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For inpainting with linear partial differential equations (PDEs) such as homogeneous or biharmonic diffusion, sophisticated data optimisation strategies have been found recently. These allow high-quality reconstructions from sparse known data. While they have been explicitly developed with compression in mind, they have not entered actual codecs so far: Storing these optimised data efficiently is a nontrivial task. Since this step is essential for any competetive codec, we propose two new compression frameworks for linear PDEs: Efficient storage of pixel locations obtained from an optimal control approach, and a stochastic strategy for a locally adaptive, tree-based grid. Suprisingly, our experiments show that homogeneous diffusion inpainting can surpass its often favoured biharmonic counterpart in compression. Last but not least, we demonstrate that our linear approach is able to beat both JPEG2000 and the nonlinear state-of-the-art in PDE-based image compression.
A major task in non-rigid shape analysis is to retrieve correspondences between two almost isometric 3D objects. An important tool for this task are geometric feature descriptors. Ideally, a feature descriptor should be invariant under isometric transformations and robust to small elastic deformations. A successful class of feature descriptors employs the spectral decomposition of the Laplace-Beltrami operator. Important examples are the heat kernel signature using the heat equation and the more recent wave kernel signature applying the Schrödinger equation from quantum mechanics.
In this work we propose a novel feature descriptor which is based on the classic wave equation that describes e.g. sound wave propagation. We explore this new model by discretizing the underlying partial differential equation. Thereby we consider two different time integration methods. By a detailed evaluation at hand of a standard shape data set we demonstrate that our approach may yield significant improvements over state of the art methods for finding correct shape correspondences.
Partial differential equations have recently been used for image compression purposes. One of the most successful frameworks solves the Laplace equation using a weighting scheme to determine the importance of individual pixels. We provide a physical interpretation of this approach in terms of the Helmholtz equation which explains its superiority. For better reconstruction quality, we subsequently formulate an optimisation task for the corresponding finite difference discretisation to maximise the influence of the physical traits of the Helmholtz equation. Our findings show that sharper contrasts and lower errors in the reconstruction are possible.
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.
The main task in three dimensional non-rigid shape correspondence is to retrieve similarities between two or more similar three dimensional objects. A useful way to tackle this problem is to construct a simplified shape representation, called feature descriptor, which is invariant under deformable transformations. A successful class of such feature descriptors is based on physical phenomena, concretely by the heat equation for the heat kernel signature and the Schrödinger equation for the wave kernel signature. Both approaches employ the spectral decomposition of the Laplace-Beltrami operator, meaning that solutions of the corresponding equations are expressed by a series expansion in terms of eigenfunctions. The feature descriptor is then computed at hand of those solutions. In this paper we explore the influence of the amount of used eigenfunctions on shape correspondence applications, as this is a crucial point with respect to accuracy and overall computational efficiency of the method. Our experimental study will be performed at hand of a standard shape data set.
Morphological levelings represent a useful tool for the decomposition of an image into cartoon and texture components. Moreover, they can be used to construct a morphological scale space. However, the classic construction of levelings is limited to the use of grey scale images, since an ordering of pixel values is required.
In this paper we propose an extension of morphological levelings to colour images. To this end, we consider the formulation of colour images as matrix fields and explore techniques based on the Loewner order for formulating morphological levelings in this setting. Using the matrix-valued colours we study realisations of levelings relying on both the completely discrete construction and the formulation using a partial differential equation. Experimental results confirm the potential of our matrix-based approaches for analysing texture in colour images and for extending the range of applications of levelings in a convenient way to colour image processing.
Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian
(2017)
Partial differential equations are well suited for dealing with image reconstruction tasks such as inpainting. One of the most successful mathematical frameworks for image reconstruction relies on variations of the Laplace equation with different boundary conditions. In this work we analyse these formulations and discuss the existence and uniqueness of solutions of corresponding boundary value problems, as well as their regularity from an analytic point of view. Our work not only sheds light on useful aspects of the well posedness of several standard problem formulations in image reconstruction but also aggregates them in a common framework. In addition, the performed analysis guides us to specify two new formulations of the classic image reconstruction problem that may give rise to new developments in image reconstruction.
The main task in three dimensional shape matching is to retrieve correspondences between two similar three dimensional objects. To this end, a suitable point descriptor which is invariant under isometric transformations is required. A commonly used descriptor class relies on the spectral decomposition of the Laplace-Beltrami operator. Important examples are the heat kernel signature and the more recent wave kernel signature. In previous works, the evaluation of the descriptor is based on eigenfunction expansions. Thereby a significant practical aspect is that computing a complete expansion is very time and memory consuming. Thus additional strategies are usually introduced that enable to employ only part of the full expansion.
In this paper we explore an alternative solution strategy. We discretise the underlying partial differential equations (PDEs) not only in space as in the mentioned approaches, but we also tackle temporal parts by using time integration methods. Thus we do not perform eigenfunction expansions and avoid the use of additional strategies and corresponding parameters. We study here the PDEs behind the heat and wave kernel signature, respectively. Our shape matching experiments show that our approach may lead to quality improvements for finding correct correspondences in comparison to the eigenfunction expansion methods.