Combinatorial Continuous Maximal Flows

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cuts algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit metrication artefacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow CCMF problem to find a null-divergence solution that exhibits no metrication artefacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.Comment: 26 page

Globally Optimal Surfaces By Continuous Maximal Flows

In this paper we consider the problem of computing globally minimal continuous curves and surfaces for image segmentation and 3D reconstruction. This is solved using a maximal flow approach expressed as a PDE model. Previously proposed techniques yield either grid-biased solutions (graph based approaches) or sub-optimal solutions (active contours and surfaces). The proposed algorithm simulates the flow of an ideal fluid with a spatially varying velocity constraint derived from image data. A proof is given that the algorithm gives the globally maximal flow at convergence, along with an implementation scheme. The globally minimal surface may be obtained trivially from its output. The new algorithm is applied to segmentation in 2D and 3D medical images and to 3D reconstruction from a stereo image pair. The results in 2D agree remarkably well with an existing planar minimal contour algorithm and the results in 3D segmentation and reconstruction demonstrate that the new algorithm is free from grid bias

Globally minimal surfaces by continuous maximal flows

In this paper we address the computation of globally minimal curves and surfaces for image segmentation and stereo reconstruction. We present a solution, simulating a continuous maximal flow by a novel system of partial differential equations. Existing methods are either grid-biased (graph-based methods) or sub-optimal (active contours and surfaces). The solution simulates the flow of an ideal fluid with isotropic velocity constraints. Velocity constraints are defined by a metric derived from image data. An auxiliary potential function is introduced to create a system of partial differential equations. It is proven that the algorithm produces a globally maximal continuous flow at convergence, and that the globally minimal surface may be obtained trivially from the auxiliary potential. The bias of minimal surface methods toward small objects is also addressed. An efficient implementation is given for the flow simulation. The globally minimal surface algorithm is applied to segmentation in 2D and 3D as well as to stereo matching. Results in 2D agree with an existing minimal contour algorithm for planar images. Results in 3D segmentation and stereo matching demonstrate that the new algorithm is robust and free from grid bias

Distance, granulometry, skeleton

In this chapter, we present a series of concepts and operators based on the notion of distance. As often with mathematical morphology, there exists more than one way to present ideas, that are simultaneously equivalent and complementary. Here, our problem is to present methods to characterize sets of points based on metric, geometry and topology considerations. An important concept is that of the skeleton, which is of fundamental importance in pattern recognition, and has many practical application

A complete characterization of the (m,n)-cubes and combinatorial applications in imaging, vision and discrete geometry

AbstractThe aim of this work is to provide a complete characterization of a (m,n)-cube. The latter are the pieces of discrete planes appearing in Theoretical Computer Science, Discrete Geometry and Combinatorics. This characterization in three dimensions is the exact equivalent of the preimage for a discrete segment as it has been introduced by McIlroy. Further this characterization, which avoids the redundancies, reduces the combinatorial problem of determining the cardinality of the (m,n)-cubes to a new combinatorial problem consisting of determining the volumic regions formed by the crossing of planes. This work can find applications in Imaging, Vision, and pattern recognition for instance

Automated heart rate estimation in fish embryo

International audienceTransparent organisms such as fish embryos are being increasingly used for environmental toxicology studies. These studies require estimating a number of physiological parameters. These estimations may be diverse in nature and can be a challenge to automate. Among these, an example is the development of reliable and repeatable automated assays for the determination of heart rates. To achieve this, most existing method rely on cyclical luminance variations, since as the heart fills and empties, it become respectively brighter and darker. However, sometimes direct measurement of the heart rate may be difficult, depending on the age of the embryo, its actual transparency, and its aspect under the microscope. It may be easier to seek an indirect measurement. In this article, we estimate the heart function parameters, such as heart frequency, either from measuring the heart motion or from blood flow in arteries. This measurement is more complex from the image analysis point of view, but it is more precise, more physically meaningful and easier to use in practice and to automate than measuring illumination changes. It may also be more informative. We illustrate on medaka embryos

Combinatorial structure of rigid transformations in 2D digital images

International audienceRigid transformations are involved in a wide range of digital image processing applications. When applied on such discrete images, rigid transformations are however usually performed in their associated continuous space, then requiring a subsequent digitization of the result. In this article, we propose to study rigid transformations of digital images as a fully discrete process. In particular, we investigate a combinatorial structure modelling the whole space of digital rigid transformations on any subset of Z^2 of size N*N. We describe this combinatorial structure, which presents a space complexity O(N^9) and we propose an algorithm enabling to build it in linear time with respect to this space complexity. This algorithm, which handles real (i.e. non-rational) values related to the continuous transformations associated to the discrete ones, is however defined in a fully discrete form, leading to exact computation