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Optimal known pixel data for inpainting in compression codecs based on partial differential equations is real-valued and thereby expensive to store. Thus, quantisation is required for efficient encoding. In this paper, we interpret the quantisation step as a clustering problem. Due to the global impact of each known pixel and correlations between spatial and tonal data, we investigate the central question, which kind of feature vectors should be used for clustering with popular strategies such as k-means. Our findings show that the number of colours can be reduced significantly without impacting the reconstruction quality. Surprisingly, these benefits are negated by an increased coding cost in compression applications.
We present a strategy for the recovery of a sparse solution of a common problem in acoustic engineering, which is the reconstruction of sound source levels and locations applying microphone array measurements. The considered task bears similarities to the basis pursuit formalism but also relies on additional model assumptions that are challenging from a mathematical point of view. Our approach reformulates the original task as a convex optimisation model. The sought solution shall be a matrix with a certain desired structure. We enforce this structure through additional constraints. By combining popular splitting algorithms and matrix differential theory in a novel framework we obtain a numerically efficient strategy. Besides a thorough theoretical consideration we also provide an experimental setup that certifies the usability of our strategy. Finally, we also address practical issues, such as the handling of inaccuracies in the measurement and corruption of the given data. We provide a post processing step that is capable of yielding an almost perfect solution in such circumstances.
Real-Time Retinal Vessel Segmentation on High-Resolution Fundus Images Using Laplacian Pyramids
(2019)
In ophthalmology, fundus images are commonly used to examine the human eye. The image data shows among others the capillary system of the retina. Recognising alternations in the retinal blood vessels is pivotal to diagnosing certain diseases. The visual inspection of those fundus images is a time-consuming process and a challenging task which has to be done by medical experts. Furthermore, rapid advances in medical imaging allow for generating fundus images of increased quality and resolution. Therefore, the support by computers for the analysis and evaluation of complex fundus image information is growing in importance and there is a corresponding need for fast and efficient algorithms.
In this paper, we present a well-engineered, robust real-time segmentation algorithm which is adapted to the recent and upcoming challenges of high resolution fundus images. Thereby we make use of the multiscale representation of the Laplacian pyramid which is fast to compute and useful for detecting coarse as well as finely branched blood vessels. It is possible to process images of size 3504×2336
pixels in 0.8 s on a standard desktop computer and 0.3 on a Nvidia Titan XP GPU. By a detailed evaluation at hand of an accessible high-resolution data set we demonstrate that our approach is competitive in quality to state of the art methods for segmenting blood vessels but much faster.
The fundamental operations of mathematical morphology are dilation and erosion. In previous works, these operations have been generalised in a discrete setting to work with fields of symmetric matrices, and also corresponding methods based on partial differential equations have been constructed. However, the existing methods for dilation and erosion in the matrix-valued setting are not overall satisfying. By construction they may violate a discrete extremum principle, which means that results may leave the convex hull of the matrices that participate in the computation. This may not be desirable from the theoretical point of view, as the corresponding property is fundamental for discrete and continuous-scale formulations of dilation and erosion in the scalar setting. Moreover, if such a principle could be established in the matrix-valued framework, this would help to make computed solutions more interpretable.
In our paper we address this issue. We show how to construct a method for matrix-valued morphological dilation and erosion that satisfies a discrete extremum principle. We validate the construction by showing experimental results on synthetic data as well as colour images, as the latter can be cast as fields of symmetric matrices.
The median filter is one of the fundamental filters in image processing. Its standard realisation relies on a rank ordering of given data which is easy to perform if the given data are scalar values. However, the generalisation of the median filter to multivariate data is a delicate issue. One of the methods of potential interest for computing a multivariate median is the convex-hull-stripping median from the statistics literature. Its definition is of purely algorithmical nature, and it offers the advantageous property of affine equivariance.
While it is a classic result that the standard median filter approximates mean curvature motion, no corresponding assertion has been established up to now for the convex-hull-stripping median. The aim of our paper is to close this gap in the literature. In order to provide a theoretical foundation for the convex-hull-stripping median of multivariate images, we investigate its continuous-scale limit. It turns out that the resulting evolution is described by the well-known partial differential equation of affine curvature motion. Thus we have established in this paper a relation between two important models from image processing and statistics. We also present some experiments that support our theoretical findings.
Shape from Shading (SFS) is a fundamental task in computer vision. By given information about the reflectance of an object’s surface and the position of the light source, the SFS problem is to reconstruct the 3D depth of the object from a single grayscale 2D input image. A modern class of SFS models relies on the property that the camera performs a perspective projection. The corresponding perspective SFS methods have been the subject of many investigations within the last years. The goal of this chapter is to give an overview of these developments. In our discussion, we focus on important model aspects, and we investigate some prominent algorithms appearing in the literature in more detail than it was done in previous works.
Solving differential equations can be realised with simple artificial neural network architectures. Several methods make use of trial solutions with different construction approaches and can provide reliable results. However, many parameters, different optimisation methods and random weight initialisation result in a non constant variance to the exact solution. To our knowledge, this variance has not been studied yet. We investigate several parameters and constant versus random weight initialisation for two solution methods to determine their reliability with the use of backpropagation and ADAM optimisation.
The fast classification of shapes is an important problem in shape analysis and of high relevance for many possible applications. In this paper, we consider the use of very fast and easy to compute statistical techniques for assessing shapes, which may for instance be useful for a first similarity search in a shape database. To this end, we construct shape signatures at hand of stochastic sampling of distances between points of interest in a given shape. By employing the Kolmogorov-Smirnov statistics we then propose to formulate the problem of shape classification as a statistical hypothesis test that enables to assess the similarity of the signature distributions. In order to illustrate some important properties of our approach, we explore the use of simple sampling techniques. At hand of experiments conducted with a variety of shapes in two dimensions, we give a discussion of potentially interesting features of the method.
The decomposition of 3D shapes into simple yet representative components is a very intriguing topic in computer vision as it is very useful for many possible applications. Superquadrics may be used with benefit to obtain an implicit representation of the 3D shapes, as they allow to represent a wide range of possible forms by few parameters. However, in the computation of the shape representation, there is often an intricate trade-off between the variation of the represented geometric forms and the accuracy in such implicit approaches. In this paper, we propose an improved loss function, and we introduce beneficial computational techniques. By comparing results obtained by our new technique to the baseline method, we demonstrate that our results are more reliable and accurate, as well as much faster to obtain.