. Fig. 9Open in figure viewerPowerPoint Results with and without incorporation of global variance for synthetic images with noise (a) Image to be treated and the initial contour, (b) Obtained results with the incorporation of global variance, (c) Obtained results with the proposed model (without the incorporation of global variance) The next experiment is devoted to conducting a qualitative evaluation of the segmentation produced by the proposed model in the presence of noise. Peak signal-to-noise ratio (PSNR, in decibel, db) is used to estimate the degree of image degradation introduced by the added noise. Fig. 10a shows the original (clean) image and the initial contour and Fig. 10b the perfect segmentation obtained on that image. The following sub-figures show the segmentation obtained by the proposed model when different kinds of noise are added to the original image. Fig. 10c shows, from the left to right, the results when adding 1, 5 and 6% of speckle noise that correspond to degraded images with PSNRs equal to 29.73, 28.37 and 28.27 db, respectively. Fig. 10d shows the results when adding 0.3, 0.6 and 1% of Gaussian noise which represent a degraded images with PSNRs equal to 30.42, 27.40 and 28.79 db, respectively. Finally, Fig. 10e depicts the results when adding 0.1, 0.9 and 6% of impulsive noise which represent degraded images with PSNRs equal to 57.40, 47.40 and 39.37 db, respectively. The last row of Fig. 10 shows the segmentation produced by the proposed model in the presence of a mixture noise of 0.1, 0.5 and 1% representing degraded images with PSNRs equal to 32.04, 29.20, and 28.53 db. The above-mentioned results show that the proposed model achieves almost perfect segmentations for noisy images with PSNR ≥30 db when additive, multiplicative or a mixture of noises affect the image. This robustness to noise is made possible because the model includes a convolution of the image with a Gaussian kernel that reduces those types of noise. The segmentation of images with impulsive (salt-and-pepper) noise is less good, but in all three cases, the object contour is correctly identified. This can be explained by the fact that Gaussian smoothing is less effective on this type (impulsive) of noise. Fig. 10Open in figure viewerPowerPoint Performance of the proposed model in the presence of difference kind of noise (a) Original clean image and initial contour, (b) Segmentation of original image. From the second row to the fifth row we depict the segmentation results in the presence of difference kind and density of noises, (c) Multiplicative noise, (d) Additive noise, (e) Impulsive noise, (f) Mixed noise The last experiment, whose results are shown in Fig. 11, indicates however, that in the presence of strong inhomogeneity, the incorporation of more statistical information improves the performance of the segmentation. The image used in this experiment contains an inhomogeneity in the intensity distribution due to a change in illumination that makes the bottom of the image dark. Fig. 11a shows the initial location of the curve. The second row shows the results obtained by Krinidis–Chatzis et al. model [6] that uses a functional based on the global means only. Here it should be mentioned that this model is a special case of the proposed model obtained when . The third row presents the results obtained by the hybrid Shyu et al. model [18] that incorporates both global means and local means. The fourth row of Fig. 11 displays the outcomes of the proposed model. Figs. 11b and c indicate that Krinidis–Chatzis et al. model fails to extract all the objects present in the image (objects located in the dark region at the bottom of the image) due to the high inhomogeneous intensity distribution. On the other hand, Figs. 11d–i indicate that the Shyu et al. model as well as the proposed model achieve much better segmentation results. The two models produce very similar results for most objects. However, the proposed algorithm outperforms Shyu et al. algorithm in the regions with strong inhomogeneity (region enclosed by white dashed line in Figs. 11d and g and zoomed into in Figs. 11e and h; this demonstrates the benefit of including the local variances in the model. Fig. 11Open in figure viewerPowerPoint Comparison of the performance of Krinidis–Chatzis et al. model [6], Shyu et al. [18] and the proposed model in segmenting and restoring highly inhomogeneous images (a) Original image with initial contour (dashed green line), (b), (c) Segmentation and restoration result by the Krinidis–Chatzis model, (d)–(f) Outcomes of Shyu et al, (e) Zoom of the ninth potato segmentation, (g)–(i) Segmentation obtained by the proposed model, (h) Zoom into the segmented ninth potato showing the slight difference between the proposed model and Shyu et al. model zoomed in (e) Figs. 11c, f and i display the restored images obtained by the Krinidis–Chatzis et al. and our proposed model, respectively. In the case of Krinidis–Chatzis model, the restored image is simply a separation between foreground and background of the image 5 Conclusion Although the segmentation via deformable models has been widely used for decades, it is still a hot research topic. In this paper, we have described a hybrid region-based active contours technique that combines a global energy defined by the global means and a local energy defined by the local means and variances. Such hybrid approach gives the model several advantages one of which is its ability to handle equally homogenous and inhomogeneous images. The novelty in this approach is the proposed algorithm that optimises the functional by applying a sweeping principle. The proposed algorithm avoids solving any partial differential equation and therefore does not include any stability condition. Furthermore, the distance regularisation term is omitted, which speeds up the optimisation process, and makes the algorithm well suited to real-time applications. 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