55,293 research outputs found
Fast algorithm for the 3-D DCT-II
Recently, many applications for three-dimensional
(3-D) image and video compression have been proposed using 3-D discrete cosine transforms (3-D DCTs). Among different types of DCTs, the type-II DCT (DCT-II) is the most used. In order to use the 3-D DCTs in practical applications, fast 3-D algorithms are essential. Therefore, in this paper, the 3-D vector-radix decimation-in-frequency (3-D VR DIF) algorithm that calculates the 3-D DCT-II directly is introduced. The mathematical analysis and the implementation of the developed algorithm are presented,
showing that this algorithm possesses a regular structure, can be implemented in-place for efficient use of memory, and is faster than the conventional row-column-frame (RCF) approach. Furthermore, an application of 3-D video compression-based 3-D DCT-II is implemented using the 3-D new algorithm. This has led to a substantial speed improvement for 3-D DCT-II-based compression systems and proved the validity of the developed algorithm
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
Generating and Searching Families of FFT Algorithms
A fundamental question of longstanding theoretical interest is to prove the
lowest exact count of real additions and multiplications required to compute a
power-of-two discrete Fourier transform (DFT). For 35 years the split-radix
algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic
operations on real numbers for a size-n DFT, and was widely believed to be the
best possible. Recent work by Van Buskirk et al. demonstrated improvements to
the split-radix operation count by using multiplier coefficients or "twiddle
factors" that are not n-th roots of unity for a size-n DFT. This paper presents
a Boolean Satisfiability-based proof of the lowest operation count for certain
classes of DFT algorithms. First, we present a novel way to choose new yet
valid twiddle factors for the nodes in flowgraphs generated by common
power-of-two fast Fourier transform algorithms, FFTs. With this new technique,
we can generate a large family of FFTs realizable by a fixed flowgraph. This
solution space of FFTs is cast as a Boolean Satisfiability problem, and a
modern Satisfiability Modulo Theory solver is applied to search for FFTs
requiring the fewest arithmetic operations. Surprisingly, we find that there
are FFTs requiring fewer operations than the split-radix even when all twiddle
factors are n-th roots of unity.Comment: Preprint submitted on March 28, 2011, to the Journal on
Satisfiability, Boolean Modeling and Computatio
Biorthogonal partners and applications
Two digital filters H(z) and F(z) are said to be biorthogonal partners of each other if their cascade H(z)F(z) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. We first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above-mentioned applications
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