Przemyslaw Musialski

Tensor Completion for Estimating Missing Values in Visual Data

In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependant relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between FaLRTC and HaLRTC the former is more efficient to obtain a low accuracy solution and the latter is preferred if a high-accuracy solution is desired.

A Survey of Urban Reconstruction

This paper provides a comprehensive overview of urban reconstruction. While there exists a considerable body of literature, this topic is still under active research. The work reviewed in this survey stems from the following three research communities: computer graphics, computer vision and photogrammetry and remote sensing. Our goal is to provide a survey that will help researchers to better position their own work in the context of existing solutions, and to help newcomers and practitioners in computer graphics to quickly gain an overview of this vast field. Further, we would like to bring the mentioned research communities to even more interdisciplinary work, since the reconstruction problem itself is by far not solved.

Tensor completion for estimating missing values in visual data

In this paper we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process, or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by laying out the theoretical foundations and then by building a working algorithm. First, we propose a definition for the tensor trace norm, that generalizes the established definition of the matrix trace norm. Second, similar to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we employ a relaxation technique to separate the dependant relationships and use the block coordinate descent (BCD) method to achieve a globally optimal solution. Our experiments show potential applications of our algorithm and the quantitative evaluation indicates that our method is more accurate and robust than heuristic approaches.

Reduced-order shape optimization using offset surfaces

Given the 2-manifold surface of a 3d object, we propose a novel method for the computation of an offset surface with varying thickness such that the solid volume between the surface and its offset satisfies a set of prescribed constraints and at the same time minimizes a given objective functional. Since the constraints as well as the objective functional can easily be adjusted to specific application requirements, our method provides a flexible and powerful tool for shape optimization. We use manifold harmonics to derive a reduced-order formulation of the optimization problem, which guarantees a smooth offset surface and speeds up the computation independently from the input mesh resolution without affecting the quality of the result. The constrained optimization problem can be solved in a numerically robust manner with commodity solvers. Furthermore, the method allows simultaneously optimizing an inner and an outer offset in order to increase the degrees of freedom. We demonstrate our method in a number of examples where we control the physical mass properties of rigid objects for the purpose of 3d printing.

Interactive Coherence-Based Façade Modeling

We propose a novel interactive framework for modeling building façades from images. Our method is based on the notion of coherence‐based editing which allows exploiting partial symmetries across the façade at any level of detail. The proposed workflow mixes manual interaction with automatic splitting and grouping operations based on unsupervised cluster analysis. In contrast to previous work, our approach leads to detailed 3d geometric models with up to several thousand regions per façade. We compare our modeling scheme to others and evaluate our approach in a user study with an experienced user and several novice users.

Non-linear shape optimization using local subspace projections

In this paper we present a novel method for non-linear shape optimization of 3d objects given by their surface representation. Our method takes advantage of the fact that various shape properties of interest give rise to underdetermined design spaces implying the existence of many good solutions. Our algorithm exploits this by performing iterative projections of the problem to local subspaces where it can be solved much more efficiently using standard numerical routines. We demonstrate how this approach can be utilized for various shape optimization tasks using different shape parameterizations. In particular, we show how to efficiently optimize natural frequencies, mass properties, as well as the structural yield strength of a solid body. Our method is flexible, easy to implement, and very fast.

A framework for interactive image color editing

We propose a new method for interactive image color replacement that creates smooth and naturally looking results with minimal user interaction. Our system expects as input a source image and rawly scribbled target color values and generates high quality results in interactive rates. To achieve this goal we introduce an algorithm that preserves pairwise distances of the signatures in the original image and simultaneously maps the color to the user defined target values. We propose efficient sub-sampling in order to reduce the computational load and adapt semi-supervised locally linear embedding to optimize the constraints in one objective function. We show the application of the algorithm on typical photographs and compare the results to other color replacement methods.

Structure completion for facade layouts

We present a method to complete missing structures in facade layouts. Starting from an abstraction of the partially observed layout as a set of shapes, we can propose one or multiple possible completed layouts. Structure completion with large missing parts is an ill-posed problem. Therefore, we combine two sources of information to derive our solution: the observed shapes and a database of complete layouts. The problem is also very difficult, because shape positions and attributes have to be estimated jointly. Our proposed solution is to break the problem into two components: a statistical model to evaluate layouts and a planning algorithm to generate candidate layouts. This ensures that the completed result is consistent with the observation and the layouts in the database.

Interactive Multi-View Facade Image Editing

We propose a system for generating high-quality approximated façade ortho-textures based on a set of perspective source photographs taken by a consumer hand-held camera. Our approach is to sample a combined orthographic approximation over the façade-plane from the input photos. In order to avoid kinks and seams which may occur on transitions between different source images, we introduce color adjustment and gradient domain stitching by solving a Poisson equation in real-time. In order to add maximum control on the one hand and easy interaction on the other, we provide several editing interactions allowing for user-guided post-processing.

Tiling of ortho-rectified facade images

Typical building facades consist of regular structures such as windows arranged in a predominantly grid-like manner. We propose a method that handles precisely such facades and assumes that there must be horizontal and vertical repetitions of similar patterns. Using a Monte Carlo sampling approach, this method is able to segment repetitive patterns on orthogonal images along the axes even if the pattern is partially occluded. Additionally, it is very fast and can be used as a preprocessing step for finer segmentation stages.