For many applications, we need to use techniques to represent convex shapes and objects. In this work, we use level set method to represent shapes and find a necessary and sufficient condition on the level set function to guarantee the convexity of the represented shapes. We take image segmentation as an example to apply our technique. Numerical algorithm is developed to solve the variational model. In order to improve the performance of segmentation for complex images, we also incorporate landmarks into the model. One option is to specify points that the object boundary must contain. Another option is to specify points that the foreground (the object) and the background must contain. Numerical experiments on different images validate the efficiency of the proposed models and algorithms. We want to emphasize that the proposed technique could be used for general shape optimization with convex shape prior. For other applications, the numerical algorithms need to be extended and modified.
In this paper, we presented an efficient algorithm to implement the regularization reconstruction of SPECT. Image reconstruction with priori assumptions is usually modeled as a constrained optimization problem. However, there is no efficient algorithm to solve it due to the large scale of the problem. In this paper, we used the superiorization of the expectation maximization (EM) iteration to implement the regularization reconstruction of SPECT. We first investigated the convergent conditions of the EM iteration in the presence of perturbations. Secondly, we designed the superiorized EM algorithm based on the convergent conditions, and then proposed a modified version of it. Furthermore, we gave two methods to generate desired perturbations for two special objective functions. Numerical experiments for SPECT reconstruction were conducted to validate the performance of the proposed algorithms. The experiments show that the superiorized EM algorithms are more stable and robust for noised projection data and initial image than the classic EM algorithm, and outperform the classic EM algorithm in terms of mean square error and visual quality of the reconstructed images.
We present a novel and computable characterization method for convex shapes. We prove that the shape convexity is equivalent to a quadratic constraint on the associated indicator function. Such a simple characterization method allows us to design efficient algorithms for various applications with convex shape prior. In order to show the effectiveness of the proposed approach, this method is incorporated with a probability-based model to extract an object with convexity prior. The Lagrange multiplier method is used to solve the proposed model. Numerical results on various images show the superiority of the proposed method.
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