The Mathematical Model And Fast Algorithm For Segmentation And Denoising Of Urinary Sediment Microscopic Image

Techniques based on partial differential equations are increasingly being used inimage processing for tasks such as noise reduction,segmentation, and object track-ing. In this dissertation,we focus on the use of P-M model,CLMC model,and ROFmodel for urinary sediment microscopic image denoise, and explicit active contoursas snakes,implicit active contours,also knows as level set techniques, such as C-V model,level set model without re-initialization,implicit geodesic active contourmodel,geometric active contour model,and a new topology-preserving level set modelfor image segmentation.There are two ways to get the PDE. One is direct derivation of the evolutionequations . PDE can also be obtained from variational problems.This is more commonin image processing. The basic idea is to minimize an energy functional .The classical algorithm for image denoising is based on least square estimate.ThisL2 norm based algorithm amounts to solving a heat equation.The main advantageof this method is that it is linear and stable,thus can be easily implemented.But itsuffers seriously from the fact that it can not keep the edges in images.Denoisingmodel based on the least squares estimate will inevitably blur the sharp edges. Toovercome this difficulty,Perona and Malik proposed a new image denoising model:where g is a positive decreasing function,and:g(0) = 1;s→∞,g(s)→0,such asg(s) = 1/(1 + s²), g(s) = e^-s2.The P-M model can remove the noise while preserving the edge at best.As thesame time,this model is an ill-posed problem.The new method was followed by Catt′eet al to tackle the ill-posed problem of the P-M model,that is CLMC model.The idea is to substitute in the diffusion coefficient g(|(?)u|) the gradient of the image (?)u by asmooth version of it (?)G_σ* u,where G_σis a smoothing kernel.The CLMC model is:This model has at least two advantages other the P-M model.If the initial datais very noisy,then the P-M model cannot distinguish between"true"edges and"false"edges created by the noise.The CLMC model avoids this drawback.This model iswell-posed.In order to increase the computational speed of the CLMC model,Weickertet al present a new efficient and reliable numerical schemes–AOS scheme.The AOSscheme of the 2-D CLMC model is:The AOS schemes have the same approximation order as their correspondingsemi-implicit schemes.They come down to solving tridiagonal linear systems of equa-tions which can be done in linear complexity with a very simple algorithm–theThomas algorithm. This scheme is easy to implement and storage effort is linearin the number of pixels.This makes this type of schemes attractive for more applica-tions.Rudin,Osher and Fetami proposed a new technique based on the minimization ofthe Total Variation(TV) norm subject to some noise constraints. This model ,ROFmodel,is expressed as:In practice,there are two ways to solve the equation,the first one is to solve theEuler-Lagrange equation directly.The second is using gradient decent method.That is:This amounts to solving a time dependent partial differential equation. As t→∞the solution converges to a steady state which is the denoised image. We presentthe AOS scheme as the numerical scheme algorithm for the fast evolution of the ROFdenoising model.The AOS scheme is:Segmentation of medical images is an important step in various applicationssuch as visualization,quantitative analysis and image-guide surgery.In the past twodecades,numerous segmentation methods have been developed for extraction of or-gan contours on medical images.Low-level segmentation methods, such as pixel-based clustering, region growing, and filter-based edge detection,require additionalpre-processing and post-processing as well as considerable amounts of expert inter-vention or information of the objects of interest.Deformable models, on the otherhand, provide an explicit representation of the boundary and the shape of the ob-ject.They combine several desirable feature such as inherent connectivity and smooth-ness ,which counteract noise and boundary irregularities, as well as the ability toincorporate knowledge about the object of interest. However, parametric deformablemodels(snake model) have two main limitations.First, in situations where the initialmodel and desired object boundary differ greatly in size and shape, the model mustbe re-parameterized dynamically to faithfully recover the object boundary.The secondlimitation is that it has difficulty dealing with topological adaptation such as split-ting or merging model parts,a useful property for recovering either multiple objectsor objects with unknown topology.This difficulty is caused by the fact that a new pa-rameterization must be constructed whenever topology change occurs, which requiressophisticated schemes.Level set deformable models ,also referred to as geometric de-formable models,provide an elegant solution to address the primary limitations of parametric deformable models.These methods have draws a great deal of attentionsince their introduction in 1988.Advantages of the contour implicit formulation ofthe deformable model over parametric formulation include: (1)no parameterizationof the contour,(2)topological flexibility,(3)good numerical stability,(4)straightforwardextension of the 2D formulation to n-D.In this dissertation,we give a general overview of the level set segmentation meth-ods in the context of urinary sediment microscopic image.Segmentation of an imageu,via active contours,also referred to as snakes,operates through an energy functionalcontrolling the deformation of an initial contour curve C(s),s∈[0,1],under the influ-ence of internal and external forces achieving a minimum energy state at high-gradientlocations.The generic energy functional for active contour model is expressed as:where (α,β)are positive parameters.The first two terms control the rigidity and elas-ticity of the contour while the last term attracts the model to high-gradient locationsin the image u. Osher and Sethian introduced the concept of geometric deformablemodels,which provide an implicit formulation of the deformable contour in a levelset framework. To introduce the concept of the level set framework we focus on theboundary-value problem of a close contour C deforming with a speed V along itsnormal direction:Their fundamental idea is, instead of tracking in time the positions of the front C(x,y),to embed the curve into a higher dimension functionφ(x,y,t),such that:The functionφevolves with the dynamic equation: The concept of active contours models for image segmentation defined in a levelset framework is called the geometric active contours.The C-V model is a region-based active contour,which were derived from the Mumford-Shah segmentation frame-work.Mumford and Shah defined a new segmentation framework performing segmen-tation of a given image u into a set of contours S and a smooth approximation f ofthe image.In a level set framework implementation the C-V model is express as:We present a new numerical scheme ,AOS scheme, for the fast evolution of theC-V segmentation model.The scheme is expressed as:We introduced a new variational formulation under the level set framework with-out re-initialization.The energy functional as:and: By calculus of variations, the G(a|^)teaux derivative(first variation) of the functionalE can be written as:The steepest descent process for minimization of the functional E is the followinggradient flow:To approximated this evolutional equation,the numerical scheme is also used AOSscheme,that is:We also introduced geometric active contours and the geodesic active contours inthis dissertation.The energy functional of the geometric active contours model underthe level set framework is expressed as:The G(a|^)teaux derivative of the functional can be written as:Geodesic active contours were introduced by Caselles et al. as a segmentationframework,derived from energy-based active contours, performing contour extractionvia the computation of geodesics,i.e. minimal distance curves in a Riemannian spacederived from the image.The energy functional of the geodesic active contours modelis:where g is a positive decreasing function.Segmentation is achieved via minimizationof this energy functional equivalent to the computation of geodesics in a Riemannianspace. Minimization of the functional is performed via the derivation of the Euler-Lagrange equation:whereκis the Euclidian curvature of the curve C and N is the unit normal vectorto the curve.Implementation with a level set framework is performed by embeddingthe curve C into a level set functionφ.In this dissertation, we present an efficient algorithm that is based on an addi-tive operator splitting(AOS).It suitable for geodesic active contours model.The AOSscheme is simple implementation,equal treatment of all axes,suitability for paral-lel computing, and straightforward generalization to higher dimensions.Experimentsshow that one can gain a speed-up by one order of magnitude compared to the widelyused explicit time discretization. The AOS scheme is :At the end of this dissertation, we propose a new topology-preserving imagesegmentation model.The necessity of designing topology-preserving processes arisesin medical imaging.In some application,the topological feature must be preservedthroughout the reconstruction process.This model is based on a level-set formulationand on the geodesic active contours.This new model maintain the other advantagesof standard geodesic deformable models.This model is expressed as:As mentioned before ,there are several way to increase the computational speedof the model including the AOS numerical scheme and narrow-band method.Thenarrow band method provides a considerable computational advantage since only a small set of grid points near the zero level set are modified during each iteration.Thenumerical scheme is :...