Level set issues for efficient image segmentation

In conventional level set methods, the slope of the level set function needs to be well controlled to maintain the numerical stability during the topology optimization process. One common solution is to regularize the level set function to be a signed distance function, which is usually achieved by periodically implementing the so called re-initialization scheme to force the level set function to gain the desired signed distance property. However, the re-initialization scheme will bring some unwanted drawbacks to the optimization process, such as zero level set drifting, time consuming etc. In addition, re-initialization is usually implemented outside the optimization loop, which will cause convergence issues. In this paper, a distance regularization functional is introduced to the structural topology optimization objective functional to ensure the signed distance property of the level set function near the structure boundaries. This functional can also keep the level set function to be constant-value at positions far away from the structural boundaries. The radial basis function (RBF) based parameterization technique together with the mathematical programming are utilized to improve the potential capability of handling multiple constraints for the topology optimization. The combination of these two techniques makes the level set based topology optimization be capable of handling complicated multi-constrained problems with higher numerical efficiency, leaving no compromise to multiple drawbacks. To demonstrate the validity of the proposed scheme, benchmark examples on minimum compliance structural optimization are employed. This type of problem is computed by the conventional level set method with the introduced distance regularization functional, the RBF based parametric level set and at last, the distance regularized RBF based parametric level set separately to demonstrate their differences.

Segmentation of stem cells is a challenging process in image processing where level set method is used extensively. This level set criterion with re-initialisation process reduces its wider range of application. Thus re-initialisation is eradicated by a method called distance regularisation in which the signed distance property is maintained near the zero level set via forward and backward (FAB) diffusion effect. A potential function is habituated to sustain the desired shape of the level set function. The initial potential function used in distance regularisation is single-well potential technique that causes excess backward diffusion and oscillation in LSF, so it is succeeded by double-well potential. But this double well technique losses the capability to conserve the regularity at some points. This problem is eliminated by a new novel method called improved diffusion rate with triple-well potential. The proposed diffusion rate can maintain the signed distance property of the level set function (LSF) and overcome the side effect of double-well potential with reduced number of iterations.

Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost.

SummaryThe parametric level set approach is an extension of the conventional level set methods for topology optimization. By parameterizing the level set function, level set methods can be directly coupled with mathematical programming to achieve better numerical robustness and computational efficiency. Moreover, the parametric level set scheme can not only inherit the primary advantages of the conventional level set methods, such as clear boundary representation and the flexibility in handling topological changes, but also alleviate some undesired features from the conventional level set methods, such as the need for reinitialization. However, in the existing radial basis function–based parametric level set method, it is difficult to identify the range of the design variables. Besides, the parametric level set evolution often struggles with large fluctuations during the optimization process. Those issues cause difficulties both in numerical stability and in material property mapping. In this paper, a cardinal basis function is constructed based on the radial basis function partition of unity collocation method to parameterize the level set function. The benefit of using cardinal basis function is that the range of the design variables can now be clearly specified as the value of the level set function. A distance regularization energy functional is also introduced, aiming to maintain the desired signed distance property during the level set evolution. With this desired feature, the level set evolution is stabilized against large fluctuations. In addition, the material properties mapped from the level set function to the finite element model can be more accurate.

We present a high performance variant of the popular geodesic active contours which are used for splitting cell clusters in microscopy images. Previously, we implemented a linear pipelined version that incorporates as many cues as possible into developing a suitable level-set speed function so that an evolving contour exactly segments a cell/nuclei blob. We use image gradients, distance maps, multiple channel information and a shape model to drive the evolution. We also developed a dedicated seeding strategy that uses the spatial coherency of the data to generate an over complete set of seeds along with a quality metric which is further used to sort out which seed should be used for a given cell. However, the computational performance of any level-set methodology is quite poor when applied to thousands of 3D data-sets each containing thousands of cells. Those data-sets are common in confocal microscopy. In this work, we explore methods to stream the algorithm in shared memory, multi-core environments. By partitioning the input and output using spatial data structures we insure the spatial coherency needed by our seeding algorithm as well as improve drastically the speed without memory overhead. Our results show speed-ups up to a factor of six.

In traditional image segmentation models based on single level set function, only two regions can be identified because different regions are identified by the signs of single level set function. Though several segmentation models with multiple regions have been proposed, but the largest number of regions that can be identified was limited by the number of embedded level set functions in them. Moreover, the more embedded level set functions, the higher the time cost, usually increasing linearly with the increase of embedded level set functions. In this paper, by introducing the segmentation-measure function, a new model for multi-regions image segmentation based on single level set function is proposed. At the same time, a new initialization function for the level set function is also proposed in order to reduce the time cost of the segmentation model. The experiment results show that the new Image segmentation mode with multiple regions proposed in this paper performs well and dramatically reduces the time cost compared with the popular model for multiple regions proposed by Vese and Chan.

In this work we discuss variants of a PDE based level set method. Traditionally interfaces are represented by the zero level set of continuous level set functions. We instead use piecewise constant level set functions, and let interfaces be represented by discontinuities. Some of the properties of the standard level set function are preserved in the proposed method. Using the methods for interface problems, we minimize a smooth locally convex functional under a constraint. We show numerical results using the methods for image segmentation.

An efficient lithographic source and mask optimization approach is developed based on localized level set methods, which is reformulated as an inverse problem by tracking the evolution of level-set functions (LSFs) embedding the level-set representation of source and mask patterns. A distance regularized level-set (DRLS) term is incorporated into the level-set formulation enabling not only stable LSF evolution and accurate computation by maintaining a signed-distance property, but also a simple and efficient narrow-band implementation. Consequently, optimization dimensionality is significantly reduced by updating the pixels in the vicinity of zero level set (narrow band) instead of all the level sets, effectively reducing computation complexity, resulting in significantly improvements in pattern fidelity convergence in terms of runtime, computation load, Euler time step and caching memory requirement which are merited by numerical simulations.

Image segmentation plays an important role in computer vision and image processing. Level set model is a classical image segmentation method. During level set evolution, almost all the level set energy minimization of image segmentation are based on the gradient descent method and the finite difference scheme. The speed of evolution is slow and easy to fall into local minima. In this paper, we propose a fast sweeping optimization algorithm to minimize global cosine fitting (GCF) model. When moving a pixel from the one side region to the another side region of evolving contour, the sweeping algorithm calculates the energy change directly and checks whether the cosine fitting energy is decreased. With this, we can avoid solving the Euler-Lagrange equation and the partial differential equation, which usually take a lot of time. Moreover, our proposal is robust to initial level set contour and it automatically handles the topological variety, algorithm automatic termination and no longer requires the reinitialization step, parameter adjustment and the distance regularization term. The experiments on real noise and synthetic images show the effectiveness of the sweeping algorithm.

Distance regularization energy terms in level set image segment model: A survey

Image segmentation and image denoising are two important and fundamental topics in the field of image processing. Geometric active contour model based on level set method can deal with the problem of image segmentation, but it does not consider the problem of image denoising. In this paper, a new diffusion equation model for noisy image segmentation is proposed by incorporating some classical diffusion equation denoising models into the segmental process. An assumption about the connection between the image intensity and level set function is given firstly. Some classical denoising models are employed to describe the evolution of level set function secondly. The final nonlinear diffusion equation model for noisy image segmentation is built thirdly. Then image segmentation and image denoising are combined in a united framework. The segmental results can be presented by level set function. Experimental results show that the new model has the advantage of noise resistance and is superior to traditional segmentation model.

The aim of this research is to construct a structural optimization method based on a level set method, the so-called convected level set method. The basic idea of this method is that the level set function is defined as a truncated smooth function obtained using a sinus filter based on a hyperbolic tangent function. The local property of the hyperbolic tangent level set function stabilizes the computation of the advection equation used for evolving the level set function and reduces the computational cost of the reinitialization that maintains the profile of the level set function. Additionally, the low computational cost characteristic enables the use of convective reinitialization, whose basic idea is that the reinitialization is incorporated in the advection equation, avoiding the need for a separate computation. Consequently, the convected level set method can avoid the need for additional computations when conducting reinitialization. The validity of our proposed method is tested with standard structural optimization problems.

We propose a variational framework for the integration of multiple competing shape priors into level set based segmentation schemes. By optimizing an appropriate cost functional with respect to both a level set function and a (vector-valued) labeling function, we jointly generate a segmentation (by the level set function) and a recognition-driven partition of the image domain (by the labeling function) which indicates where to enforce certain shape priors. Our framework fundamentally extends previous work on shape priors in level set segmentation by directly addressing the central question of where to apply which prior. It allows for the seamless integration of numerous shape priors such that--while segmenting both multiple known and unknown objects--the level set process may selectively use specific shape knowledge for simultaneously enhancing segmentation and recognizing shape.

We propose a new multiphase level set model for image segmentation, using Mumford-Shah techniques in the piecewise-constant case. The proposed model is also a generalization of our active contour model without edges based on a two-phase segmentation. In order to handle multiple individual segments and complex topologies (such as multiple junctions), we propose a new and efficient multiphase representation by level sets: the necessary number of level set functions is considerably reduced (we need only n level set functions to represent 2/sup n/ phases or segments), and in addition we have overcome the problems of vacuum and overlap, naturally arising in multiphase problems. Finally, we show how the model can be used in image segmentation, for synthetic and real (possible noisy) pictures.

Image segmentation is one of the most fundamental and important areas in the field of image processing and computer vision. The traditional level set methods need initialize the level set function as a distance function. If the initial contour is selected inappropriate, we may not get the desired ideal segmentation result. In order to solve the problem of level set automation initial, we proposed a new image segmentation algorithm based on level set and marker extraction. First, we extract the internal mark as level set initial curve by using Extended-minima transform. And then, through using the local binary fitting active contour model, we evolve the labeled image to get the final segmentation result. The simulation results show that this method has low computing complexity than the traditional level set method requirements, and can effectively solve the initialization problem of level set.