@article {12433,
	title = {A Blur-robust Descriptor with Applications to Face Recognition},
	journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
	volume = {PP},
	year = {2012},
	month = {2012/01/10/},
	pages = {1 - 1},
	abstract = {Understanding the effect of blur is an important problem in unconstrained visual analysis. We address this problem in the context of image-based recognition, by a fusion of image-formation models, and differential geometric tools. First, we discuss the space spanned by blurred versions of an image and then under certain assumptions, provide a differential geometric analysis of that space. More specifically, we create a subspace resulting from convolution of an image with a complete set of orthonormal basis functions of a pre-specified maximum size (that can represent an arbitrary blur kernel within that size), and show that the corresponding subspaces created from a clean image and its blurred versions are equal under the ideal case of zero noise, and some assumptions on the properties of blur kernels. We then study the practical utility of this subspace representation for the problem of direct recognition of blurred faces, by viewing the subspaces as points on the Grassmann manifold and present methods to perform recognition for cases where the blur is both homogenous and spatially varying. We empirically analyze the effect of noise, as well as the presence of other facial variations between the gallery and probe images, and provide comparisons with existing approaches on standard datasets.},
	keywords = {Blur, convolution, Face, face recognition, Grassmann manifold, Kernel, Manifolds, NOISE, PROBES},
	isbn = {0162-8828},
	doi = {10.1109/TPAMI.2012.15},
	author = {Gopalan,R. and Taheri, S. and Turaga,P. and Chellapa, Rama}
}
@article {14886,
	title = {Lambertian reflectance and linear subspaces},
	journal = {Pattern Analysis and Machine Intelligence, IEEE Transactions on},
	volume = {25},
	year = {2003},
	month = {2003/02//},
	pages = {218 - 233},
	abstract = {We prove that the set of all Lambertian reflectance functions (the mapping from surface normals to intensities) obtained with arbitrary distant light sources lies close to a 9D linear subspace. This implies that, in general, the set of images of a convex Lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low-dimensional linear subspace, explaining prior empirical results. We also provide a simple analytic characterization of this linear space. We obtain these results by representing lighting using spherical harmonics and describing the effects of Lambertian materials as the analog of a convolution. These results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions. We also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4D linear space. We apply these algorithms to perform face recognition by finding the 3D model that best matches a 2D query image.},
	keywords = {2D, 4D, 9D, analog;, analytic, characterization;, convex, convolution, distant, functions;, harmonics;, image, image;, intensities;, Lambertian, light, lighting, linear, methods;, nonnegative, normals;, object, optimization;, programming;, query, recognition;, reflectance;, reflectivity;, set;, sources;, space;, spherical, subspace;, subspaces;, surface},
	isbn = {0162-8828},
	doi = {10.1109/TPAMI.2003.1177153},
	author = {Basri,R. and Jacobs, David W.}
}
@article {15596,
	title = {Approximating large convolutions in digital images},
	journal = {Image Processing, IEEE Transactions on},
	volume = {10},
	year = {2001},
	month = {2001/12//},
	pages = {1826 - 1835},
	abstract = {Computing discrete two-dimensional (2-D) convolutions is an important problem in image processing. In mathematical morphology, an important variant is that of computing binary convolutions, where the kernel of the convolution is a 0-1 valued function. This operation can be quite costly, especially when large kernels are involved. We present an algorithm for computing convolutions of this form, where the kernel of the binary convolution is derived from a convex polygon. Because the kernel is a geometric object, we allow the algorithm some flexibility in how it elects to digitize the convex kernel at each placement, as long as the digitization satisfies certain reasonable requirements. We say that such a convolution is valid. Given this flexibility we show that it is possible to compute binary convolutions more efficiently than would normally be possible for large kernels. Our main result is an algorithm which, given an m times;n image and a k-sided convex polygonal kernel K, computes a valid convolution in O(kmn) time. Unlike standard algorithms for computing correlations and convolutions, the running time is independent of the area or perimeter of K, and our techniques do not rely on computing fast Fourier transforms. Our algorithm is based on a novel use of Bresenham{\textquoteright}s (1965) line-drawing algorithm and prefix-sums to update the convolution incrementally as the kernel is moved from one position to another across the image},
	keywords = {2D, algorithm;binary, approximation;mathematical, convolution, convolution;binary, convolutions;Bresenham{\textquoteright}s, convolutions;convex, convolutions;geometric, images;discrete, kernel;convex, kernel;digital, line-drawing, morphology;, morphology;approximation, object;image, polygonal, processing;large, processing;mathematical, theory;convolution;image, two-dimensional},
	isbn = {1057-7149},
	doi = {10.1109/83.974567},
	author = {Mount, Dave and Kanungo,T. and Netanyahu,N. S and Piatko,C. and Silverman,R. and Wu,A. Y}
}
@article {15800,
	title = {Restoring Images Degraded by Spatially Variant Blur},
	journal = {SIAM Journal on Scientific Computing},
	volume = {19},
	year = {1998},
	month = {1998///},
	pages = {1063 - 1082},
	abstract = {Restoration of images that have been blurred by the effects of a Gaussian blurring function is an ill-posed but well-studied problem. Any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation. Fast and effective algorithms then exist for determining the original image by preconditioned iterative methods. If the blurring function is spatially variant, however, then the problem is more difficult. In this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support. This assumption leads to a discrete problem involving a banded matrix. We devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm. Numerical examples are given, related to the Hubble Space Telescope (HST) Wide-Field/Planetary Camera. The algorithms that we develop are applicable to other ill-posed integral equations as well.},
	keywords = {convolution, discrete ill-posed problems, first-kind integral equations, image restoration, regularization, spatially variant point spread function},
	doi = {10.1137/S106482759528507X},
	url = {http://link.aip.org/link/?SCE/19/1063/1},
	author = {Nagy,James G. and O{\textquoteright}Leary, Dianne P.}
}
