Extract Object Boundaries in Noisy Images using Level Set. Final Report

Transcription

1 Extact Object Boundaies in Noisy Images using Level Set by: Quming Zhou Final Repot Submitted to Pofesso Bian Evans EE381K Multidimensional Digital Signal Pocessing May 10, 003 Abstact Finding object contous in noisy images is a challenging task because of the amophous natue of the object and the lack of shap boundaies. Classical edge-based segmentation methods have the dawback of not connecting edge segments to fom a distinct and meaningful bounday. Many level set appoaches, which can deal with changes of topology and the pesence of cones, have been developed to extact object boundaies. Pevious eseaches have used image gadient, edge stength, aea minimization and egion intensity to define the speed function. Howeve, no pape mentions the edge/gadient diection. Ou appoach will incopoate diection and magnitude in the speed function.

2 1. Intoduction Many compute vision applications involve the decomposition of image into egions with homogeneous popeties, which ae elated to the natue of the application. The bounday of an object is an impotant featue fo the object detection, classification and tacking. Edge based appoaches ae not suitable fo bounday extaction in noisy images [1]. They will detect edges that ae not pat of an object s bounday o miss pats of a bounday when the intensity contast is weak. In geneal, additional effot is needed to connect the incomplete edges into a distinct and meaningful object bounday. Seveal appoaches have been poposed to extact object boundaies in images using closed cuves. Roughly speaking, thee ae two types of bounday seach appoaches. One uses a closed contou epesented by a paameteized cuve. The poblem of finding the desiable contou is posed as an enegy minimization poblem. The classical Eule-Lagange fomulation of the active contou is called snake []. This kind of method elies on an initial guess of the bounda image featues and paametes. Moeove, its pefomance suffes fom the change of topology and the pesence of cones. To ovecome these poblems, the level set appoach has been poposed [3]. The guiding pinciple of level set methods is to descibe a closed cuve γ in R as the zeo level set of a highe dimensional function Φ( x, in R 3. Instead of popagating the cuve γ diectl we conside the evolution of function Φ ( x, with a speed function F and extact the zeo level set of points to obtain the bounday cuve. Since level set methods epesent the cuve in an implicit fom, they geatly simplify the management of the contou evolution, especially fo handling topological changes. Most of the challenges in level set methods esult fom the need to constuct an adequate model fo the speed function.. Backgound We conside the geneation of a family of contous. Let an initial cuve 0 undego defomation in a Euclidean plane. Let ( x, denote the family of cuves geneated by the popagation of 0 in the outwad nomal diection N with the speed F. We ignoe the tangential velocity because it does not influence the geomety of the defomation, but only its paameteization [4]. The cuve velocity t ( x, is denoted by ( x, = FN, (1) t whee F is a scala function and N is a unit nomal vecto.

3 Accoding to the level set method, we can expess the closed cuve ( in an implicit fom as ( x, = {( x, Φ( x, = 0}, o Φ( (, ) = 0. () By the chain ule, Φ t + Φ t = Φ t + Φ FN = Φ t + Φ F Φ Φ = 0, yielding the movement equation of cuves, Φ t + F Φ = 0, with Φ ( x, t = 0) = 0. (3) The speed function is essentially a deceasing function of a set of featues. These featues should have vey high values at the final shape bounday. In geneal, speed function models can be classified as edge-based and egion-based..1 Speed Function Due to Image Gadient Caselle et al. [5] poposed the geometic active contou followed by Malladi et al. [6]. The model poposed by Caselle and Malladi was based on the following speed function: F ( a + ε k) / 1+ G I, (4) = σ whee k is the cuvatue of the cuve, a, ε and p ae constants and 1+ Gσ I is the edge gadient using a Gaussian filte G σ with a known standad deviation σ. Since the stop citeion is the magnitude of the gadient, the speed slows down at stong edges. The dawback of this model is that it only detects objects with edges defined by stong gadients. F is neve small enough to stop the cuve evolution in a noisy image and the cuve may extend beyond the bounday. Moeove the pulling back foce is not stong hence it may not be able to pull back the expanding contou if it wee to popagate and coss the desied bounday.. Speed Function Due to Region Intensity Chan et al. [7] poposed an active model based on Mumfod-Shan segmentation technique and the level set method. Thei model can extact objects whose boundaies ae not necessaily defined by gadient o with vey smooth boundaies. They intoduced the enegy function F c, c, ), defined by ( 1 C F( c1, c, C) = u Length( C) + v Aea( inside( C)) + λ 1 u0( x, c1 dxdy + λ u0( x, c dxdy (5) inside( C) outside( C) whee u, v 0, λ, λ 0 ae fixed paametes, u ( x, ) is the intensity of pixel (x,, C is the 0 1 > 0 y cuve, while the constants c 1 and c depending on C, ae the aveage of u 0 inside o outside the

4 cuve C. Finding the object bounday tuns out to be the minimization of the enegy F c, c, ). ( 1 C Fo the level set fomulation of the vaiation active contou model, they deduced the associate Eule-Lagange equation fo Φ as Φ Φ = δ ε ( Φ)[ u div( ) v λ1 ( u0 c1) + λ ( u 0 c ) ] = 0 in (0, ) Ω, t Φ δ Φ( 0, x, = Φ 0 ( x, in Ω, ε ( Φ) Φ = 0 on Ω (6) Φ n whee n denotes the exteio nomal to the bounday Ω, Φ / n denotes the nomal deivation of Φ at the bounda and δ (Φ) is the egulation function. ε The poblem with this method is that we have to estimate the intensity distibution of the egion; howeve, the distibution model may degade in a noisy image. 3: The Poposed Method The poposed appoach uses the edge diection as well as the gadient magnitude. A stong edge can stop the cuve evolution by its gadient magnitude. A weak edge can halt the cuve by its edge map diection, which points towad the closest bounday. The esult of evolution will be a cuve that goes though the most homogenous egion to fit the object bounday. When the speed function is small, the evolution pocess ceases. 3.1 Edge Map The motivation fo the edge map comes fom the fact that the magnitude of the intensity gadient cannot estict the level set flow completely and the edge diection will help us localize the edge. We intoduce a concept, edge map, to epesent the gadient magnitude and diection. Edge flow detects the image boundaies by identifying the location, which has non-zeo edge flow coming fom two opposite diections. We exploe the isotopic and linea chaacteistics of Gaussian filtes to obtain the edge map, which accounts fo the local edge gadients and thei neighbohood. A D isotopic Gaussian filte with standad deviation σ is applied to the image I ( x,. The smoothed image is denoted by I σ ( x,. Futhe, the gadient images g x ( x, and ( x, ae computed by the fist ode diffeence of I σ ( x, along x-axis and y-axis espectively. Then, the local edge vecto at pixel s = ( x s, ys ) along the oientation θ is a linea combination as E( s, θ ) = ( g ( s)cos( θ ) + g ( s)sin( θ )) θ. (7) x y whee gx(s) and gy(s) ae the gadients, fo pixel s, along the x-axis and y-axis espectively. g y

5 E ( s, θ ) is the local edge vecto along the oientation θ. It gives us the local intensity change, which is widely used in edge detectos. Ou edge map fo pixel s is defined by M ( s) = E( s, θ ) dθ. (8) θ + π θ The integation ange paamete θ is now estimated. Without loss of genealit fo the pixel s = x s, y ), we compute the intensity diffeence with pixel s = ( x + d cosθ, y d sinθ ) as ( s Diff ( s, θ ) = Iσ ( xs + d cosθ, ys + d sinθ ) Iσ ( xs + ys ), (9) s s + whee d = 5σ.We assume that Diff ( s, θ ) is usually no less than Diff ( s, θ + π ) when the bounday is a distance d away fom the pixel s in the diection θ. Howeve it is still not enough to know whee the bounday is exactly. To quantify the pediction of the bounda an index P ( s, θ ) is assigned to evey pixel with the same offset distance, d, fom the pixel s by Diff ( s, θ ) Diff ( s, θ + π ) P ( s, θ ) =. (10) Diff ( s, θ ) + Diff ( s, θ + π ) A lage index value implies a bounday located in that diection. We choose to maximize the integation of P ( s, θ ) in the coesponding half plane: θ = ag max θ θ + π ( θ P s, θ ) dθ. (11) The edge map M (s) θ in ode is a vecto pointing towad the closest bounday pixel with its magnitude epesenting the total gadient enegy in the half plane Figue 1 shows an example of the edge map foσ = 1. Each aow indicates the magnitude and the diection of a pixel. The cycle points ae the edge pixels obtained by the Canny edge detecto. As we can see, the diection of the edge map points to its neaest bounday as its magnitude vaies with the distance fom the bounday. Fig. 1. Edge map fo each pixel.

6 3. Speed Function We define ou speed function in the outwad nomal diection of the cuve as v F = ( c εk) / g( M ), (1) whee c and εae constants, k is the cuvatue and g(m v ) is a scaling function of the edge map M v. Physicall c denotes an expansion tem, ε k plays the smoothing ole and g(m v ) is a stop citeion. The value of g(m v ) depends on the magnitude of the edge map as well as its diection. We give its fomula as follows. Let θ be the angle between the edge map and the outwad nomal vecto N. Then, 1+ M if cosθ >= 0 g( M ) = 1+ M if 0.5 <= cosθ < 0 1+ (3 ~ 5) M othewise The ability to slow down the speed function vaies with the diection of the edge map. An edge map with a low magnitude value in the diection opposite to the outwad cuve nomal diection will have a halting ability compaable to that of the stong edge map. Thus, the speed function has values close to zeo nea high image gadients o edges. 4. Segmentation of Themal Images The geneal famewok intoduced in section 3 is applied to segment themal images. The objective is to constuct bounday elements of the given stuctue in the image. We tested themal images, fom vaious applications such as medicine, defense and suveillance, with excellent esults. We make no assumption about the object s shape, but use only thee o fou andom points inside o aound the inteesting object as the initial points. Initial pixels locate the place whee the evolution begins and povide some gadient infomation. The use of moe initial pixels educes the total segmentation time but has only a small effect on the final esult. 4.1 Examples of Themal Images Segmenting themal medical images is a means of identifying diseased tissues. Once diseased tissue has been segmented, it is useful to compae it with the nomal tissue and see how it changes with pathology. Fo example, utilization of themal imaging has been an effective method in the evaluation of vascula disease. Figue shows a patient with vascula disease of the legs. The inceased flow of blood though the vessel poduces moe heat, which is ecodable with a themal imaging pocedue. Themal imaging povides clues to the potential of developing

7 vascula disease, which may lead to stoke o cance. An unsatisfied esult is shown in Figue (f) with a geneal level set method poposed by [6]. The speed function without gadient diections cannot maintain the closed cuve along the object bounday. Fig. (a) Fig. (b) Fig. (c) Fig. (d) Fig. (e) Fig. (f) Fig.. (a) Thee initial pixels. (b-d) Evolution of the bounday. (e) Final segmentation esult. (f) Evolution without consideing gadient diections. Figue 3(a) shows a themal image of a at and fou calibating emittes. The fou cicles coespond to fou emittes (only thee ae eally appaen at diffeent tempeatues, and the oblong shape is a live at. The vaiability in shape adds to the segmentation challenge. The pupose of this expeiment is to measue the themal tempeatue of the at. Figue 3(b) shows the boundaies of the at and the fou themal emittes. The efficacy of this technique is eally phenomenal, since using an edge opeato on the image yields nowhee nea a complete contou fo the fouth emitte. Fig. 3 (a) Fig.3 (b) Fig. 3. (a) A themal image of a at and fou calibating emittes (only thee ae visible). (b) Segments of the at and emittes. Figues 4(a) shows an image taken fom a senso mounted on a helicopte. The esult shows an example that ou appoach captues the cones. Figue 4(b) demonstates how ou

8 appoach deals with noisy pats in the object. The cuve flows aound the noisy pats, because the cuve always looks fo the elatively homogenous egion aound its cuent position. Afte being isolated by the cuve, the noisy pats ae emoved. This pocess implies that the cuve knows whee the noisy pats ae duing its popagation. The esult of ou method is shown in Figue 4(d). The computed bounday captues the cones faithfully. In contast, snake-based appoaches tend to smooth the cones of solid objects. The bounday esulting fom the snake poposed by Kass et al. [] is shown in Figue 4(f), with the initial bounday shown in Figue 4(e). The final bounday poduced by the snake appoach looks too smooth because the fist and the second deivatives ae used as constaints in the classical snake []. Fig. 4 (a) Fig. 4 (b) Fig. 4 (c) Fig. 4 (d) Fig. 4 (e) Fig. 4 (f) 4. Appoach Compaison The impotance of the gadient diection in the speed function is emphasized in ou pape. The intoduction of the gadient diection as defined in ou pape, ovecomes the disadvantages in the geneal level set methods, which ae summaized in [8]. a) The speed function may not tun out to be zeo in multiple objects segmentation. Fig.3 (a) shows a themal image of a at and fou cicula calibation emittes with vey diffeent intensities. The speed function of the bighte emittes can easily be educed to zeo while the dak emittes with low contast fom the backgound ae likely to be missed by the evolving cuve unde the same model paametes. The active contou meets anothe poblem in segmenting the at and emittes, i.e. diffeent shapes. Additional cae is to be taken to set diffeent

9 model paametes fo the at and emittes sepaately. In contast, ou poposed method segments all the five objects using the same paametes. b) Embedding the object. If one object has one o moe objects located inside, the geneal level set method and the active contou will not captue all objects of inteest. Ou poposed cuve flows aound the noisy pats o the embedded objects, because the cuve always looks fo the elatively homogenous egion aound its cuent position. Afte being isolated by the cuve, the noisy pats o embedded objects ae located. This pocess implies that the cuve knows the noisy pats o embedded objects duing its popagation. Fig.4(c) shows how the noisy pats o the embedded objects ae located even though we emove them in Fig.4 (d). c) Gaps in Boundaies. Gaps in Boundaies ae not a poblem in the active contou model because the smoothing estiction and intenal iteate values make the contou complete. Howeve, they ae the dawback of the level set method when applied to noisy images. The contou in the level set method is in an implicit fom, which may simply leak though gaps. 5. Conclusion In this pape we have pesented a level set appoach to segment themal images. The edge map is intoduced as the main component of the speed function. The edge map points towad the neaest bounday and its magnitude epesents the total gadient enegy in the half plane. The poposed appoach uses both the edge diection and the gadient magnitude to ovecome the poblems esulting fom weak edges. As shown in ou expeiments, ou appoach has seveal desiable featues besides those of the geneal level set method. Good boundaies can be extacted fom many kinds of themal images with vey few initial pixels inside o aound the object; the final esult is elatively independent of the initial guess; adding moe initial pixels can educe the total segmentation time; the paametes set in ou expeiments can wok fo themal images fom vaious applications; and the cuve knows whee the isolated noisy pats ae.

10 Refeences [1] J. Cann A Computational Appoach to Edge Detection, IEEE Tans. PAMI, vol. 8, , Nov [] M. Kass, A. Witkin and D. Tezopoulos, Snake: Active Contou Models, Int. J. Comput. Vis., vol. 1, pp , [3] S. Oshe and J. A. Sethian, Font Popagation with Cuvatue-Dependent Speed: Algoithms Based on Hamilton-Jacobi fomulation, J. Computat. Phys., vol.79, pp.1-49, [4] G. Sapio, Geometic Patial Diffeence Equations and Image Analysis, Cambidge Univesity Pess, Cambidge, UK, 000. [5] V. Caselles, F. Catte and F. Dibos, A Geometic Model fo Active Contou in Image Pocessing, Numeische Mathematik, vol. 66, no. 1, pp. 1-31, [6] R. Malladi, J. A. Sethian and B. C. Vemui, Shape Modeling with Font Popagation: A Level Set Appoach, IEEE Tans. PAMI, vol. 17, no., pp , Feb [7] T. F. Chan and L. A. Vese, Active Contou without Edges, IEEE Tans. Image Pocessing, vol. 10, no., pp , Feb [8] J. S. Sui, K.Liu, S. Singh, S.N.Laxminaayan, X.Zeng and L.Reden, Shape Recove Algoithms Using Level Sets in -D/3-D Medical Imagey: a State-of-the-At Review, IEEE Tans. Infomation Technology in Biomedicine, vol.6, no.1, pp.8-8, Mach 00.