On Nearly Orthogonal Lattice Bases and Random Lattices

Abstract

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 π 3 radians. 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.

JPEG compression history estimation for color images

“…Further, based on studies in [3], [10], we can model the DCT coefficients using a zero-mean Laplacian distribution (7) We have assumed that the parameter is known; in practice, we estimate from the observed decompressed image for each DCT frequency as described later in this section. From (7), we have (8) and hence (9) Now, assuming that the round-off error is independent of and , 's distribution is obtained by convolving the distributions for and (see …”

“…Recently, [7] and [8] See [8] for the proof and further details. Theorem 2 guarantees that in , a weakly orthogonal basis with nearly equal length vectors is related to every weakly orthogonal basis by a unimodular matrix with small elements.…”

“…We also demonstrate that the JPEG CH information is encoded in the nearly orthogonal bases that span the DCT lattices. Recently, [7] and [8] derived the geometric conditions for a lattice basis to contain the shortest nonzero lattice vector and the conditions to characterize such bases' uniqueness. Using these recent insights, and using novel applications of existing lattice algorithms, we estimate the color image's CH, namely, the affine color transform and the quantization tables.…”

JPEG compression history estimation for color images

“…As desired, this iteration is guaranteed to converge to a locally optimal that minimizes the round-off error between the observations and the closest points on the lattice spanned by . 8 In summary, we fuse LLL with noise attenuation as follows. 2) Estimating the 's: In the absence of round-offs, there exist 's such that is exactly diagonal (see Section VIII-A2).…”

“…Similar to the quantization step-size estimation described in Section IV, we estimate 's column norms by solving a penalized least-squares cost function, as shown by (28) at the bottom of the page) 8 Convergence follows because both (25) and (27) monotonically reduce kD 0 B I k .…”