Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Dictionary Learning-based Inpainting on Triangular Meshes

The problem of inpainting consists of filling missing or damaged regions in images and videos in such a way that the filling pattern does not produce artifacts that deviate from the original data. In addition to restoring the missing data, the inpainting technique can also be used to remove undesired objects. In this work, we address the problem of inpainting on surfaces through a new method based on dictionary learning and sparse coding. Our method learns the dictionary through the subdivision of the mesh into patches and rebuilds the mesh via a method of reconstruction inspired by the Non-local Means method on the computed sparse codes. One of the advantages of our method is that it is capable of filling the missing regions and simultaneously removes noise and enhances important features of the mesh. Moreover, the inpainting result is globally coherent as the representation based on the dictionaries captures all the geometric information in the transformed domain. We present two variations of the method: a direct one, in which the model is reconstructed and restored directly from the representation in the transformed domain and a second one, adaptive, in which the missing regions are recreated iteratively through the successive propagation of the sparse code computed in the hole boundaries, which guides the local reconstructions. The second method produces better results for large regions because the sparse codes of the patches are adapted according to the sparse codes of the boundary patches. Finally, we present and analyze experimental results that demonstrate the performance of our method compared to the literature

Robust Modular Feature-Based Terrain-Aided Visual Navigation and Mapping

The visual feature-based Terrain-Aided Navigation (TAN) system presented in this thesis addresses the problem of constraining inertial drift introduced into the location estimate of Unmanned Aerial Vehicles (UAVs) in GPS-denied environment. The presented TAN system utilises salient visual features representing semantic or human-interpretable objects (roads, forest and water boundaries) from onboard aerial imagery and associates them to a database of reference features created a-priori, through application of the same feature detection algorithms to satellite imagery. Correlation of the detected features with the reference features via a series of the robust data association steps allows a localisation solution to be achieved with a finite absolute bound precision defined by the certainty of the reference dataset. The feature-based Visual Navigation System (VNS) presented in this thesis was originally developed for a navigation application using simulated multi-year satellite image datasets. The extension of the system application into the mapping domain, in turn, has been based on the real (not simulated) flight data and imagery. In the mapping study the full potential of the system, being a versatile tool for enhancing the accuracy of the information derived from the aerial imagery has been demonstrated. Not only have the visual features, such as road networks, shorelines and water bodies, been used to obtain a position ’fix’, they have also been used in reverse for accurate mapping of vehicles detected on the roads into an inertial space with improved precision. Combined correction of the geo-coding errors and improved aircraft localisation formed a robust solution to the defense mapping application. A system of the proposed design will provide a complete independent navigation solution to an autonomous UAV and additionally give it object tracking capability

A Bezier curve-based generic shape encoder

Existing Bezier curve based shape description techniques primarily focus upon determining a set of pertinent Control Points (CP) to represent a particular shape contour. While many different approaches have been proposed, none adequately consider domain specific information about the shape contour like its gradualness and sharpness, in the CP generation process which can potentially result in large distortions in the object’s shape representation. This paper introduces a novel Bezier Curve-based Generic Shape Encoder (BCGSE) that partitions an object contour into contiguous segments based upon its cornerity, before generating the CP for each segment using relevant shape curvature information. In addition, while CP encoding has generally been ignored, BCGSE embeds an efficient vertex-based encoding strategy exploiting the latent equidistance between consecutive CP. A nonlinear optimisation technique is also presented to enable the encoder is automatically adapt to bit-rate constraints. The performance of the BCGSE framework has been rigorously tested on a variety of diverse arbitrary shapes from both a distortion and requisite bit-rate perspective, with qualitative and quantitative results corroborating its superiority over existing shape descriptors

A topological approach for segmenting human body shape

Segmentation of a 3D human body, is a very challenging problem in applications exploiting human scan data. To tackle this problem, the paper proposes a topological approach based on the discrete Reeb graph (DRG) which is an extension of the classical Reeb graph to handle unorganized clouds of 3D points. The essence of the approach concerns detecting critical nodes in the DRG, thereby permitting the extraction of branches that represent parts of the body. Because the human body shape representation is built upon global topological features that are preserved so long as the whole structure of the human body does not change, our approach is quite robust against noise, holes, irregular sampling, frame change and posture variation. Experimental results performed on real scan data demonstrate the validity of our method

PROFILE user's guide

User information for program PROFILE, an aerodynamics design utility for refining, plotting, and tabulating airfoil profiles is provided. The theory and implementation details for two of the more complex options are also presented. These are the REFINE option, for smoothing curvature in selected regions while retaining or seeking some specified thickness ratio, and the OPTIMIZE option, which seeks a specified curvature distribution. REFINE uses linear techniques to manipulate ordinates via the central difference approximation to second derivatives, while OPTIMIZE works directly with curvature using nonlinear least squares techniques. Use of programs QPLOT and BPLOT is also described, since all of the plots provided by PROFILE (airfoil coordinates, curvature distributions) are achieved via the general purpose QPLOT utility. BPLOT illustrates (again, via QPLOT) the shape functions used by two of PROFILE's options. The programs were designed and implemented for the Applied Aerodynamics Branch at NASA Ames Research Center, Moffett Field, California, and written in FORTRAN and run on a VAX-11/780 under VMS

A hierarchical curve-based approach to the analysis of manifold data

One of the data structures generated by medical imaging technology is high resolution point clouds representing anatomical surfaces. Stereophotogrammetry and laser scanning are two widely available sources of this kind of data. A standardised surface representation is required to provide a meaningful correspondence across different images as a basis for statistical analysis. Point locations with anatomical definitions, referred to as landmarks, have been the traditional approach. Landmarks can also be taken as the starting point for more general surface representations, often using templates which are warped on to an observed surface by matching landmark positions and subsequent local adjustment of the surface. The aim of the present paper is to provide a new approach which places anatomical curves at the heart of the surface representation and its analysis. Curves provide intermediate structures which capture the principal features of the manifold (surface) of interest through its ridges and valleys. As landmarks are often available these are used as anchoring points, but surface curvature information is the principal guide in estimating the curve locations. The surface patches between these curves are relatively flat and can be represented in a standardised manner by appropriate surface transects to give a complete surface model. This new approach does not require the use of a template, reference sample or any external information to guide the method and, when compared with a surface based approach, the estimation of curves is shown to have improved performance. In addition, examples involving applications to mussel shells and human faces show that the analysis of curve information can deliver more targeted and effective insight than the use of full surface information