FROST-BRDF: A Fast and Robust Optimal Sampling Technique for BRDF Acquisition

Efficient and accurate BRDF acquisition of real world materials is a challenging research problem that requires sampling millions of incident light and viewing directions. To accelerate the acquisition process, one needs to find a minimal set of sampling directions such that the recovery of the full BRDF is accurate and robust given such samples. In this paper, we formulate BRDF acquisition as a compressed sensing problem, where the sensing operator is one that performs sub-sampling of the BRDF signal according to a set of optimal sample directions. To solve this problem, we propose the Fast and Robust Optimal Sampling Technique (FROST) for designing a provably optimal sub-sampling operator that places light-view samples such that the recovery error is minimized. FROST casts the problem of designing an optimal sub-sampling operator for compressed sensing into a sparse representation formulation under the Multiple Measurement Vector (MMV) signal model. The proposed reformulation is exact, i.e. without any approximations, hence it converts an intractable combinatorial problem into one that can be solved with standard optimization techniques. As a result, FROST is accompanied by strong theoretical guarantees from the field of compressed sensing. We perform a thorough analysis of FROST-BRDF using a 10-fold cross-validation with publicly available BRDF datasets and show significant advantages compared to the state-of-the-art with respect to reconstruction quality. Finally, FROST is simple, both conceptually and in terms of implementation, it produces consistent results at each run, and it is at least two orders of magnitude faster than the prior art.Comment: Submitted to IEEE Transactions on Visualization and Computer Graphics (IEEE TVCG

Robust Orthogonal Complement Principal Component Analysis

Recently, the robustification of principal component analysis has attracted lots of attention from statisticians, engineers and computer scientists. In this work we study the type of outliers that are not necessarily apparent in the original observation space but can seriously affect the principal subspace estimation. Based on a mathematical formulation of such transformed outliers, a novel robust orthogonal complement principal component analysis (ROC-PCA) is proposed. The framework combines the popular sparsity-enforcing and low rank regularization techniques to deal with row-wise outliers as well as element-wise outliers. A non-asymptotic oracle inequality guarantees the accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle the computational challenges, an efficient algorithm is developed on the basis of Stiefel manifold optimization and iterative thresholding. Furthermore, a batch variant is proposed to significantly reduce the cost in ultra high dimensions. The paper also points out a pitfall of a common practice of SVD reduction in robust PCA. Experiments show the effectiveness and efficiency of ROC-PCA in both synthetic and real data

Ultrasound Signal Processing: From Models to Deep Learning

Medical ultrasound imaging relies heavily on high-quality signal processing algorithms to provide reliable and interpretable image reconstructions. Hand-crafted reconstruction methods, often based on approximations of the underlying measurement model, are useful in practice, but notoriously fall behind in terms of image quality. More sophisticated solutions, based on statistical modelling, careful parameter tuning, or through increased model complexity, can be sensitive to different environments. Recently, deep learning based methods have gained popularity, which are optimized in a data-driven fashion. These model-agnostic methods often rely on generic model structures, and require vast training data to converge to a robust solution. A relatively new paradigm combines the power of the two: leveraging data-driven deep learning, as well as exploiting domain knowledge. These model-based solutions yield high robustness, and require less trainable parameters and training data than conventional neural networks. In this work we provide an overview of these methods from the recent literature, and discuss a wide variety of ultrasound applications. We aim to inspire the reader to further research in this area, and to address the opportunities within the field of ultrasound signal processing. We conclude with a future perspective on these model-based deep learning techniques for medical ultrasound applications

Sparse and Redundant Representations for Inverse Problems and Recognition

Sparse and redundant representation of data enables the description of signals as linear combinations of a few atoms from a dictionary. In this dissertation, we study applications of sparse and redundant representations in inverse problems and object recognition. Furthermore, we propose two novel imaging modalities based on the recently introduced theory of Compressed Sensing (CS). This dissertation consists of four major parts. In the first part of the dissertation, we study a new type of deconvolution algorithm that is based on estimating the image from a shearlet decomposition. Shearlets provide a multi-directional and multi-scale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. We develop a deconvolution algorithm that allows for the approximation inversion operator to be controlled on a multi-scale and multi-directional basis. Furthermore, we develop a method for the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation method. In the second part of the dissertation, we study a reconstruction method that recovers highly undersampled images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. Our method makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. We will demonstrate by experiments that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors. In the third part of the dissertation, we introduce a novel Synthetic Aperture Radar (SAR) imaging modality which can provide a high resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. We demonstrate that this new imaging scheme, requires no new hardware components and allows the aperture to be compressed. Also, it presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements. The last part of the dissertation deals with object recognition based on learning dictionaries for simultaneous sparse signal approximations and feature extraction. A dictionary is learned for each object class based on given training examples which minimize the representation error with a sparseness constraint. A novel test image is then projected onto the span of the atoms in each learned dictionary. The residual vectors along with the coefficients are then used for recognition. Applications to illumination robust face recognition and automatic target recognition are presented

Learning from High-Dimensional Multivariate Signals.

Modern measurement systems monitor a growing number of variables at low cost. In the problem of characterizing the observed measurements, budget limitations usually constrain the number n of samples that one can acquire, leading to situations where the number p of variables is much larger than n. In this situation, classical statistical methods, founded on the assumption that n is large and p is fixed, fail both in theory and in practice. A successful approach to overcome this problem is to assume a parsimonious generative model characterized by a number k of parameters, where k is much smaller than p. In this dissertation we develop algorithms to fit low-dimensional generative models and extract relevant information from high-dimensional, multivariate signals. First, we define extensions of the well-known Scalar Shrinkage-Thresholding Operator, that we name Multidimensional and Generalized Shrinkage-Thresholding Operators, and show that these extensions arise in numerous algorithms for structured-sparse linear and non-linear regression. Using convex optimization techniques, we show that these operators, defined as the solutions to a class of convex, non-differentiable, optimization problems have an equivalent convex, low-dimensional reformulation. Our equivalence results shed light on the behavior of a general class of penalties that includes classical sparsity-inducing penalties such as the LASSO and the Group LASSO. In addition, our reformulation leads in some cases to new efficient algorithms for a variety of high-dimensional penalized estimation problems. Second, we introduce two new classes of low-dimensional factor models that account for temporal shifts commonly occurring in multivariate signals. Our first contribution, called Order Preserving Factor Analysis, can be seen as an extension of the non-negative, sparse matrix factorization model to allow for order-preserving temporal translations in the data. We develop an efficient descent algorithm to fit this model using techniques from convex and non-convex optimization. Our second contribution extends Principal Component Analysis to the analysis of observations suffering from circular shifts, and we call it Misaligned Principal Component Analysis. We quantify the effect of the misalignments in the spectrum of the sample covariance matrix in the high-dimensional regime and develop simple algorithms to jointly estimate the principal components and the misalignment parameters.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91544/1/atibaup_1.pd