arXiv subjects

Pedro Felzenszwalb

Publications and source records attributed to Pedro Felzenszwalb.

11 recordsLinked to original sources

arXiv2026

Multiscale Super Resolution without Image Priors

We address the ambiguities in the super-resolution problem under translation. We demonstrate that combinations of low-resolution images at different scales can be used to make the super-resolution problem well posed. Such differences in scale can be achieved using sensors with different pixel sizes (as demonstrated here) or by varying the effective pixel size through changes in optical magnification (e.g., using a zoom lens). We show that images acquired with pairwise coprime pixel sizes lead to a system with a stable inverse, and furthermore, that super-resolution images can be reconstructed efficiently using Fourier domain techniques or iterative least squares methods. Our mathematical analysis provides an expression for the expected error of the least squares reconstruction for large signals assuming i.i.d. noise that elucidates the noise-resolution tradeoff. These results are validated through both one- and two-dimensional experiments that leverage charge-coupled device (CCD) hardware binning to explore reconstructions over a large range of effective pixel sizes. Finally, two-dimensional reconstructions for a series of targets are used to demonstrate the advantages of multiscale super-resolution, and implications of these results for common imaging systems are discussed.

cs.CV↗

arXiv2026

Projected Subgradient Ascent for Convex Maximization

We consider the problem of maximizing a convex function over a closed convex set in a real Hilbert space. For linear functions, we show that a single orthogonal projection suffices to obtain an approximate solution. For continuous convex functions over convex sets, we show that projected subgradient ascent converges to a first-order stationary point when using arbitrarily large step sizes. Taking the step sizes to infinity leads to a deterministic variant of the conditional gradient algorithm, and iterated linear optimization as a special case.

math.OC↗

arXiv2025

Super-Resolution with Structured Motion

We consider the limits of super-resolution using imaging constraints. Due to various theoretical and practical limitations, reconstruction-based methods have been largely restricted to small increases in resolution. In addition, motion-blur is usually seen as a nuisance that impedes super-resolution. We show that by using high-precision motion information, sparse image priors, and convex optimization, it is possible to increase resolution by large factors. A key operation in super-resolution is deconvolution with a box. In general, convolution with a box is not invertible. However, we obtain perfect reconstructions of sparse signals using convex optimization. We also show that motion blur can be helpful for super-resolution. We demonstrate that using pseudo-random motion it is possible to reconstruct a high-resolution target using a single low-resolution image. We present numerical experiments with simulated data and results with real data captured by a camera mounted on a computer controlled stage.

cs.CV↗

arXiv2024

Deconvolution with a Box

Deconvolution with a box (square wave) is a key operation for super-resolution with pixel-shift cameras. In general convolution with a box is not invertible. However, we can obtain perfect reconstructions of sparse signals using convex optimization. We give a direct proof that improves on the reconstruction bound that follows from previous results. We also show our bound is tight and matches an information theoretic limit.

math.NA↗

arXiv2022

Convex Combination Belief Propagation Algorithms

We present new message passing algorithms for performing inference with graphical models. Our methods are designed for the most difficult inference problems where loopy belief propagation and other heuristics fail to converge. Belief propagation is guaranteed to converge when the underlying graphical model is acyclic, but can fail to converge and is sensitive to initialization when the underlying graph has complex topology. This paper describes modifications to the standard belief propagation algorithms that lead to methods that converge to unique solutions on graphical models with arbitrary topology and potential functions.

cs.AI↗

arXiv2022

Clustering with Semidefinite Programming and Fixed Point Iteration

We introduce a novel method for clustering using a semidefinite programming (SDP) relaxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using iterated linear optimization. We show the vertices of the Max k-Cut relaxation correspond to partitions of the data into at most k sets. We also show the vertices are attractive fixed points of iterated linear optimization. Each step of this iterative process solves a relaxation of the closest vertex problem and leads to a new clustering problem where the underlying clusters are more clearly defined. Our experiments show that using fixed point iteration for rounding the Max k-Cut SDP relaxation leads to significantly better results when compared to randomized rounding.

math.OC↗

arXiv2021

Direct Estimation of Appearance Models for Segmentation

Image segmentation algorithms often depend on appearance models that characterize the distribution of pixel values in different image regions. We describe a new approach for estimating appearance models directly from an image, without explicit consideration of the pixels that make up each region. Our approach is based on novel algebraic expressions that relate local image statistics to the appearance of spatially coherent regions. We describe two algorithms that can use the aforementioned algebraic expressions to estimate appearance models directly from an image. The first algorithm solves a system of linear and quadratic equations using a least squares formulation. The second algorithm is a spectral method based on an eigenvector computation. We present experimental results that demonstrate the proposed methods work well in practice and lead to effective image segmentation algorithms.

cs.CV↗

arXiv2021

Iterated Linear Optimization

We introduce a fixed point iteration process built on optimization of a linear function over a compact domain. We prove the process always converges to a fixed point and explore the set of fixed points in various convex sets. In particular, we consider elliptopes and derive an algebraic characterization of their fixed points. We show that the attractive fixed points of an elliptope are exactly its vertices. Finally, we discuss how fixed point iteration can be used for rounding the solution of a semidefinite programming relaxation.

math.OC↗

arXiv2019

Generalized Majorization-Minimization

Non-convex optimization is ubiquitous in machine learning. Majorization-Minimization (MM) is a powerful iterative procedure for optimizing non-convex functions that works by optimizing a sequence of bounds on the function. In MM, the bound at each iteration is required to \emph{touch} the objective function at the optimizer of the previous bound. We show that this touching constraint is unnecessary and overly restrictive. We generalize MM by relaxing this constraint, and propose a new optimization framework, named Generalized Majorization-Minimization (G-MM), that is more flexible. For instance, G-MM can incorporate application-specific biases into the optimization procedure without changing the objective function. We derive G-MM algorithms for several latent variable models and show empirically that they consistently outperform their MM counterparts in optimizing non-convex objectives. In particular, G-MM algorithms appear to be less sensitive to initialization.

cs.CV↗

arXiv2019

Flow-firing processes

We consider a discrete non-deterministic flow-firing process for rerouting flow on the edges of a planar complex. The process is an instance of higher-dimensional chip-firing. In the flow-firing process, flow on the edges of a complex is repeatedly diverted across the faces of the complex. For non-conservative initial configurations we show this process never terminates. For conservative initial flows we show the process terminates after a finite number of rerouting steps, but there are many possible final configurations reachable from a single initial state. Finally, for conservative initial flows around a topological hole we show the process terminates at a unique final configuration. In this case the process exhibits global confluence despite not satisfying local confluence.

math.CO↗

arXiv2015

Automatic Discovery and Optimization of Parts for Image Classification

Part-based representations have been shown to be very useful for image classification. Learning part-based models is often viewed as a two-stage problem. First, a collection of informative parts is discovered, using heuristics that promote part distinctiveness and diversity, and then classifiers are trained on the vector of part responses. In this paper we unify the two stages and learn the image classifiers and a set of shared parts jointly. We generate an initial pool of parts by randomly sampling part candidates and selecting a good subset using L1/L2 regularization. All steps are driven "directly" by the same objective namely the classification loss on a training set. This lets us do away with engineered heuristics. We also introduce the notion of "negative parts", intended as parts that are negatively correlated with one or more classes. Negative parts are complementary to the parts discovered by other methods, which look only for positive correlations.

cs.CV↗