A fast method to segment images with additive intensity value

A Fast Method to Segment Images with Additive Intensity Value LAU TZE SIONG (B.Sc.(Hons.),NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgements I would like to express my heartfelt gratitude to Assistant Professor Andy Yip for accepting me as his graduate student I would also like to take this opportunity to thank him for his help and guidance throughout this project His advice on different aspects of life, not just academically, influenced me greatly and is much appreciated i Contents Preliminaries 1.1 Soft Additive Model 1.2 Existence of solutions for the Soft Additive Model Methods to optimize the soft additive model 10 2.1 Augmented Lagrangian method on level sets 11 2.2 Lagged Curvature Method 17 2.2.1 Formulation of Outer Iterations 2.2.2 Augmented Lagrangian Method on the Inner Iterations 25 Solutions of subproblems 17 31 3.1 Solution for the augmented Lagrangian method on level sets 33 3.1.1 Problem (2.3), (2.17) and (2.19) 34 3.1.2 Problem (2.4) 36 3.1.3 Problem (2.5) 36 3.1.4 Problem (2.6,2.7) 37 3.1.5 Problem (2.8,2.9) 38 ii CONTENTS 3.2 Solutions for the lagged curvature method 39 3.2.1 Problem (2.23) 40 3.2.2 Problem (2.24)-(2.26) 40 Numerical Results 42 Conclusion 65 iii Summary We consider the problem of segmenting a pair of overlapping objects whose intensity level in the intersection is approximately the sum of individual objects We assume that the image domain Ω = [0, N ] × [0, M ] contains two overlapping objects O1 ⊆ Ω and O2 ⊆ Ω and consider images u ∶ Ω → R such that ⎧ ⎪ ⎪ ⎪c10 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪c ⎪ 01 ⎪ ⎪ u(x, y) ≈ ⎨ ⎪ ⎪ ⎪c10 + c01 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪c ⎪ ⎪ 00 ⎪ ⎩ if (x, y) ∈ O1 /O2 if (x, y) ∈ O2 /O1 (1) if (x, y) ∈ O1 ∩ O2 if (x, y) ∈ Ω/(O1 ∪ O2 ) The identification of the true objects O1 and O2 from a given image u is called an additive segmentation problem A segmentation of an image u is a pair of objects {E1 , E2 } such that E1 , E2 ⊆ Ω and {E1 /E2 , E2 /E1 , E1 ∩ E2 , Ω/(E1 ∪ E2 )} forms a partition of Ω with E1 , E2 approximating the true objects O1 , O2 The real-world applications of this model include X-ray images [1], magnetic resonance angiography images [14, 7] and microscopy imiv CONTENTS ages recording protein expression levels [11] which standard segmentation models does not work In the paper [16], the authors proposed to solve the additive segmentation problem by looking for a segmentation {E1 , E2 } and a set of constants c = (c10 , c01 , c11 , c00 ) that minimize the soft additive energy This energy contains a curvature term Applying the gradient descent method to the model leads to a fourth-order Euler-Lagrange equation which is often difficult to solve efficiently In this thesis, we present two methods to optimize the soft additive model In the first method, we adapt the augmented Lagrangian method developed in [25] to optimize the Euler’s elastica to solve the Euler-Lagrange equations In the second method, we formulate a new Euler-Lagrange equation by placing the terms resulting from the curvature term in the Euler-Lagrange equation one step behind the rest and call it the lagged Euler-Lagrange equation In each step, we formulate a constrained convex minimization problem whose minimizer is a solution of the lagged Euler-Lagrange equation Each of these constrained convex minimization problems can be solved by applying the augmented Lagrangian method [10, 24] The subproblems arising from the augmented Lagrangian method can be solved directly by either an explicit formula or by applying the Discrete Cosine Transform The solution of the Euler-Lagrange equation is achieved by allowing the iterative map to converge to a fixed point This thesis is organized as follows We first review the soft additive model and some of its results in Chapter In Chapter 2, we give details of the v CONTENTS adaptation of the augmented Lagrangian method to solve the soft additive model and also the lagged curvature method In Chapter 3, we provide solutions for the unconstrained minimization problems occurring in the algorithms developed The numerical results are given in Chapter and the thesis is summarized in Chapter vi List of Figures 4.1 Solutions for Image 47 4.2 Solutions for Image 48 4.3 Solutions for Image 49 4.4 Solutions for Image RHip 50 4.5 Solutions for Image Vessel 51 4.6 Solutions for Image Arm 52 4.7 Comparison of algorithms with respect to energy for Image 53 4.8 Comparison of algorithms with respect to energy for Image 53 4.9 Comparison of algorithms with respect to energy for Image 54 4.10 Comparison of algorithms with respect to energy for Image RHip 54 4.11 Comparison of algorithms with respect to energy for Image Vessel 55 4.12 Comparison of algorithms with respect to energy for Image Arm 55 4.13 Comparison of segmentation errors for Image 56 4.14 Comparison of segmentation errors for Image 56 4.15 Comparison of segmentation errors for Image 57 4.16 Comparison of segmentation errors for Image RHip 57 vii LIST OF FIGURES 4.17 Comparison of segmentation errors for Image Vessel 58 4.18 Comparison of segmentation errors for Image Arm 58 4.19 Segmentations for Image 59 4.20 Segmentations for Image image2 60 4.21 Segmentations for Image image3 61 4.22 Segmentations for Image RHip 62 4.23 Segmentations for Image Vessel 63 4.24 Segmentations for Image arm 64 viii Chapter Preliminaries 1.1 Soft Additive Model A closed plane curve is a map γ ∶ [0, 1] → R2 such that γ(0) = γ(1), and is continuous for every t ∈ [0, 1] It is said to be regular if dγ dt dγ dt exists ≠ for each t ∈ [0, 1] We denote the arc length parameter by s and γ ′ , γ ′′ denotes the first and second derivative of γ with respect to s If the nth derivative γ (n) exists and is continuous, we say that γ is a curve of class C n , and we write γ ∈ C n We denote C ∞ = ∩∞ C n We also denote the curvature of a curve as n=1 κ = γ ′′ Given a Lebesgue measurable set E ⊆ R2 , we denote its boundary by ∂E We say that a bounded open set E is of the class C ∞ if and only if its boundary ∂E is a closed plane curve of class C ∞ A signed distance function of a set E is a function Dist(E) ∶ Ω → R defined as Dist(E)(x) ≜ (−1)χE (x) inf{∣x − y∣ ∶ Symmetric difference from Suggested Segmentation CHAPTER NUMERICAL RESULTS Comparison of Algorithms for image1 10 Lagged Curvature Augmented Lagrangian Gradient Descent 10 10 100 200 300 400 time in seconds 500 600 Symmetric difference from Suggested Segmentation Figure 4.13: Comparison of segmentation errors for Image Comparison of Algorithms for image2 10 Lagged Curvature Augmented Lagrangian Gradient Descent 10 10 10 10 10 10 200 400 600 time in seconds 800 1000 Figure 4.14: Comparison of segmentation errors for Image 56 Symmetric difference from Suggested Segmentation Comparison of Algorithms for image3 10 Lagged Curvature Augmented Lagrangian Gradient Descent 10 10 200 400 600 time in seconds 800 1000 Figure 4.15: Comparison of segmentation errors for Image Symmetric difference from Suggested Segmentation Comparison of Algorithms for RHip Lagged Curvature Augmented Lagrangian Gradient Descent 3.5 10 3.4 10 3.3 10 3.2 10 50 100 150 200 250 time in seconds 300 350 400 Figure 4.16: Comparison of segmentation errors for Image RHip 57 Symmetric difference from Suggested Segmentation CHAPTER NUMERICAL RESULTS Comparison of Algorithms for Vessel Lagged Curvature Augmented Lagrangian Gradient Descent 3.5 10 3.4 10 3.3 10 3.2 10 50 100 150 200 250 time in seconds 300 350 400 Symmetric difference from Suggested Segmentation Figure 4.17: Comparison of segmentation errors for Image Vessel Comparison of Algorithms for arm 10 Lagged Curvature Augmented Lagrangian Gradient Descent 10 10 50 100 150 200 250 time in seconds 300 350 400 Figure 4.18: Comparison of segmentation errors for Image Arm 58 Gradient Descent Augmented Lagrangian Lagged Curvature 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 200 250 100 150 200 250 100 150 200 250 50 100 150 200 250 150 200 150 50 150 100 50 200 250 100 150 200 50 50 100 150 50 50 100 100 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 Figure 4.19: Comparison of Segmentations provided by the gradient descent method, augmented Lagrangian method and the lagged curvature method on level sets when applied Image at t = 50s(first row),100s(second row), 200s(third row), 400s(forth row) and 600s(fifth row) 59 CHAPTER NUMERICAL RESULTS Gradient Descent Augmented Lagrangian Lagged Curvature 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 200 250 100 150 200 250 100 150 200 250 50 100 150 200 250 150 200 150 50 150 100 50 200 250 100 150 200 50 50 100 150 50 50 100 100 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 Figure 4.20: Comparison of Segmentations provided by the gradient descent method, augmented Lagrangian method and the lagged curvature method on level sets when applied to Image at t = 100s(first row),200s(second row), 400s(third row),600s(forth row)and 1000s(fifth row) 60 Gradient Descent Augmented Lagrangian Lagged Curvature 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 200 250 100 150 200 250 100 150 200 250 50 100 150 200 250 150 200 150 50 150 100 50 200 250 100 150 200 50 50 100 150 50 50 100 100 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 50 50 50 100 100 100 150 150 150 200 200 200 250 250 50 100 150 200 250 250 50 100 150 200 250 Figure 4.21: Comparison of Segmentations provided by the gradient descent method, augmented Lagrangian method and the lagged curvature method on level sets when applied to Image t = 100s(first row), 200s(second row), 400s(third row), 600s(forth row) and 1000s(fifth row) 61 CHAPTER NUMERICAL RESULTS Gradient Descent Augmented Lagrangian Lagged Curvature 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 80 100 80 100 120 40 60 80 100 120 40 60 80 100 120 20 40 60 80 100 120 80 100 60 60 80 40 20 60 120 40 60 100 20 40 80 20 20 40 60 20 20 40 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 Figure 4.22: Comparison of Segmentations provided by the gradient descent method, augmented Lagrangian method and the lagged curvature method on level sets when applied to the Xray of a right hip (RHip) at t = 50s(first row), 100s(second row), 150s(third row), 200s(forth row) and 400s(fifth row) 62 Gradient Descent Augmented Lagrangian Lagged Curvature 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 80 100 80 100 120 40 60 80 100 120 40 60 80 100 120 20 40 60 80 100 120 80 100 60 60 80 40 20 60 120 40 60 100 20 40 80 20 20 40 60 20 20 40 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 20 20 20 40 40 40 60 60 60 80 80 80 100 100 100 120 120 20 40 60 80 100 120 120 20 40 60 80 100 120 Figure 4.23: Comparison of Segmentations provided by the gradient descent method, augmented Lagrangian method and the lagged curvature method on level sets when applied to the MRA of two overlapping blood vessel (Image Vessel) at t = 50s(first row), 100s(second row), 150s(third row), 200s(forth row) and 400s(fifth row) 63 CHAPTER NUMERICAL RESULTS Gradient Descent Augmented Lagrangian Lagged Curvature 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 100 90 100 10 20 30 40 50 60 70 80 90 100 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 100 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 80 90 100 70 80 90 60 70 80 50 60 70 40 50 60 30 40 50 20 30 40 10 20 30 10 10 20 10 10 100 90 100 10 20 30 40 50 60 70 80 90 100 100 10 20 30 40 50 60 70 80 90 100 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 80 90 90 100 90 100 10 20 30 40 50 60 70 80 90 100 100 10 20 30 40 50 60 70 80 90 100 10 10 10 20 20 20 30 30 30 40 40 40 50 50 50 60 60 60 70 70 70 80 80 90 90 90 100 100 100 10 20 30 40 50 60 70 80 90 100 80 10 20 30 40 50 60 70 80 90 100 Figure 4.24: Comparison of Segmentations provided by the gradient descent method, augmented Lagrangian method and the lagged curvature method on level sets when applied to the Xray of an arm (Image arm) at t = 50s(first row), 100s(second row), 150s(third row), 200s(forth row) and 400s(fifth row) 64 Chapter Conclusion We have provided two methods to solve the soft additive model In the augmented Lagrangian method on level sets, we modified the idea of the augmented Lagrangian method applied to solve problems related to the Euler’s Elastica[25] to solve the soft additive model In the lagged curvature method, the problem of solving the Euler-Lagrange equations of the soft additive functional is approached by solving a sequence of lagged Euler-Lagrange equations This sequence of lagged Euler-Lagrange equations turns out to be the Euler-Lagrange equations of a sequence of minimization problems Using a similar method demonstrated in [21], we can find the minimizer to each minimization problem in the sequence by solving a convex problem Finally, these convex problems can be solved by the augmented Lagrangian method and it turns out that the subproblems that arises are easy to solve 65 CHAPTER CONCLUSION From the numerical examples we have provided, the augmented Lagrangian method on level sets does improve the computational time of the gradient descent method In most cases, there is a period of time where it has better performance in energy minimization than the gradient descent method However, it can be seen in the real images that the gradient descent method eventually catches up and out-performs the augmented Lagrangian method on 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additive, the value of γ can be lowered to give... THE AUGMENTED LAGRANGIAN METHOD ON LEVEL SETS 3.1 Solution for the augmented Lagrangian method on level sets In the case of the augmented Lagrangian method on level sets, the augmented Lagrangian... we adapt the augmented Lagrangian method developed in [25] to optimize the Euler’s elastica to solve the Euler-Lagrange equations In the second method, we formulate a new Euler-Lagrange equation