Browsing by Author "Dash, Sanjeeb"

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    JPEG Compression History Estimation for Color Images
    (2006) Neelamani, Ramesh; de Queiroz, Ricardo; Fan, Zhigang; Dash, Sanjeeb; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
    We routinely encounter digital color images that were previously JPEG-compressed. En route to the image's current representation, the previous JPEG compression's various settings—termed its JPEG compression history (CH)—are often discarded after the JPEG decompression step. Given a JPEG-decompressed color image, this paper aims to estimate its lost JPEG CH. We observe that the previous JPEG compression's quantization step introduces a lattice structure in the discrete cosine transform (DCT) domain. This paper proposes two approaches that exploit this structure to solve the JPEG Compression History Estimation (CHEst) problem. First, we design a statistical dictionary-based CHEst algorithm that tests the various CHs in a dictionary and selects the maximum a posteriori estimate. Second, for cases where the DCT coefficients closely conform to a 3-D parallelepiped lattice, we design a blind lattice-based CHEst algorithm. The blind algorithm exploits the fact that the JPEG CH is encoded in the nearly orthogonal bases for the 3-D lattice and employs novel lattice algorithms and recent results on nearly orthogonal lattice bases to estimate the CH. Both algorithms provide robust JPEG CHEst performance in practice. Simulations demonstrate that JPEG CHEst can be extremely useful in JPEG recompression; the estimated CH allows us to recompress a JPEG-decompressed image with minimal distortion (large signal-to-noise-ratio) and simultaneously achieve a small file-size.
  • Thumbnail Image
    Item
    On Nearly Orthogonal Lattice Bases
    (2005-07-01) Neelamani, Ramesh; Dash, Sanjeeb; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)
    We study "nearly orthogonal" lattice bases, or bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is greater than 60°. We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, if the lengths of the basis vectors are "nearly equal", then the basis is the unique nearly orthogonal lattice basis, up to multiplication of basis vectors by ±1. These results are motivated by an application involving JPEG image compression.
  • Thumbnail Image
    Item
    On the matrix cuts of Lovasz and Schrijver and their use in integer programming
    (2001) Dash, Sanjeeb; Cook, William J.
    An important approach to solving many discrete optimization problems is to associate the discrete set (over which we wish to optimize) with the 0-1 vectors in a given polyhedron and to derive linear inequalities valid for these 0-1 vectors from a linear inequality system defining the polyhedron. Lovasz and Schrijver (1991) described a family of operators, called the matrix-cut operators, which generate strong valid inequalities, called matrix cuts, for the 0-1 vectors in a polyhedron. This family includes the commutative, semidefinite and division operators; each operator can be applied iteratively to obtain, in n iterations for polyhedra in n-space, the convex hull of 0-1 vectors. We study the complexity of matrix-cut based methods for solving 0-1 integer linear programs. We first prove bounds on the (rank) number of iterations required to obtain the integer hull. We show that the upper bound of n, mentioned above, can be attained in the case of the semidefinite operator, answering a question of Goemans. We also determine the semidefinite rank of the standard linear relaxation of the traveling salesman polytope up to a constant factor. We study the use of the semidefinite operator in solving numerical instances and present results on some combinatorial examples and also on a few instances from the MIPLIB test set. Finally, we examine the lengths of cutting-plane proofs based on matrix cuts. We answer a question of Pudlak on such proofs, and prove an exponential lower bound on the length of cutting-plane proofs based on one class of matrix cuts.
  • Thumbnail Image
    Item
    On the Matrix Cuts of Lovász and Schrijver and Their Use in Integer Programming
    (2001-03) Dash, Sanjeeb
    An important approach to solving many discrete optimization problems is to associate the discrete set (over which we wish to optimize) with the 0-1 vectors in a given polyhedron and to derive linear inequalities valid for these 0-1 vectors from a linear inequality system defining the polyhedron. Lovász and Schrijver (1991) described a family of operators, called the matrix-cut operators, which generate strong valid inequalities, called matrix cuts, forthe 0-1 vectors in a polyhedron. This family includes the commutative, semidefinite and division operators; each operator can be applied iteratively to obtain, in n iterations for polyhedra in n-space, the convex hull of 0-1 vectors. We study the complexity of matrix-cut based methods for solving 0-1 integer linear programs. We first prove bounds on the (rank) number of iterations required to obtain the integer hull. We show that the upper bound of n, mentioned above, can be attained in the case of the semidefinite operator, answering a question of Goemans. We also determine the semidefinite rank of the standard linear relaxation of the traveling salesman polytope up to a constant factor. We study the use of the semidefinite operator in solving numerical instances and present results on some combinatorial examples and also on a few instances from the MIPLIB test set. Finally, we examine the lengths of cutting-plane proofs based on matrix cuts. We answer a question of Pudlák on such proofs, and prove an exponential lower bound on the length of cutting-plane proofs based on one class of matrix cuts.