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On Constrained Optimization Approach To Object Segmentation CHIA HAN, XUN WANG, FENG GAO, ZHIGANG PENG, XIAOKUN LI, LEI HE, WILLIAM WEE Department of Electrical & Computer Engineering and Computer Science University of Cincinnati Cincinnati, OH 45221-0030 USA Abstract: - A general contour constrained optimization approach is formulated to extract a target contour in an image for various applications. These different approaches of mean field annealing method, variational method, and evolutionary strategy are derived to provide global and local optimal solutions using level set numerical implementations. Impositions of constraints to characterize the contour interior features are employed on different specific applications. Contour shape models using either the thin plate spline matching method or the implicit polynomial representation method can be added into the optimization process to further improve contour extraction results. A set of illustrative examples for the applications of the approaches on biomedical images are presented. Key-Words: - Image Segmentation, Deformable Contour Methods, Constrained Optimization. 1 Introduction Intelligent systems with any visual ability for object understanding need to both possess object recognition capability at the high level of abstraction and provide flexible and robust capability to extract corresponding meaningful contours or surfaces of the object of interest at the low level of image processing. A major research activity in our Artificial Intelligence and Computer Vision Laboratory at the University of Cincinnati has been to solve image-based object understanding problems by using a framework that is built on Deformable Contour Methods (DCM) with a constrained energy minimization formulation of contour extraction [1][2][3][4][5][6]. Object understanding encompasses a wide gamma of concepts and processing methodologies in computer vision and pattern recognition, which may include object recognition, object interior structure description, object motion tracking and dynamic behavior of its parts, and visible surface recognition with occlusion due to multiple parts. All the processing are possible when starting with a successful search for a good object surface or contour delineating the target object in the image. From the traditional 2-D image processing point of view, given that each object has an isolatable and discernable boundary, contour extraction simply consists of the segmentation step in image processing, which is to partition an image into regions corresponding to the areas of interest and to extract their boundary accurately. The extracted contour is then used in subsequent operations to provide vital information for obtaining quantitative measurements such as area or volume, and for determining qualitative features for latter stage of multi-modality image registration, classification and interpretation. Thus, contour extraction, when considered in a broader context, becomes the essential step in object understanding. Not only does it form the basis of segmentation, i.e., the isolation of the object from the background, but also contains the ability to represent the characteristics needed for object recognition and labeling. When a framework that allows the integration of the latter capability of recognition within the process of abstracting the desired contour, then it forms the core of the object understanding. In this paper we will present such a framework, which is built upon the contour energy minimization formulation. It allows that the knowledge of understanding the object be represented in terms of the features and dynamic behavior of its constituent parts of interest and be expressed as constraints of its optimization index. But before we get to this point, we should review the basics of segmentation. The problem of image segmentation, especially extracting from images the structures that are found in nature, is by no means trivial. For instance, in biomedical images, there are many difficult challenges that are well known to the image processing research community. They include complex shape, inhomogeneous brightness distribution both at the interior of the region of interest and along the boundary region, very blur contour segments, and gaps (missing edges, e.g., due to drop of imaging signal during acquisition). Instances of such difficulties can be seen in the sample images, shown in Figure 1, of different subjects and from various imaging techniques: brain MRI images have complex shapes with inhomogeneous brightness distributions in both the interior and along the contour of the gray and white matters; and ultrasound heart images are rather noisy with fuzzy boundary and gaps in many portions of the contour. It is clear from these pictures that an image-based system with the capability for an automatic and robust segmentation and accurate contour extraction would make a great impact in delivering the critical functionalities of diagnosis, treatment, and surgery planning in the clinical applications and generating useful digital anatomical atlas from sources, such as Visible Human Project [7] in both clinical and educational applications. Fig. 1 Sample biomedical images. These hard image segmentation problems are not easily handled by the existing schemes, which are primarily boundary (edge)-based and regionbased segmentation methods. The edge-based approach is conceptually sound and there are many edge extraction methods to choose from. However, it has the contour linking problem when there are either too many edge points or too few edge points with gaps. The region-based approach is based on the region growing methods. There are also many similarity or dissimilarity functions to select from. But the same problems of contour linking, disjoint regions, and inhomogeneous intensity distribution plague this approach as well. A more elaborated and sophisticated approach has also been developed in the last decade. This is the deformable (also known as active or evolving) contour approach [8]. This approach includes the snakes [8][9][10][11], level sets [12][13][14][15], and non-deterministic methods [16][17][18], and thereafter will be referred to as deformable contour methods (DCMs). The methods in this category have less of the contour linking problems as they make use of the notion of closed contour. Thus, an interior/exterior structure could be established and a global energy minimization function for global control over local variation could be formulated. The reasoning behind these methods also conforms to the natural notion of the life dynamic evolution process. This evolutionary process always emanating from the interior, as applied to image segmentation, would provide an invaluable insight in distinguishing the connected segments. For instance, if many seeds grow into many plants, any leaf in dense foliage can still be traced to its own stem, even when the connection to its origin may be obscured by other leaves or complex local structures. The reverse tracking is also true; given the right stem is selected at the beginning, following the connection would lead to the desired leaf. But at the present time, any of the DCMs, when applied indiscriminately, have difficulty searching for the global minimum contour from an arbitrary interior point. It also has difficulty with gaps and very blur contour segments, complex shape, inhomogeneous interior, and lastly, it may be sensitive to the initial location. However, most of these difficulties can be overcome, provided that we have the understanding of their presence and know how to tackle them properly once they appear. The segmentation problem would become one of the image understanding (IU) problems and the many techniques used in IU could be applied. When each of the methods in the above major approaches faces some of the aforementioned difficulties and fails, special measure based on the knowledge of the problem domain is usually needed to tackle specific failures: additional image dependent pre and post processing, major algorithmic reformulation, and even the semi-automatic approach with a direct human intervention are generally introduced in attempt to overcome the problems. However, we should note that this approach normally involves distinct methods and processes that are applied separately; what is needed is a more integrated approach that offers robustness necessary to automatically handle the typical difficulties by keeping the strength while avoiding the shortcomings of the existing methods. 2 Problem Formulation The contour searching and extraction from an image using a deformable contour strategy can be formulated as a constrained optimization problem as follows: Let be an open domain subset of 2 and I ( x, y ) : be the image intensity function. Our problem is to find a close contour C ( q, t ) enclosing region c(t) at time t, such that the contour energy (1) E (C (q, t )) g ( I (C (q, t )) )ds C is the minimum under the constraint D( x, y ) TV for all ( x, y) C (t ) , where 0 TV is a constant, s is the arc length, q is the contour parameter, and I ( x, y ) is the image brightness at ( x, y ) . Also, 1 . (2) g ( I (C (q, t )) ) 2 1 I (C (q, t )) I (C (q, t )) is the gradient of I ( x, y ) with ( x, y ) on C ( q, t ) . The D( x, y ) in the constraint includes any contour interior brightness characterization features, such as smoothness, texture, and other structural features. Contour shape matching and fittings specifications can be added to the contour energy. 2.1 Interior characterization constraint We may consider any positive interior characteristic function as a modeling function of the interior region with D(x, y)>0, such as D( x, y) k1 A( x. y) k2 B( x, y) (3) with k1, k2 0 and k1+k2=1. That is, any interior structural information, including texture, can be incorporated in the deformation search as long as it can be expressed by a single positive potential function. From a pattern recognition or discrimination view point, the proposed algorithm provides an iterative partition or discrimination scheme of having D(x, y) as a similarity measure characterizing the interior with a similarity threshold TV to explicitly isolate the interior region from its surrounding rather than generating a decision rule. For example, for a smooth interior characterization, one may consider a function D( x, y ) that consists of two components: D( x, y ) A( x, y ) * B( x, y ) , with A( x, y ) being a measurement of smoothness expressed as a function of point gradient, and B ( x, y ) a measurement of smoothness as a potential function of deviation from an average brightness value I0 of the interior region (s ) . More specifically, these two components can be defined as follows: A( x, y ) 1 1 G*I ( x, y ) (4) 2 I ( x , y ) I 0 and B( x, y ) e (5) Therefore, D(x, y), as defined here, can be interpreted as a homogeneity measure of smoothness at an interior point (x, y). The large value of the gradient and the large deviation of the point brightness from I0 produce a much smaller value of D(x, y). In other words, the threshold value TV in the constraint is the lowest smoothness value allowed in C . Thus, the contour optimization of E in Eq. (1) would be subjected to this interior homogeneity constraint and the constraint D( x, y ) TV can be thought of as an interior description of the contour C ( q, t ) and D( x, y ) can be redefined to model the interior C . In summary, proper imposition of constraints to characterize the contour interior features may be employed effectively on different specific applications. In addition to the modeling of the interior, modeling of the contour itself may further assist the contour extraction process. We will see in the next two subsections how two contour shape models are added to the optimization process to improve contour extraction results. They are the thin plate spline matching method and the implicit polynomial representation method. 2.2 Thin plate spline shape matching The basic idea of the thin plate spline shape matching is the calculation of the matching error between a current contour and a model contour, where the model contour is transformed to the current contour space using thin plate spline method to obtain the coordinate transformations, f x and f y . To facilitate the computation of the coordinate transformations, the matching distance is approximated by the matching error between two contour point sets sampled from the respective continuous contours. More detailed discussion of the above results can be found in [4]. 2.2.1 Modified problem formulation Our problem then is to find C(q,t) such that the energy function E( ) is minimized, subject to the region constraint D( ). E (C (q, t ), M g (q ), f x , f y ) w g ( I (C (q, t )) )ds 0i , j 0i j n C ( q ,t ) (1 w) cs (C (q, t ), M (q ), f x , f y ), (6) g 0 w 1 D( x, y) TV xi y j a 00 a10 x a 01 y a 20 x 2 a11 xy a 0 n y n (11) X A if (x, y) c(t) where 1 2 dist (( x(q), y(q)), ( f x (u(q), v(q)), f y (u(q), v(q)))) dq (7) 0 ( x(q ), y (q )) and (u (q), v(q)) are contour points on the continuous contour functions, (q ) and M (q) , respectively. dist 2 (( x(q), y (q)), ( f x (u (q), v(q)), f y (u (q), v(q))) (8) ( x(q) f x (u (q), v(q))) 2 ( y (q) f y (u (q), v(q))) 2 2.2.2 Solution The solution includes two alternating procedures: curve evolution and shape matching. The formula for curve evolution in the first procedure can be defined as follows. T A a00 a10 a01 a0n and X 1 x y x 2 xy y n . A minimum least square approach using the contour point set is applied to derive the coefficient set A. The invariant feature set is then derived for matching. The same operation on the model contour to produce the invariant feature set is also executed. The matching operation is the calculation of the Mahalanobis distance. Detailed discussion can be found in [5]. T 2.3.1 Modified problem formulation Let the polynomial representation of C(q,t) be the coefficient vector A. Z f ( A) ( x, y) : f ( x, y) X T A 0 Our problem then is to find C(q,t) such that E (C (q, t )) w g ( I (C (q, t )) )ds C ( q ,t ) C (q, t ) w kg( I ) (g N ) N t (1 w) D f ( x, y ) 1[ D( x, y ) TV ] N ij T where M g (q) is the shape model contour, cs ((q), M g (q), f x , f y ) a f ( x, y ) (12) (1 w)( * ) T g ( * ), 0 w 1 * (9) with is minimized subject to the following constraints: i) D( x, y) TV if (x, y) c(t) 1 dist ( x(q, t ), y(q, t ), Z f ( A))dq T f , D f ( x, y ) ii) dist 2 (( x, y ), T ( M g ) if ( x, y ) is inside T ( M g ) , if ( x, y ) T ( M g ) 0 dist 2 (( x, y ), T ( M g ) if ( x, y ) is outside T ( M g ) where ( * )T g* ( * ) represents the similarity between the two shapes represented by the shape features from A and the model shape features * from the model coefficient vector A* derived from fitting shape model M g (q) . g* is the information T (M g ) ( f x 0 (10) (u (q ), v(q)), f y (u (q), v(q ))) : (u (q), v(q)) M g (q) 1 0 is the Lagrange multiplier, k is the contour curvature and N is the normal direction. In the shape matching procedure the goal is to compute f x and f y by minimizing the matching error between two equally spaced point sets sampled from (q ) and M (q) . Here, thin plate spline method is used to find the solution. 2 matrix of * . T f 0 is a constant. 2.3.2 Solution The solution includes two alternating procedures: curve evolution and shape matching. The resulting curve evolution solution to this formulation is: C (q, t ) 1 [ D( x, y ) TV ] N (13) t wkg( I ) w(g N ) 2 D f ( x, y ) N 2.3 Implicit polynomial shape representation In here, a close contour is represented by a polynomial of degree n in ( x, y ) as with dist 2 (( x, y ), Z f ( A)) , ( x, y ) inside Z f ( A) D f ( x, y ) 0 , ( x, y ) Z f ( A) 2 dist (( x, y ), Z f ( A)), ( x, y ) outside Z f ( A) and 2 0 is the Lagrange multiplier. The shape matching procedure consists of holding C(q,t) constant and minimizing Eq. (12) by adjusting A. A 2 X 3TL X 3 L (1 w)Q( A* ) (1 w)Q( A ) A 1 * * 2 X 3TL b the average intensity and the standard deviation inside the deforming contour C (q, t , Ti ) at 2.4 Pattern recognition consideration As the recognition is now integrated in the contour extraction process, the pattern recognition tasks need to be considered. In this consideration, a pattern class is defined as a set of different shape models. Classification of a resultant contour shape output, from either one of the methods as presented in Sections 2.2 or 2.3, would be decided by its shape distance measure between the obtained shape and a given class model. Once the constraints are defined, both the interior and the contour model, we are ready to present the contour optimization approaches using the formulated constraints. temperature Ti . 1C (Ti ) is the Lagrange multiplier. (1) Curve evolution solution: Variational based deformable contour searching, as shown in Equation (14), with an MFA parameter estimation scheme, is used. C (q, t , T ) (14) t 1* (T )D( x, y, T ) TV kg( I ) g N N Level set numerical method is used. (2) Interior characterization and constraint ( I ( x, y) I 0 ) 2 D( x, y) TV for all 2 ( x, y) C (t ) , where 0 is a constant. The MFA method treats the estimations of I 0 and 2 as functions of annealing temperature Ti , i 1,n . D( x, y, Ti ) 3. Contour Optimization Approaches This section presents the concepts for three constrained optimization contour extraction methods as well as their highlights. These approaches, which provide global and local optimal solutions using the level set numerical implementations for extracting target contours in images, include: the mean field annealing method, variational method, and evolution strategy. For each of the three approaches, a summary of the methods features, such as the curve evolution formulation, the interior characterization and the constraint, analysis, and parameter estimation, is included. 3.1 Mean field annealing method The Mean Field Annealing (MFA) method, as the name indicates, uses the mean field annealing approach to avoid local minima and to provide a fast and efficient global approach to search for constrained global optimal solutions to the problem of object boundary extraction. Using this method, with a given contour energy function, different target boundaries can be modeled as constrained global optimal solutions under different constraints expressed as a set of parameters characterizing the target contour interior structures. The method derivation and a detailed discussion can be found in [6]. The formulas have the following notation. Ti is the annealing temperature. I 0C (Ti ) and C2 (Ti ) are Thus the constraint becomes ( I ( x, y ) I 0 (Ti )) TV and I 0 and 2 (Ti ) 2 2 become I 0 (Ti ) and 2 (Ti ) . (3) Analysis and optimality * Lagrange approach * Global solution (4) MFA parameter estimations: I 0 (Ti ) I 0C (Ti ) 2 (Ti ) C2 (Ti ) Ti T 1* (Ti ) 1C (Ti ) i As an illustrative example, three target boundaries in a synthetic image are modeled as constrained global energy minimum contours with different constraint parameters and are successfully located using the derived algorithm. A conventional variational based deformable contour method with the same energy function and constraint fails to achieve the same task. Experimental evaluations and comparisons with other methods on ultrasound pig heart, MRI knee, and CT kidney images where gaps, blur contour segments having complex shape and inhomogeneous interiors have been conducted with most favorable results [6]. 3.2 Variational Method The variational method first takes a Lagrange approach to the constrained optimization problem and then uses a derivative based approach to derive the curve evolution solution. As a result, the interior constraint is reflected in this curve evolution solution and act as a “balloon force” that can be used to characterize any interior property and/or structure including the “balloon force” used in the geodesic snake [15] as a special case. The method derivation and a detailed discussion can be found in [2]. Here is a summary of the highlights for this method. (1) Curve evolution solution: Variational based deformable contour searching with Equation (15) is used. 0 is a constant. C (q, t ) ([ D( x, y ) TV ] (15) t kg( I ) (g N )) N Level set numerical method is used. (2) Interior characterization and constraint D( x, y ) TV for all ( x, y) C (t ) . D( x, y ) can be any function characterizing the interior; (3) Analysis and optimality * Lagrange approach * Local solution interior; D( x, y ) TV for all ( x, y) C (t ) . (3) Analysis and optimality * Not a Lagrange approach * Global solution is not guaranteed N(b1, b2) indicates the Gaussian random variable with mean ‘b1’ and standard deviation ‘b2’. Z1e Z with 1 Z1 I ( x, y ) M ( x, y ) 1 and 1 2 . M(x, y) is the local average intensity of size 3 by 3 with (x, y) being the center. 4 Applications with Sample Results The comparison of these three different constrained optimization approaches are made on the following set of biomedical images: microscopic blood cell in Fig. 2a, an MRI sagittal brain corpus callosum image in Fig. 3a, two axial MRI brain images in Fig. 4a and 5a, and a noisy CT kidney image in Fig. 6a. The results of applying the three approaches are shown in Figures 2-6: (b) the MFA method, (c) the variational method, and (d) the evolution strategy, respectively. 3.3 Evolution Strategy The evolution strategy [19] is to mimic the process of evolution in nature, the deriving process for emergence of complex and well-adapted organic structure. This evolution strategy is used to deform C(s, t) until an optimum is reached. It is a recursive process in which a population of individuals, the parents, mutates and recombines to generate a large population of offspring. These offspring are then evaluated according to a fitness function and as a result, a best subset of offspring is selected to replace the existing parents. There are three main factors to consider: (a) The representation of individual contours (represented by states), (b) variations of states, and (c) the selection scheme. The method derivation and a detailed discussion can be found in [1]. The highlights of this method are: (1) Curve evolution solution: Generation of a small population of candidate contours and application of evolution strategy to iteratively select a subset of contours with lowest energies C (q, t ) (16) ( D( x, y )(1 N (0,1)) TV ) N t Level set numerical method is used. (2) Interior characterization and constraint D( x, y ) can be any function characterizing the (a) (b) (c) (d) Fig. 2 Blood cell microscopic image. (a) original image. Results from (b) MFA method, (c) variational, and (d) evolution method. (a) (b) (c) (d) Fig. 5 Axial MRI brain image-2. (a) original, (b) MFA, (c) variational, (d) evolution method. (c) (d) Fig. 3 Brain corpus callosum image. (a) original, (b) MFA, (c) variational, (d) evolution method. (a) (b) (c) (d) Fig. 4 Axial MRI brain image-1. (a) original, (b) MFA, (c) variational, (d) evolution method. (a) (b) Fig. 4 Axial MRI brain image-2. One of the implementation details is to select the initial points. The three dark points in Fig. 3a and Fig. 6a are the locations where the initial points were chosen for the experiments. (a) (b) (c) (d) Fig. 6 Noisy CT kidney image. (a) original, (b) MFA, (c) variational, (d) evolution method. Overall, from the sample results of Fig. 2 through Fig. 6, we can see that MFA method has the best results comparing to those of the variational method and evolution strategy. One major reason is that the MFA is a global optimization method thus is more robust to image noises and undesired local energy minima. As explained in Section 1, image understanding can be greatly enhanced in applications where object models can be incorporated into the object segmentation and recognition process. Two contour models were introduced in Section 2. The following subsections will illustrate the results of using the models in augmenting the image segmentation and recognition. 4.1 Results using the thin plate spline model The fruit image shown in Fig. 7a, containing two apples, one banana, one cucumber, and one pear on a 144 by 108 image, is obtained from [20] and used here for illustration. We first use the apple model on the algorithm with an initial seed region inside each fruit object. The algorithm runs 5 times, one on each fruit. The resulting five contours are shown in Fig. 7b. The matching measure is based on the shape matching distance values, in terms of bending energy Eb , as shown in column one of Table 1. Apple II Banana Cucumber Apple I Pear (a) (b) selecting the row minimum value as indicated by (*). As one can see in Fig. 7, both Apple I and Apple II contours are rather nicely segmented even with Apple II touching the cucumber. Without the apple model, one would have some difficulty separating the two. The experiment is repeated for the banana model, the cucumber model and the pear model with the resulting contours shown in Figs. 7c, 7d and 7e, and Eb values in columns 2, 3 and 4, of Table 1, respectively. The banana in Fig. 7c has the best segmentation result noticing the dark banana skin cover on the upper right side of the banana. The cucumber in Fig. 7d has the best segmentation among all other models in Figs. 7b, 7c and Fig. 7e. Similar result can be reported on the pear in Fig. 7e. All are successfully recognized. The algorithm has also been successfully applied on Fig. 8a, which is an image of size 300 by 400 consisting of two bananas partially occluded by a cucumber. (c) Fig. 8. Fruit image-2, with overlapping objects. (e) (d) Fig. 7 Fruit image. (a) original image. Results using (b) the apple model, (c) the banana model, (d) the cucumber model, and (e) the pear model. Model Bending Energy Apple Banana Cucumber Pear Apple I 0.627* 1.361 1.276 1.773 Apple II 2.381 2.897 2.483 Banana 1.699* 4.88 0.498* 0.700 0.557 Cucumber 4.605 6.737 Pear 2.948 2.224 1.919* 2.349 1.954* Object 3.477 Table 1. Resulting Performance Matrix Our recognition decision is based on the minimum Eb value on each object in using different model assumptions as shown in Table 1, i.e., 4.2 Results using the implicit polynomial model Leaf image is used in the experiments with the implicit polynomial model as presented in Section 2.3. There are two parts to our experiment. Part 1 is to illustrate the segmentation capability of the algorithm. Here, we use the leaf, marked with an “X1”, as the model shown in Figure 9a. The final segmentation result of leaf with a “Y12” mark in Figure 9a is shown in Figure 9b. Figure 9c shows the resulting segmentation contour using the same velocity equation of Eq (13) without using the model. Similarly, using “X2” of Figure 10a as model provides a segmentation result of “Y22”on Figure 10b. As for the illustration of the model selection result, we compute the shape distance measure Dm value of 4 leaves data, “Y11” and “Y12” for class 1 in Figure 9a, and “Y21” and “Y22” for class 2 in Figure 10a. The resulting Dm using different models deforming targets and classification decisions are shown in Table 2. By selecting the lowest Dm value, all classifications are correct in 4 classes. 5 Conclusion (a) (b) (c) Fig. 9 Leaf-1 image. (a) original. (b) Segmented Leaf Y12 using Leaf X1 as a model. (c) Without model. (a) (b) Fig. 10 Leaf-1 image. (a) original. (b) Segmented Leaf Y22 using Leaf X2 model. Data(Deforming Model) Y11 (Model1) Y11 (Model2) Y12 (Model1) Y12 (Model2) Y21 (Model2) Y21 (Model1) Y22 (Model2) Y22 (Model1) DM 14.1 30.7 15.7 38.0 16.0 30.6 12.9 22.4 Class Decision Class 1 Class 1 Class 2 Class 2 Table2 Model Selection Result This paper presented a general contour constrained optimization approach to object segmentation. By integrating knowledge of image pattern structures, both the contour interior and contour itself, in terms of constraints and models into the formulation of the solution for image processing, this approach sets up a framework for a more holistic approach to object understanding. From a computational point of view, three different approaches of mean field annealing method, variational method, and evolutionary strategy are presented to provide global and local optimal solutions using level set numerical implementations. Constraints that characterize the contour interior features are employed on different specific applications. 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