A nearly-mlogn time solver for SDD linear systems

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$ such that $A\bar{x} = b$ for some (unknown) vector $\bar{x}$, our algorithm computes a vector $x$ such that $||{x}-\bar{x}||_A < \epsilon ||\bar{x}||_A$ {$||\cdot||_A$ denotes the A-norm} in time $${\tilde O}(m\log n \log (1/\epsilon)).$$ The solver utilizes in a standard way a `preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties. We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in time $\tilde{O}(m\log{n})$, a factor of $O(\log{n})$ faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.Comment: to appear in FOCS1

Efficient Poisson Image Editing

Image composition refers to the process of composing two or more images to create a natural output image. It is one of the important techniques in image processing. In this paper, two efficient methods for composing color images are proposed. In the proposed methods, the Poisson equation is solved using image pyramid and divide-and-conquer methods. The proposed methods are more efficient than other existing image composition methods. They reduce the time taken in the composition process while achieving almost identical results using the previous image composition methods. In the proposed methods, the Poisson equation is solved after converting it to a linear system using different methods. The results show that the time for composing color images is decreased using the proposed methods

Multi-scale Image Harmonization

Traditional image compositing techniques, such as alpha matting and gradient domain compositing, are used to create composites that have plausible boundaries. But when applied to images taken from different sources or shot under different conditions, these tech- niques can produce unrealistic results. In this work, we present a framework that explicitly matches the visual appearance of images through a process we call image harmonization, before blending them. At the heart of this framework is a multi-scale technique that allows us to transfer the appearance of one image to another. We show that by carefully manipulating the scales of a pyramid decomposition of an image, we can match contrast, texture, noise, and blur, while avoiding image artifacts. The output composite can then be reconstructed from the modified pyramid coefficients while enforcing both alpha-based and seamless boundary constraints. We show how the proposed framework can be used to produce realistic composites with minimal user interaction in a number of different scenarios.Engineering and Applied Science

User-assisted intrinsic images

For many computational photography applications, the lighting and materials in the scene are critical pieces of information. We seek to obtain intrinsic images, which decompose a photo into the product of an illumination component that represents lighting effects and a reflectance component that is the color of the observed material. This is an under-constrained problem and automatic methods are challenged by complex natural images. We describe a new approach that enables users to guide an optimization with simple indications such as regions of constant reflectance or illumination. Based on a simple assumption on local reflectance distributions, we derive a new propagation energy that enables a closed form solution using linear least-squares. We achieve fast performance by introducing a novel downsampling that preserves local color distributions. We demonstrate intrinsic image decomposition on a variety of images and show applications.National Science Foundation (U.S.) (NSF CAREER award 0447561)Institut national de recherche en informatique et en automatique (France) (Associate Research Team “Flexible Rendering”)Microsoft Research (New Faculty Fellowship)Alfred P. Sloan Foundation (Research Fellowship)Quanta Computer, Inc. (MIT-Quanta T Party