Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image, unless it satisfies certain constraints (e.g., bandlimitedness). In this paper, we study the problem of recovering shape images with smooth boundaries from a set of samples. Thus, the reconstructed image is constrained to regenerate the same samples (consistency), as well as forming a shape (bilevel) image. We initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We also introduce a requirement (called reducibility) that guarantees equivalence between the two problems. We illustrate that the reducibility property effectively sets a requirement on the minimum sampling density. We also evaluate the performance of the relaxed alternative in various numerical experiments.
The shape image recovery can also be viewed as fitting boundary curves to the interpolated gray-scale image (high-resolution version of the measurements). Such methods are widely known as segmentation techniques that fit deformable curves to gray-scale images, and include active contour algorithms also known as snakes. Based on the curve models, they are classified as point snakes @cite_1 , geodesic snakes @cite_22 @cite_26 and parametric snakes @cite_13 @cite_18 . In all cases, the segmentation algorithm is formulated by minimizing a snake energy functional that depends on the gray-scale image and the model of the boundary curves. However, it does not take the PSF into account @cite_6 . As a consequence, the resulting binary image is likely to fail the consistency requirements.
@cite_18 @cite_26 @cite_22 @cite_1 @cite_6 @cite_13
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The active contour snake model is one of the most successful variational models in image segmentation. It consists of evolving a contour in images toward the boundaries of objects. Its success is based on strong mathematical properties and efficient numerical schemes based on the level set method. The only drawback of this model is the existence of local minima in the active contour energy, which makes the initial guess critical to get satisfactory results. In this paper, we propose to solve this problem by determining a global minimum of the active contour model. Our approach is based on the unification of image segmentation and image denoising tasks into a global minimization framework. More precisely, we propose to unify three well-known image variational models, namely the snake model, the Rudin---Osher---Fatemi denoising model and the Mumford---Shah segmentation model. We will establish theorems with proofs to determine the existence of a global minimum of the active contour model. From a numerical point of view, we propose a new practical way to solve the active contour propagation problem toward object boundaries through a dual formulation of the minimization problem. The dual formulation, easy to implement, allows us a fast global minimization of the snake energy. It avoids the usual drawback in the level set approach that consists of initializing the active contour in a distance function and re-initializing it periodically during the evolution, which is time-consuming. We apply our segmentation algorithms on synthetic and real-world images, such as texture images and medical images, to emphasize the performances of our model compared with other segmentation models. Abstract The use of energy-minimizing curves, known as “snakes,” to extract features of interest in images has been introduced by Kass, Witkin & Terzopoulos ( Int. J. Comput. Vision 1, 1987, 321–331). We present a model of deformation which solves some of the problems encountered with the original method. The external forces that push the curve to the edges are modified to give more stable results. The original snake, when it is not close enough to contours, is not attracted by them and straightens to a line. Our model makes the curve behave like a balloon which is inflated by an additional force. The initial curve need no longer be close to the solution to converge. The curve passes over weak edges and is stopped only if the edge is strong. We give examples of extracting a ventricle in medical images. We have also made a first step toward 3D object reconstruction, by tracking the extracted contour on a series of successive cross sections. This paper presents a novel deformable-model-based algorithm for fully automated detection of optic disk boundary in fundus images. The proposed method improves and extends the original snake (deforming-only technique) in two aspects: clustering and smoothing update. The contour points are first self-separated into edge-point group or uncertain-point group by clustering after each deformation, and these contour points are then updated by different criteria based on different groups. The updating process combines both the local and global information of the contour to achieve the balance of contour stability and accuracy. The modifications make the proposed algorithm more accurate and robust to blood vessel occlusions, noises, ill-defined edges and fuzzy contour shapes. The comparative results show that the proposed method can estimate the disk boundaries of 100 test images closer to the groundtruth, as measured by mean distance to closest point (MDCP) <3 pixels, with the better success rate when compared to those obtained by gradient vector flow snake (GVF-snake) and modified active shape models (ASM) We present a new family of snakes that satisfy the property of multiresolution by exploiting subdivision schemes. We show in a generic way how to construct such snakes based on an admissible subdivision mask. We derive the necessary energy formulations and provide the formulas for their efficient computation. Depending on the choice of the mask, such models have the ability to reproduce trigonometric or polynomial curves. They can also be designed to be interpolating, a property that is useful in user-interactive applications. We provide explicit examples of subdivision snakes and illustrate their use for the segmentation of bioimages. We show that they are robust in the presence of noise and provide a multiresolution algorithm to enlarge their basin of attraction, which decreases their dependence on initialization compared to singleresolution snakes. We show the advantages of the proposed model in terms of computation and segmentation of structures with different sizes.