Image Segmentation Based on Active Contours without Edges

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1 Image Segmentation Based on Active Contours without Edges Anca Morar Faculty of Automatic Control and Comuters University POLITEHNICA of Bucharest Florica Moldoveanu Faculty of Automatic Control and Comuters University POLITEHNICA of Bucharest Eduard Grӧller Institute of Comuter Grahics and Algorithms Vienna University of Technology Abstract There are a lot of image segmentation techniques that try to differentiate between background and object ixels, but many of them fail to discriminate between different objects that are close to each other. Some image characteristics like low contrast between background and foreground or inhomogeneity within the objects increase the difficulty of correctly segmenting images. We designed a new segmentation algorithm based on active contours without edges. We also used other image rocessing techniques such as nonlinear anisotroic diffusion and adative thresholding in order to overcome the images roblems stated above. Our algorithm was tested on very noisy images, and the results were comared to those obtained with known methods, like segmentation using active contours without edges and grah cuts. The new technique led to very good results, but the time comlexity was a drawback. However, this drawback was significantly reduced with the use of grahical rogramming. Our segmentation method has been successfully integrated in a software alication whose aim is to segment the bones from CT datasets, extract the femur and roduce ersonalized rostheses in hi arthrolasty. Active contours without edges, image segmentation, nonlinear anisotroic diffusion, arallel image rocessing I. INTRODUCTION Image rocessing is a oular technique in many domains, like medicine, security and surveillance, traffic control and image editing. The human eye can easily distinguish imortant characteristics even in very oor quality images. The aim of image-rocessing software-alications is to interret images in a similar (or even better) way as comared to a human being, but only faster. The results obtained by comuters can be very imressive, for examle, the 3D visualization of a CT dataset, with the highlighting of some imortant characteristics, like different human tissues. Often these automatic rocessing algorithms may lead to errors. The challenge in this domain is to discover fast and fully automatic image rocessing algorithms, or algorithms that require only little human intervention. The research resented in this aer concentrates on the segmentation of oor quality images, like CT images, that have a granular asect, small contrast between foreground and background and inhomogeneities regarding the intensity within the foreground. The second chater discusses the state of the art in image segmentation. The third chater describes a series of image rocessing techniques that were used in our algorithm and in the segmentation methods that were comared to our algorithm. Next, we state our motivation for designing a new segmentation technique. The fifth chater resents the stes of our new algorithm. In the sixth chater we focus on imlementation details both on the CPU and on the GPU with CUDA. In the last chater we resent a comarison between the results obtained with our algorithm and the results roduced by other segmentation algorithms. II. RELATED WORK There are a lot of image segmentation techniques, some based on intensity or texture, others on gradient or shae characteristics. Some of the methods that have roven to lead to good results in the segmentation of oor quality images are briefly resented in this section. Kass et al. [] introduce the concet of snakes, or active contours. Snakes are energy-minimizing slines guided by external constraint forces and influenced by image forces that ull them toward features such as lines and edges. Chan and Vese [] roose active contours without edges. It is a new model for active contours, which is based on techniques of curve evolution, the Mumford-Shah functional for segmentation, and level sets. This method will be detailed in the next chater. Boykov and Veksler [3] describe the use of grah cuts in comuter vision and grahics through theories and alications. In image segmentation, a grah is created from the image or the set of images. The grah construction and the characteristics that divide the ixels into two disjoint arts, i.e., the background and the foreground, will be detailed in the next section. In grey scale mathematical morholog the watershed transform, originally roosed by Digabel and Lantuejoul [4] and later imroved by Buecher and Lantuejoul [5], is considered to lead to very good results in image segmentation. Roerdink and Meijster [6] wrote a review of several definitions and algorithms of the watershed transform. They describe in this review the geograhic idea behind the watershed transform as that of a landscae or toograhic relief which is flooded by water. Watersheds are the dividing lines of the domains of attraction of rain falling over the region. When the water level has reached the highest eak in the landscae, the rocess is stoed, and the result is a landscae artitioned into basins searated by dams, called watershed lines. Porwik and Lisowska [7] resent the use of the Haarwavelet transform in digital image rocessing. Their aer describes a method of image analysis by means of the wavelet-

2 Haar sectrum. Glavasova et al. [8] discuss the wavelet transform for feature extraction, based on texture analysis, for the final goal of image segmentation. There are a lot of other segmentation algorithms, but most of them are based to some extent on one of the techniques mentioned above. The challenges in image segmentation come from the roblems reviously stated. The small contrast between foreground and background makes it difficult to extract all the edges or lines that are used for examle in active contours, grah cuts and watersheds. The inhomogeneities within the objects rove to be a drawback for active contours without edges, because this method tries to minimize the differences within the foreground and the background. The lack of a texture attern could be a roblem for the wavelet transform based segmentation. Our algorithm takes into account the imerfections of oor quality images (like CTs), not only differentiating between objects and background, but also between different objects. III. IMAGE PROCESSING TECHNIQUES For the understanding and motivation of our algorithm, we shortly describe in this section the existing techniques that we used in our image segmentation. We also resent some details regarding the other segmentation methods that were comared to our algorithm. A. Noise Reduction Using Gaussian Filtering One of the most common tyes of noise is Gaussian noise [9]. Gaussian noise is modeled using a robability density function. In our imlementation we used a convolution mask aroximating a D Gaussian distribution function. B. Nonlinear Anisotroic Diffusion Nonlinear anisotroic diffusion [, ] reduces noise, but reserves the image information. We resent in this subsection some concets of diffusion filtering. Diffusion is a hysical rocess that balances the concentration changes of a certain substance. Having a concentration distribution u, Flick s law states that the concentration gradient determines a flow: j = D u. () D is the diffusion tensor, a ositive-definite symmetric matrix. Diffusion reresents mass transort without destroying or creating new mass. From the continuity equation we can derive the following equation: j j t u = div ( j) = +, () x y where t denotes time, and t u, the deviation of u with resect to t. From () and () we can deduce the following result: u = div ( D u ). (3) t In image rocessing the image intensity can be seen as concentration, and image noise as concentration inhomogeneities. These inhomogeneities can be smoothed through diffusion. In order to reserve edges, the filtering will be accomlished in the following manner: no diffusion across edges diffusion arallel to the edges. The diffusion tensor is a x matrix comuted with the following exression: λ [ v v ] [ v v ] T D = λ. (4) The eigenvectors v and v rovide the local diffusion orientations. Their corresonding eigenvalues give the contrast along these directions. The first eigenvector denotes the direction of maximum variation. Therefore, v reresents the direction arallel to the image gradient. The other eigenvector is determined considering the orthogonality roerty: u v = and v u v y = v x. (5) The eigenvalues are comuted so that diffusion is inhibited across the edges, and activated arallel to the edges: ( ) λ and λ. (6) = g u = The first eigenvalue is determined using the following exression roosed by Perona and Malik []: g ( ) m u = ex ( ) C m u / λ. (7) The constant C m is comuted so that the diffusion is big for u [, λ ] and small for u ( λ, ). λ denotes the threshold for the gradient. Values bigger than the threshold determine edges. In our imlementation we have chosen λ = 4 and m = 4. The value of m determines C m = After comuting the eigenvectors and the eigenvalues of the diffusion tensor D D D =, equation (3) is solved D D with finite differences. t u can be relaced through a forward difference aroximation. The obtained exlicit scheme allows the iterative comutation of several versions of the image:

3 u u( x, s) = u( x, s ) + t [ ( D ) x x, (8) u u u + ( D ) + ( D ) + ( D )] x y y x y y where t denotes the time ste size, and u(x,s) reresents the image at time t s = s t. The standard scheme for the aroximation of satial derivatives is based on central differences. Fig. resents the results of alying nonlinear anisotroic diffusion on a CT image. background. Q reresents the set of all the n-links, f is the label assigned to ixel ( object or background ) and P denotes the set of all non-terminal nodes. In order to define the border term V,q we use the observation of Boykov and Veksler [3] that ixels with high image gradient would imly low cost of n-links and vice-versa. This is why we comute the gradient of the image using a Sobel filter. For smoother borders, we decided to aly the Sobel filter after convolving the image with a Gaussian filter and a nonlinear anisotroic diffusion filter. If the absolute difference between the gradient magnitude of two neighboring ixels and q is greater than a given threshold k, V,q is directly roortional with that difference. If there is a small variation of the gradient, the value of V,q, given by the constant v, is high. The border term can be defined as follows: Figure. Nonlinear anisotroic diffusion filtering of a CT image at different moments in time (different number of iterations) C. Grah Cuts The mechanisms of grah cuts for image segmentation are given in more detail. This rovides some insight into the grah construction algorithm used in the imlementation that was comared to our algorithm. In a tutorial illustrating grah cuts in the context of comuter vision and grahics, Boykov and Veksler [3] exlain general theoretical roerties that motivate their use. Let G = V, E be a grah comosed of a set of nodes V and a set of oriented edges E. The set of nodes V = { s, t} P contains two nodes called terminals, i.e., the source s and the sink t, as well as a set of non-terminal nodes P. Every edge in the grah is assigned a nonnegative weight w(,q). An edge is called t-link if it connects a non-terminal node with a terminal one. It is called n-link if it connects two non-terminal nodes. An s/t cut is a artitioning of the grah nodes into two disjoint sets S and T in a manner that the source s is in S and the sink t in T. Starting from an image, the grah is built in a manner that every ixel in the image defines a non-terminal node of the grah. The cost of an n-link is based on the likeliness of the neighboring non-terminal nodes. The cost of a t-link is based on the likeliness of the connected non-terminal node and the terminal one. After the cut, some ixels belong to the source (being labeled as object ) and others to the sink ( background ). The minimal cut minimizes the energy function introduced by Boykov and Veksler [3]:, q (, q ) Q E ( f ) = V ( f, f ) + D ( f ), (9) q P where D is the er-ixel term that reflects the enalty of assigning the label f to ixel. V,q is the border term that encourages satial coherence within the objects and the V, q = u f ( ) u v, otherwise f ( q), if u f ( ) u f ( q) > k () where u f is the image obtained after alying the Gaussian filter and the nonlinear anisotroic diffusion filter. This definition encourages borders in regions where there are abrut variations of the gradient magnitude. Boykov and Jolly [3] designed a technique for general urose interactive segmentation of N-dimensional images. In their workflow, the user marks certain ixels as object or background. We extend their aroach by marking certain ixels based on their intensities. We introduce two thresholds, T low and T high, that determine the most robable background and foreground ixels. If the intensity of a ixel is greater than T high, the ixel is most likely an object ixel. If its intensity is lower than T low, there is a high robability that the ixel belongs to the background. If the ixel does not belong to any of the two categories, the er-ixel term deends on its intensity relative to T low and T high : w, if f = " object" and u ( ) > Thigh, if f = " background" and u ( ) > Thigh w, if f = " background" and u ( ) < Tlow, if f = " object" and u ( ) < Tlow. () u( ) Tlow D = w if f = object ( f ), " " Thigh Tlow and Tlow u ( ) Thigh u( ) T low w w, if f = " background" Thigh Tlow and Tlow u ( ) Thigh where u denotes the initial image. The constant w assures a high cost for t-links that connect a non-terminal node and a terminal one that have the same label (either object or background ).

4 Our algorithm was comared to two segmentation methods based on grah cuts. In the first imlementation, the set of n- links Q contains only 4-connected neighbors. The second imlementation can be used only for volumetric data, because it considers also connections of ixels from adjacent slices. The first method is further called D grah cuts (D GC), because it segments the images individually. The second one is being referred to as 3D grah cuts (3D GC), because it can be alied only to 3D images (like CT datasets). D. Active Contours without Edges This section resents the active contours model without edges. Part of the method is included in the workflow of our algorithm. This technique is also used as a comarison to our segmentation. In their aer, Chan and Vese [] roose a new active contour model for object detection in a D image. The stoing term for the involved curve evolution rocess does not deend on the image gradient, but is related to a articular segmentation of the image. An evolving curve C, in the image sace, can be defined as the frontier of a subset ω of (ω and C= ω). ω reresents the region occuied by foreground ixels. inside(c) denotes the region ω and outside(c) denotes the region \ ω. The image u is assumed to be comosed of two regions of aroximately constant intensities c (the intensity of the object) and c (the intensity of the background). If the object s boundary is C, then inside of C the intensity value should be equal to c. Outside of C, the intensity value should be equal to c. Chan and Vese [] introduce the following energy: F ( c, c, C ) = µ Length ( C ) + ν Area ( inside ( C )) + () η u ( x, c + η u ( x, c inside ( C ) outside ( C ) where µ, ν, η, η > are fixed arameters. The length of the curve, Length(C), and the area of the region inside C, Area(inside(C)), are two regularizing terms. Chan and Vese [] set ν =, η = and η =. We kee the values of ν, η and η defined by them, but we also set µ to. We exlain in the fifth chater the reason for omitting the length term. The image segmentation into foreground and background is accomlished by solving the minimization roblem inf F ( c, c, C ). Let C be defined as the zero level set c, c, C of a Lischitz function φ : R, so that: C = ω = {( x, : φ ( x. = } inside ( C ) = ω = {( x, : φ ( x. > }. outside ( C ) = \ ω = {( x, : φ ( x. < } Using the Heaviside function H and the one-dimensional Dirac measure δ defined by Chan and Vese [], the energy F c, c, C ) = F ( c, c, ) can be exressed as follows: ( φ F ( c, c, C ) = u ( x, c u ( x, c H ( φ ( x, ) +. (3) ( H ( φ ( x, )) The constants c and c can be exressed relative to φ : c ( φ ) = c ( φ ) = u H ( φ ( x, ) H ( φ ( x, ) ( H ( φ ( x, )) and (4) u ( H ( φ ( x, )). (5) The evolution of φ can be arametrized as follows: φ = δ ( φ )[( u c ) + ( u c ) ] =. (6) t The segmentation algorithm follows an iterative method. Knowing φ ( x, s) at time t s = s t, c( φ, s) and c ( φ, s) can be comuted by using (4) and (5). Then, φ ( x, s + ) can be comuted by the following discretization and linearization of (6) in φ : φ ( x, s + ) = φ ( x, s) + t δ ( φ ( x, s)) (7) [ ( u ( x, c ( s)) + ( u ( x, c ( s)) ] where t denotes the time ste size and u ( x, reresents the initial image. IV. MOTIVATION In this chater we give the reasons for designing a new segmentation method. The algorithms described in the revious chater were tested on ten noisy CT datasets. The aim of the segmentations was to differentiate between bones and other tissues, but also to discriminate between different bones. Fig. resents an image before (a) and after being rocessed with D grah cuts (c), 3D grah cuts (d) and active contours without edges (e). Comared to the manual segmentation (b), the results show that, even if the most ixels are correctly labeled as object or background, there are some roblems with differentiating between bones. As reviously mentioned, our segmentation is art of a bigger roject, whose aim is to obtain rototyes for ersonalized imlants in hi arthrolasty. Since one of the requirements of automatically roducing an artificial imlant is

5 to extract the femoral bone and to differentiate it from other bones, we decided to imlement a new segmentation technique. Our algorithm segments the objects from the background quite well, without connecting different objects. In the examle rovided in Fig., it can be observed that our segmentation (f) was the closest to the outut of the manual segmentation. In chater seven we resent the results of a comarison with the other segmentation algorithms on ten datasets, in order to demonstrate the quality of our algorithm in more detail. interreted as a single object. This is the reason we decided to set µ = in (), and to use (7) for the comutation of φ. Most of the tested images had a lot of noise. In order to remove this noise we smooth the initial image u with a Gaussian filter. The image segmentation that divides the ixels based on the value of φ relative to is too rough in the sense that it leads to a binary image (the white ixels belong to the foreground and the black ixels belong to the background). The original active contours aroach without edges also requires a large number of iterations in order to get to a stable configuration (the final segmentation). An alternative to converting the image into a binary segmented one could be a segmentation where the active contours model reresents only an enhancement ste. Figure. Comarison of different CT image segmentation methods: (a) original image, (b) image segmented by a human secialist, (c) image segmented with D grah cuts, (d) image segmented with 3D grah cuts, (e) image segmented with active contours without edges, (f) image segmented with our algorithm The algorithm has been exemlified using CT images, but it can be alied on all kinds of grayscale images that have the following characteristics: The foreground has a higher intensity than the background The images contain multile objects that are ositioned close to each other and should not be connected There are big inhomogeneities within the foreground. V. SEGMENTATION ALGORITHM This chater resents our new segmentation technique. The stes of the algorithm are identified in the workflow of Fig. 3 and are described in detail in the following subsections. Ste : Gaussian and Active Contours (G + AC) As mentioned before, our segmentation is based on the active contours model by Chan and Vese []. Here we state our reasons for making some changes to the original algorithm. The first change was the omission of the length term in equation (). We tested the original algorithm on images where two different objects were ositioned very close to each other. The outut image contained a single object, comosed of the two initial objects. An exlanation could be the one given by Chan and Vese [] regarding the use of the length term as a scale arameter. If the constant µ from () is small, then also smaller objects will be detected. If it is larger, then only larger objects are detected, or objects that are groued together. We do not want different objects close to each other to be Figure 3. Image segmentation workflow After comuting the values of φ based on (7), in the last iteration we normalize the values of φ to [-,]. In the original active contours aroach without edges, if φ <, then the segmented image u ( x, = and if φ, u ( x, = I. max The biggest change to the original algorithm is in comuting the outut image by normalizing φ to [,I max ], where I max is the maximum level of intensity (55 in our case): ( + φ ( x, ) I u( x, = max. (8) With our aroach, the outut image u will have shades of gra with an enhanced contrast between foreground and background. Fig. 4(b) resents a CT image after alying the first ste from our algorithm.

6 Ste : Gaussian and Nonlinear Anisotroic Diffusion (G+NAD) For the removal of small discontinuities within the object (and esecially near the borders) we aly a Gaussian filter on the image u. We also aly a nonlinear anisotroic diffusion filter, considering the image gradient, in order to avoid connecting different objects. Fig. 4(c) resents the outut image u after alying these two smoothing filters. filter removes ixels that could wrongly connect different objects. Choosing a window of neighbors W of size nxn for the current ixel we comute the average intensit multilied by a fixed arameter α : u ( q) q W M = n α. (9) M is a threshold for dividing the ixels into foreground and background. If u () < M, then is labeled as background ixel. If α =, there are still some ixels wrongly considered as art of the foreground. This is why we set a slightly higher threshold with α = + α, where α > is very close to. Fig. 4(d) shows the result u 3 of alying the adative thresholding to image u. Figure 4. Outut images in the image segmentation stes: (a) original image, (b) image enhanced with active contours without edges (Ste ), (c) image smoothed with nonlinear anisotroic diffusion (Ste ), (d) result of adative thresholding (Ste 3), (e) outut image after alying an adative threshold on the original image (Ste 4), (f) image obtained by combining the images u 3 and u 4 (Ste 5), (g) image resulting from connecting the foregorund ixels into slice islands and removing those islands with a low intensity (Ste 6), (h) final image after alying the hole filling ste (Ste 7) Ste 3: Adative thresholding (AT) This sub-section states the reasons for introducing another rocessing ste, an adative thresholding. The intensities of the image u are between and I max. A simle segmentation would be to searate the ixels based on their intensity relative to I max /. But there are two roblems with this segmentation: Some ixels with intensity close to the threshold I max / but lower than I max / should belong to the foreground. We set the interval to search for these ixels to [I max / I,I max /], where I is a fixed arameter. There are some ixels whose intensities are close to the threshold I max / but greater than I max / which should belong to the background. We set the interval to search for these ixels to [I max /,I max / + I]. In order to eliminate the above roblems, we use two thresholds T = I max / I (low) and T = I max / + I (high). All the ixels with intensity value lower than T are labeled as background ixels (the intensity is set to ). All the ixels with intensity value greater than T are considered foreground ixels (their intensity remains unchanged). For the ixels that belong to the interval [T,T ] we aly an adative threshold. This Ste 4: Adative thresholding and Gaussian (AT+G) Due to Ste 3, there are fewer wrongly labeled ixels. However, in case of objects that are very close to each other, or small contrast between foreground and background, there could still be ixels that connect different objects. For this roblem we decided to aly another adative threshold, but this time, on the initial image u. We also smooth the outut with a Gaussian filter. This adative threshold does not use T and T but is alied to all the image ixels. Alying the second adative threshold assures the removal of the ixels that could have been wrongly labeled as object because of the smoothing stes (the two Gaussian filtering and the nonlinear anisotroic diffusion). Fig. 4(e) resents the outut image u 4 after alying the adative threshold on the initial image u. Ste 5: Combining images (CI) The next ste consists of combining the images u 3 and u 4 in order to obtain an image with very few or no ixels that connect different objects. For the current ixel, if u 3 ()> and u 4 ()<T 3 (where T 3 is an exerimentally determined threshold), then the combined image u 5 ()=, else u 5 ()=u 3 (). Fig. 4(f) shows the outut image u 5 after combining images u 3 and u 4. Ste 6: Island extraction (IE) The low value of the threshold T from Ste 3 removes small discontinuities within the objects. However, this also adds wrongly labeled ixels. We treat this roblem in the following manner: the foreground ixels that are connected are groued into islands. For every island, the average intensity I avg is comuted. If I avg <T 4, where T 4 is a fixed arameter, then the current island is considered to belong to the background. This way we are assured that only those low intensity ixels which are connected to a large number of high intensity ixels are considered to belong to the foreground. Fig. 4(g) resents the outut image u 6 after removing the low intensity islands from image u 5. Ste 7: Hole filling (HF) The last ste in the segmentation rocess solves the roblem of inhomogeneities within the object. Starting from the first black ixel in the image (from uer left to lower

7 right) we do a breadth-first-search (BFS) in order to visit all the background ixels connected to the first one. All the other ixels that have not been visited in the BFS are considered to be art of the foreground, as in Fig. 4(h). This ste is called hole filling because it re-labels all the background ixels that are located inside a closed foreground island. The roblem of this ste is in not differentiating between inhomogeneities within an object and real holes. This drawback can be overcome in the following way: if we can be sure that a ixel belongs to a real hole, and not an inhomogeneity within the object, this ixel can be set as another seed oint for the BFS that searches for all the connected background ixels. VI. IMPLEMENTATION The segmentation algorithm was imlemented both on the CPU and on the GPU. Even for relatively small datasets, the CPU imlementation takes a lot of time. On the other hand, the CUDA architecture [4] rovides the ossibility of running the same instructions for each ixel in a arallel manner, decreasing the comuting times considerably. Based on the imlementation of the Sobel filter from the CUDA SDK examle [5] and the aer by Bojsen-Hansen [6], we arallelized almost all the stes in our algorithm. Table shows all the rocessing stes that were imlemented in CUDA. It also rovides a comarison between the running times on a dataset of 56 images (each of size 5 ) for the imlementation on the CPU and on the GPU. We set the size of the Gaussian filters to 9x9 and the size of the neighborhood window for the adative thresholds to x. The maximum number of iterations for active contours without edges is 5 and the maximum number of iterations for nonlinear anisotroic diffusion is. The tests were made on an i7-6k 3.4 GHz rocessor with 8GB RAM and an Nvidia GeForce GTX 59 GPU card with.5 GB RAM. The island extraction and the hole filling ste were not imlemented on the GPU, because they are based on recursion (BFS) and cannot be intuitively aroached in a arallel manner. TABLE I. COMPUTING TIMES FOR OUR ALGORITHM ON THE CPU AND THE GPU Ste in the algorithm Time on CPU (sec) Time on GPU (sec) Ste : G + AC Ste : G + NAD Ste 3: AT Ste 4: AT + G Ste 5: CI.4. Stes The CUDA imlementation has a great imact on the seed of the segmentation rocess. Thus, with the hel of the GPGPU aradigm we now have a quite fast algorithm. In the next chater we will see how accurate this algorithm is. VII. RESULTS Our segmentation algorithm was tested on ten CT datasets with different number of slices, each of size 5x5. The arameters from the fifth chater were the same for all the images: I = 3 => T = 97.5 and T = 57.5; α = / 9 => α.345 ; T 3 = 5 and T 4 =. The initial curve C is a circle located in the center of the image with a radius of : φ = ( x 56 ) + ( y 56 ) +. Our imlementation was comared with three other segmentation methods: Active contours without edges, described in subsection III.D, with the following arameters: the maximum number of iterations for comuting φ is, and the arameters from () are η =, η =, µ =. 55 and ν =. 55. The segmentations using D and 3D grah cuts, described in subsection III.C, with the following arameters: k = 3, v = 5 and w = 3. Deending on the difficulty of correctly labeling the foreground (bones) and the background ixels (other tissues), the CT datasets were divided into two categories: low and high difficulty. The images segmented by the four algorithms were comared with the segmentation made by a human secialist. The tests consisted of counting the correctly labeled ixels, the foreground ixels that were wrongly labeled as background ixels (false negatives), the background ixels that were wrongly labeled as foreground ixels (false ositives), and the number of images where two different objects were wrongly connected. If F is the number of correctly labeled foreground ixels, and F is the number of false negatives, then the false negative ercentage is E F = F / (F + F ). Similarl the false ositive ercentage is B F = B / (B + B ), where B is the number of correctly labeled background ixels and B is the number of false ositives. Table resents the false ositive and the false negative error for the low difficulty datasets. TABLE II. Alg. ERRORS IN LABELING PIXELS FOR THE LOW DIFFICULTY DATASETS (%) Data Data Data3 Data4 Data5 B F E F B F E F B F E F B F E F B F E F Our alg D GC D GC AC Table 3 resents the ercentage of images where two different objects were wrongly connected, for the low difficulty datasets. TABLE III. PERCENTAGE OF IMAGES WHERE DIFFERENT OBJECTS WERE WRONGLY CONNECTED FOR THE LOW DIFFICULTY DATASETS Percentage of images (%) Algorithm Data Data Data3 Data4 Data5 Our algorithm.4 D grah cuts D grah cuts Active contours

8 The high difficulty datasets have more noise, more inhomogeneities within the objects, and a bigger change in intensity between different images of the same dataset. Table 4 resents a comarison regarding the false ositive and false negative errors between our algorithm and the other segmentation algorithms, for the high difficulty datasets. TABLE IV. ERRORS IN LABELING PIXELS FOR THE HIGH DIFFICULTY DATASETS (%) Data6 Data7 Data8 Data9 Data Alg. B F E F B F E F B F E F B F E F B F E F Our alg D GC D GC AC Table 5 resents the comarison between the segmentation algorithms regarding the ercentage of images were different objects are wrongly connected, for the high difficulty datasets. TABLE V. PERCENTAGE OF IMAGES WHERE DIFFERENT OBJECTS WERE WRONGLY CONNECTED FOR THE HIGH DIFFICULTY DATASETS Percentage of images (%) Algorithm Data6 Data7 Data8 Data9 Data Our algorithm D grah cuts D grah cuts Active contours The results in comuting the ixel labeling error for our algorithm were comarable to or even better than the other algorithms results. The big difference can be observed in the ercentage of images where different objects were wrongly connected. Our algorithm discriminated quite well between different objects, even on noisy images. This does not hold for the other algorithms. In order to obtain an overview of the differences between the four segmentation algorithms, we have comuted, for all the datasets, the average of the false negative and false ositive error in labeling the ixels, and the average ercentage of images where different objects were wrongly connected. These differences can be seen in Fig. 5. Figure 5. Comarison between our algorithm and other segmentation algorithms regarding the average of the false ositive and false negative error in ixel labeling and the average ercentage of images where different objects were wrongly connected Even if the average error in ixel labeling from the active contours without edges segmentation is comarable to the result obtained with our algorithm, the average ercentage of images where different objects are wrongly connected shows that our imlementation is suerior. From the tests described in the revious section and from Fig. 5 we can draw the conclusion that our algorithm is 98% and 99% accurate regarding the labeling of background and foreground ixels, resectivel and 96% accurate in discriminating between different objects in CT images. Also, we can observe that in seven cases out of ten, our algorithm differentiated erfectly between objects. The ercentage of slices where different objects were wrongly connected is % in these cases. 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