CENTER-POINT MODEL OF DEFORMABLE SURFACE

Transcription

1 CENTER-POINT MODEL OF DEFORMABLE SURFACE Piotr M. Szczypinki Iintitute of Electronic, Technical Univerity of Lodz, Poland Abtract: Key word: Center-point model of deformable urface for egmentation of 3D image i preented. Mobility of each node, element compoing the urface, i contrained in the model to one direction. Alo an original formula for image influence computation i propoed. The model aim at algorithm implification and reduction of computational time needed for egmentation of 3D imaging data acquired from magnetic reonance or computer tomography canner. egmentation; deformable urface; center-point model 1. INTRODUCTION Medical imaging technique together with computerized image analyi revolutionized medicine. They have become important tool in medical diagnoi urgical planning and urgical imulation. One of important image analyi operation i egmentation: a election of image fragment repreenting object of tudy uch a an anatomical organ, it ditinguihable part or pathological entity. In many cae thee object are oval and differ from urrounding with phyical propertie uch a hardne or water content. Medical imaging technique meaure uch phyical propertie and preent them a a 2D or a 3D image, where the meaurement i uually repreented by gray-cale intenity. Therefore, image region related to uch object differ from the background with their intenity. The center-point model (CPM) of deformable urface i deigned with thi kind of image in mind. The CPM aim at computational implification of egmentation algorithm baed on deformable model and reducing computational time needed for egmentation of 3D image.

2 2 Piotr M. Szczypinki 2. MODEL STRUCTURE Deformable model for egmentation of digital image uually comprie node. Neighboring node are connected to form dicrete curve 1 or urface 2, 3. Image analyi uing deformable model i a proce of iteration called matching proce which ytematically diplace node and deform the model tructure to fit a boundary of object of interet. Uual implementation of deformab1e model 1 aume that each node i aigned vector of coordinate in the pace of the invetigated image. In contrat to thi idea, CPM node i aigned to fixed half-line 4. Half-line originate from an arbitrary choen center point, are radiantly arranged and emi-equally paced around thi point. Thu, the poition of every node i given by it ditance r from center point. (a) (b) (c) (d) Figure 1. Teelation of center-point model of deformable urface: icoahedron (a), firt ubdiviion of icoahedron face (b), dual meh with three-connected node (c), half-line defining direction of node movement in center-point model Teelation of CPM urface i baed on ubdiviion of icoahedron face (figure 1.a). An icoahedron ha a regular tructure in an Euclidean 3D pace. It ha 12 vertice, where each vertex i connected to five other vertice forming 20 equilateral triangular face. Conecutive ubdiviion of triangular face of icoahedron 2 into maller triangle (figure 1.b) increae the number and denity of node or vertice. Every new vertex formed in thi way i connected to ix other vertice, while the original 12 vertice remain five-connected. However, alo a dual (complementary) form of a meh (figure 1.c), in which all the node are three-connected, may be applied. The center of reultant meh tructure i placed at the coordinate of an arbitrary choen center-point. The half-line contraining the node movement begin in the center-point and contain one of the meh vertice (figure 1.d). Chooing the right number of ubdiviion and the number of node i computation time v. final reult reolution tradeoff. For application of MRI data egmentation node eem adequate.

3 CENTER-POINT MODEL OF DEFORMABLE SURFACE 3 3. MATCHING PROCESS The node ha to be puhed toward deired poition near the egmentation boundary. For thi reaon, an image influence vector i computed according to local propertie of the image at node poition. On the other hand, moothne of deformable model mut be retained, which i atified by modeling tenion within urface. The deformable model matching i a proce of ucceive node diplacement intended for finding location where balance between image influence and internal tenion i obtained. The following three ubection decribe the approach implemented in CPM. 3.1 Image Influence In the current approache node are uually attracted with large image gradient magnitude. Hence, initial poition of a deformable urface hould be relatively cloe to the deired olution, i.e. within range of the gradient magnitude function lope. Thi problem i partly olved in a balloon model 6 where ome preure force inflate the model and make it grow until it node reach location of high gradient magnitude. Unfortunately, if the intenity gradient of the boundary i mall the balloon will grow without topping at all. Moreover, balloon model are vulnerable to image artifact and noie, which can caue them to fold and bend incorrectly (figure 2.a). (a) (b) (c) Figure 2. Example of magnetic reonance imaging data egmentation with balloon model (α=0.02, β=0.001, ξ=0.02, τ=16, i=80) (a), center-point model and linear formula of tenion computation (α=0.05, β=0, ξ=0.08, τ=16, i=25) (b), center-point model and formula of tenion computation with linear and cubic component (α=0.05, β=0.001, ξ=0.08, τ=16, i=25) (c) Let u aume that an object of interet differ from the image background with it brightne intenity and that ome threhold value of intenity can be found, which roughly eparate the object from it urrounding. Therefore, a imple formula of image influence force can be applied: ( I r ) f = ξ ) τ (

4 4 Piotr M. Szczypinki where ξ i a coefficient defining magnitude and turn of image influence force, I(r ) i an image intenity at coordinate of a node and τ i the image intenity threhold. If the object i brighter than the image background, parameter ξ hould be poitive, otherwie negative. Thi way, force f p puhe the node outide when the node i inide the object and pull inide otherwie. (a) (b) (c) (d) (e) (f) (g) Figure 3. Phae of magnetic reonance data egmentation proce with center-point model: cro-ection through input data (a), model initialized inide the object of interet (b), intermediate phae of matching proce (c), reult of cerebrum egmentation (ξ=0.06, τ=44, α=0.05, β=0.001, i=250) (d), initialization of urface urrounding the object (e), intermediate phae (f) and reult of egmentation (α=0.05, β=0.001, ξ=0.08, τ=16, i=25) (g) Conequently, there i no need for computation of an image gradient, which uually require coniderable computation time. Furthermore, the urface can be initialized in two way: inide the object, to grow jut like in balloon model, or it can urround the object at firt and then hrink and tighten around it. Thi property i quite ueful when two different object, one inide another, are to be egmented (figure 3). 3.2 Tenion Tenion i uually modeled by mean of the thin-plate energy function 2, 5 or by the linear low-pa filtering of node coordinate vector. In either cae, the effective tenion force influencing the node i a linear function of the node coordinate and coordinate of it neighbor.

5 CENTER-POINT MODEL OF DEFORMABLE SURFACE 5 Unfortunately, when applying the linear function for tenion computation, ome node tend to top on local and trong image diturbance, uch a MRI artifact. Eventually, they produce pike protruding from elewhere-mooth urface (figure 2.b). To avoid uch an effect, a nonlinear cubic component i added to the tenion force formula. Still, the formula i imple and reult in a reduction of computational time. The combined formula i: g = α n N n N r n rn n N r β r n N 3 The equation conit of a linear and a cubic component, where N i a neighborhood of node. The bracketed part i a difference between average coordinate of node within the neighborhood and coordinate of node itelf. If an individual node i topped on ome image diturbance and it tay far from neighboring node, the nonlinear component caue neighboring node to pull it out much tronger than the linear component alone would do (compare figure 2.b and 2.c). 3.3 Node diplacement For implicity the following equation for computation of node diplacement i applied in CPM. An index i refer to dicrete time. ( i+ 1) ( i) ( i) ( i) = r + g f r + 4. RESULTS The CPM of deformable urface wa teted on biomedical image repreenting human head. Data wa acquired from a MRI canner. Reult preented in figure 2.d and 2.g are obtained with CPM compoed of 1524 node. The proce of egmentation require 25 to 250 iteration (120 m 1.2 uing a PC with an Intel Centrino MT 1.6 GHz proceor). The input data are compried of 64 human head cro-ection, 256x256 pixel each, with 256 gray level and ratio of voxel dimenion 1:1:2. The CPM wa compared with a balloon model. Tenion force in a balloon model i computed with the ame equation a propoed for CPM, with a coordinate r ubtituted by vector of 3D pace coordinate. Alo

6 6 Piotr M. Szczypinki image influence force i computed the ame way a for CPM. The direction of the force influencing the node i pecified by average of normal vector of face including thi node. It turned out that CPM i at leat 20 time fater then the implemented balloon model. The reaon i not only higher number of equation to be computed per iteration but alo a requirement of decreaing image influence parameter ξ (uually time) to prevent urface from undeired folding over (figure 2.a). 5. SUMMARY AND CONCLUSIONS The problem of urface matching in CPM i reduced from 3D pace to 1D pace. In addition, the image influence formula i implified. Conequently, the model i more efficient in comparion with model baed on gradient computation. The weaknee of CPM are a prior need for knowledge on approximate center of volume to be egmented and limited complexity of hape that can be attained. Although the model i not limited to convex hape and ome concave hape can be egmented, the bet reult are obtained for oval object. However, in uch cae CPM can be ueful for preliminary, rough egmentation providing valuable initialization data for other model producing more precie reult. REFERENCES 1. M. Ka, A. Witkin, D. Terauzopoulo, Snake: Active Contour Model, Int. J. of Computer Viion, vol. 1, no. 4, 1988, pp T. McInerney, D. Terauzopoulo, A Dynamic Finite Element Surface Model for Segmentation and Tracking in Multidimenional Medical Image with Application to Cardiac 4D Image Analyi, Computerized Medical Imaging and Graphic, vol. 19, no. 1, 1995, pp T. McInerney, D. Terauzopoulo, Topology Adaptive Deformable Surface for Medical Image Volume Segmentation, IEEE Tran. on Medical Imaging, vol. 18, no. 10, October 1999, pp P. Szczypinki, P. Strumillo, Application of an Active Contour Model for Extraction of Fuzzy and Broken Image Edge, Machine Graphic & Viion, vol.5, no.4, 1996, pp W. Neuenchwander, P. Fua, O. Kuebler, From Ziplock Snake to Velcro Surface, Automatic Extraction of Man Made Object from Aerial and Space Image, Monte Verita, 1995 Birkhaeuer Verlag Bael, pp L. D. Cohen, On Active Contour Model and Balloon, CVGIP: Image Undertanding, vol. 17, no. 2, March 1991, pp