An improved active contour model based on level set method

Transcription

1 ( ) Journal of East China Normal University (Natural Science) No. 1 Jan. 015 : (015) , (, 0006) :,., (SPF),.,,.,,,,. : ; ; ; : O948 : A DOI: /j.issn An improved active contour model based on level set method WU Zhang, ZHU Min (Computing Center, East China Normal University, Shanghai 0006, China) Abstract: By using the local regional information which has the ability to enhance the image, an improved active contour model based on level set method is proposed. Defining a novel SPF function with a nonnegative kernel function and local intensity clustering property, the novel model could draw upon intensity information in local regions at a controllable scale. In addition, the penalizing term which can be called distance regularization term drives the motion of the zero level set toward desired locations. Experimental results for real and synthetic images show the desirable proposed method performances and the model efficiency on intensity inhomogeneities and weak boundaries. Key words: an improved active contour model based on level set method; weak boundary; intensity inhomogeneity; image segmentation 0 ( ) [1]. : : : 863 EB (013AA01A11) :,,,. naruto wu@16.com. :,,,,. mzhu@cc.ecnu.edu.cn.

2 16 ( ) 015,,,. : []. Chan Vese [3-4] Mumford-Shan (CV ),,., CV,. Zhang (SPF), (SPF ) [5].,,, CV., Li (ICP ) [6],, SPF,,, ;,, ICP [6]., SPF,,, SPF SPF.,,,. : 1, CV CV, ;, 3, ;, CV Ω, I : Ω R. C Ω, I. F (C, c 1, c ) = µ L (C) + λ 1 inside(c) I c 1 dx + λ outside(c) I c dx. (1) : L (C) C ; c 1 c, ; µ 0, λ 1, λ 0. C, (1), I. u C, {C u (x) = 0 }. u, C Ω Ω 1 = {u (x) > 0} Ω = {u (x) < 0}.,, (1) Heaviside H (u) = (1 1 + π arctan ε) u. δ (u) = dh(u) du = 1 π ε ε +u Dirac, (1) u, F (C, c 1, c ) =µ δ (u) dx Ω + λ 1 I c 1 H (u)dx + λ I c (1 H (u)) dx. () Ω Ω

3 1, : 163, u, c 1, c (), c 1 = Ω IH (u)dx 1 Ω 1 H (u)dx, c Ω = I (1 H (u))dx Ω (1 H (u))dx. (3) c 1, c I C. c 1, c, u (), [ ( ) ] t = δ µdiv λ 1 (I c 1 ) + λ (I c ). (4) CV, (3) (4),,. 1. SPF Zhang (SPF), [5], SPF. [5] SPF I (x) c1+c spf (I (x)) = max ( I (x) c 1+c ), x Ω ( ( ) ) = spf (I (x)) div + α + spf (I (x)), t, c 1, c CV (3). x Ω [5] SBGRLS, t = spf (I (x)) α, x Ω. (6), α,,.,,, CV,. 1.3 ICP Li (Intensity Clustering Property) [6 7],, ICP., Ω, I : Ω R. J,, c 1 c ; b ; n Gaussian. I I = bj + n. y Ω, r y O y = {x : x y r}, Ω 1, Ω O y Ω i, i = 1,., y O y x b(x) b(y), x O y Ω i, i = 1,, b(x)j(x) b(y)c i. I : x O y Ω i, i = 1,, I (x) = b (y)c i + n (x). I i y = {I(x) : x O y Ω i } m i b(y)c i, n m i Gaussian, I 1 y, I y m i b(y)c i, i = 1,. O y m i b(y)c i, i = 1,, (5)

4 164 ( ) 015. O y, λ 1 = λ = 1, (1) c 1 c ε y = i=1 Ω i O y I (x) m i dx i=1 Ω i O y I (x) b (y)c i dx (7) (7) m i b(y)c i, i = 1,, m i b(y)c i, i = 1, (7). [7], K : R [0, + ). K, x y, K (y x), 0, ε y = = K (y x) I (x) b (y)c i dx Ω i O y i=1 i=1 Ω i K (y x) I (x) b (y)c i dx. (8), ε y, ε y {O y Ω 1 } {O y Ω } O y, ε y,. Ω Ω 1, Ω, ε y Ω y., Ω y ε y, ε = ε y dy,. ε = i=1 Ω i K (y x) I (x) b (y)c i dxdy. (9),,,,. SPF, SPF,,. SPF ICP.,.,,.,. SPF CV,. ICP, SPF,. CV, (9) (1),,, ( b (y) K (y x)(c 1 c ) I (x) c ) 1 + c b (y). (10) (c 1 c ) b (y) K (y x)(i (x) b (y)(c 1 + c )/),,., Hamilton-Jacobi

5 1, : 165 ( b (y) K (y x) I (x) c ) 1 + c b (y). (11) b (y) K (y x) (I (x) b (y)(c 1 + c )/) < 0, Hamilton-Jacobi,.,., b K, (11),., SPF spf Kb (I (x)) = K (y x) b (y) ( I (x) c1+c b (y) ) max ( K (y x) b (y) ( I (x) c1+c b (y) ) ), x Ω. (1) SPF,. c 1, c min(i) c 1, c max(i), min (I) < b (y)(c 1 + c )/ < max(i), (1) SPF I (x), SPF, ( ( ) ) t = spfkb (I (x)) div + α + spf Kb (I (x)), x Ω. (13),,, u, u 1 [5,8]. SPF I (x),. [5] SBGFRLS : (13) div(/ ),,. u, [] div(/ ), P (u) = 1 ( 1) dx. (14) u. [9-10], δ (u),, () ( ) =µ div + spf Kb (I (x)) α t, µ 0, ν > 0. + spf Kb (I (x)) + ν ( u div ( )). (15), (15),, spf Kb (I(x))., SPF ( t = µ u div, Gaussian ) + spf Kb (I (x)) α + ν ( ( )) u div. (16) K (u) = { 1 a e u /σ, u ρ 0,. (17)

6 166 ( ) 015 : a, K (u) = 1, σ Gaussian ; ρ O y., O y ρ. c 1, c b c 1 = Ω (b K)IH (u)dy 1 Ω 1 (b K) H (u)dy, c = Ω (b K)I (1 H (u))dy Ω (b K) (1 H (u)) dy, b = (I (c 1H (u) + c (1 H (u)))) K (c 1 H (u) +. c (1 H (u))) K (18) Intel(R)Core(TM) Duo CPU E7400, Windows 7.80 GHz 4.00 GB RAM PC, MATLAB 013 a., SPF µ = , ν = 1, t = 0.1, α. 3.,. Step1:. ρ (x, y) Ω 0 Ω 0 u (x, y, t = 0) = 0 (x, y) Ω 0 (19) ρ (x, y) Ω Ω 0, ρ > 0, Ω 0 Ω, Ω 0. Step: c 1 (u n i,j ) c (u n i,j ). Step3: ( ) ( ( ))] u n+1 i,j = u n i,j [µ + t u div + spf Kb (I (x)) α + ν u div. Step4: u,, Step ,,. 1(a) 1(c) 1(e) SPF ICP. 1(b) 1(d) SPF 80 ICP 150.,. 1(f) 50, s.,.,,, SPF.

7 1, : SPF ICP (a) SPF ; (b) SPF 80 ; (c) ICP ; (d) ICP 150 ; (e) ; (f) 50, α 35 Fig. 1 Applications of Zhang s model, Li s model and our method to an inhomogeneous image (a) Initialization of Zhang s model; (b) 80 iterations of Zhang s model;(c) Initialization of Li s model;(d) 150 iterations of Li s model;(e) Initialization of our method; (f) 50 iterations of our method, the parameter α = 35, 80., 60 80,,. (a) (c) (e) SPF ICP. (b) SPF,,,, s. (d) ICP

8 168 ( ) 015,,, s. (f),, s, ICP.,,,,,. SPF ICP (a) SPF ; (b)spf 80, α = 0; (c) ICP ; (d) ICP 80 ; (e) ; (f) 80, α 15 Fig. Applications of Zhang s model and our method to a real cell image (a) Initialization of Zhang s model; (b) 80 iterations of Zhang s model. The parameter α = 0; (c) Initialization of Li s model; (b) 100 iterations of Li s model; (e) Initialization of our method; (f) 80 iterations of our method, the parameter α = 15

9 1, : ,. 3(a) SPF ICP. 3(b) 3(c) 3(d) ,,.,,,. 3 SPF ICP (a) ; (b) SPF 80 ; (c) ICP 100 ; (d) 45, α = 35 Fig. 3 Applications of Zhang s model, Li s model and our method to an synthetic image with Gaussian noise (a) Initialization of three methods; (b) 80 iterations of Zhang s model; (c) 50 iterations of Li s model; (d) 45 iterations of our method, the parameter α = (a) SPF ICP. 4(b) 4(c) 4(d) ,. 4(e) 4(f) 4(g) 4(b) 4(c) 4(d)., SPF,, 4(e). 4(f), 4(g), SPF., SPF

10 170 ( ) 015 ICP. 4 SPF ICP (a) ; (b) SPF 80 ; (c) ICP 100 ; (d) 45, α = 40; (e) 4(b) ; (f) 4(c) ; (g) 4(d) Fig. 4 Applications of Zhang s model, Li s model and our method to an synthetic image with Gaussian noise (a) Initialization of three methods; (b) 80 iterations of Zhang s model; (c) 50 iterations of Li s model; (d) 45 iterations of our method. The parameter α = 40; (e) Enlarge the sharp corners in Fig. 4(b); (f) Enlarge the sharp corners in Fig. 4(c); (g) Enlarge the sharp corners in Fig. 4(d) SBGFRLS, SPF,,, ICP. SPF, ICP, ICP,, :,,. 1 3 Tab. 1 Three models computational efficiency / /s 1 SPF ICP ( ) SPF ICP ( ) SPF ICP ( ) SPF ICP ( )

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