Optimization of Parametric Discrete Orthogonal Transforms and Applications.
domain methods constitute an important class of signal processing techniques, important
in applications such as image and video compression, noise removal,information hiding
and pattern recognition.Natural signals exhibiting high proximal correlations typically
admit transform domain representations with compact energy distributions. Given the
appropriate transform, such ... read morerepresentations capture the intrinsic features present in
the signal of interest, enabling data compression necessary for signal discrimination or
pattern recognition. Consequently, the proper selection of a transform basis is critical
to the performance of algorithms designed to accomplish these tasks. Typically, signal
dependent decompositions, such as the Karhunen-Loeve transform(KLT),are considered
optimal in terms of parsimony. However, the computational complexity of the KLT is
O(N^3), N is the size of the input data. This is very high and thereby limiting its use
in practical applications. Nearly optimal are signal independent,deterministic
transforms such as the Discrete Cosine Transform (DCT) provides more flexible signal
representations have sought to improve performance without sacrificing computational
efficiency.The complexity of such deterministic transform is O(Nlog2N). The Discrete
Walsh-Hadamard transform(WHT) and discrete slant-Hadamard transform(SHT) are the
canonical examples of non-sinusoidal transforms arising from these efforts.
Non-sinusoidal discrete transforms may be quickly computed using fast algorithms.
However,representations performance of non-sinusoidal transforms still lags the DCT and
even the KLT when dealing with real-world signals. Thus, the motivation of this work is
to preserve the computational advantage of these transforms when operating on a broad
class of signals via optimal parameter selection. Generalizations of classical discrete
orthogonal transforms offer many advantages in terms of parsimonious signal
representation and efficient signal processing. Here, we propose new signal
representations defined via variable Givens rotations of existing transforms and indexed
by sets of parameters. We investigate a unified framework for constructing such families
of planar rotation based parametric transforms. Such a formulation yields new insight
into existing orthogonal transforms such as the Hadamard, Haar, Slant, Slant-Haar and
Slantlet transforms; all may be viewed as special cases under the unified scheme.
Consequently, fast algorithms derived herein for generalized transforms may also be
applied to existing methods. Key to our approach is the selection of parameters
optimized to desirable properties of the signals to be analyzed. Sparsity measures are
shown to be effective criteria in parameter optimization.We also develop efficient
parameter pursuit algorithms by exploiting intrinsic properties of the implementation
structures of existing fast algorithms and the statistical structures of the given input
signals. Furthermore, two new classes of transforms, the parametric Slant-Hadamard
transform and the parametric Discrete Cosine Transform,are proposed admitting novel
applications such as face recognition algorithm and digital watermarking scheme. The
optimized parametric transform is also validated by studying relative mean square error
performance and rate distortion performance in image compression
Thesis (Ph.D.)--Tufts University, 2012.
Submitted to the Dept. of Electrical Engineering.
Advisor: Joseph Noonan.
Committee: Joseph Noonan, Sos Agaian, Brian Tracey, and John Hogan.
Keyword: Electrical engineering.read less