Preface
This book is a continuation of a study begun in [111] in which we developed and
applied computational electromagnetics and inverse techniques to quantitative nondestructive evaluation and materials characterization. While model-based inverse
methods using a nonlinear least-squares algorithm (NLSE) were developed in [111],
we develop two new ’voxel-based’ algorithms in this book: the bilinear conjugategradient and set-theoretic algorithms.
NLSE is suitable when there are only a few parameters that are required to
define a model. There are, of course, any number of practical problems that cannot
be simply defined in terms of a few variables. For example, an actual fatigue
crack may not be well-represented by a simple slot or EDM notch that could
be parameterized for NLSE. Actual cracks may emanate from a number of sites
(‘multisite damage’) or may simply meander through the structure, so that they are
not so easily parametrized. This suggests that we need to reconstruct the anomaly
voxel-by-voxel, which is what the set-theoretic and bilinear conjugate-gradient
algorithms are designed to do.
By ’voxel-based’, we mean that each voxel in the grid of the anomalous region
is to be reconstructed in order to determine the anomaly, rather than to model the
anomaly at the outset as a canonical structure that is defined by a few parameters
that are to be reconstructed. In this sense, voxel-based algorithms are an attempt to
eliminate the ’curse of dimensionality’ malady that is present in NLSE due to the
huge interpolation grid that is required for large problems [111].
We have studied two such voxel-based algorithms: the bilinear conjugategradient algorithm and the set-theoretic algorithm. The conjugate-gradient algorithm is rather well-known in the literature and reconstructs the grid in the usual
manner of considering the mutual interaction of each voxel on the other, much as
in solving the forward problem with VIC-3D ® . On the contrary, the set-theoretic
algorithm reconstructs each voxel independently of the other (after a first stage
of linear data processing), i.e., it is non-Markovian and yields estimation-theoretic
quality estimates of each voxel as well. Furthermore, the set-theoretic algorithm
introduces some interesting statistical notions that are relevant to the research theme
v
This book is a continuation of a study begun in [111] in which we developed and
applied computational electromagnetics and inverse techniques to quantitative nondestructive evaluation and materials characterization. While model-based inverse
methods using a nonlinear least-squares algorithm (NLSE) were developed in [111],
we develop two new ’voxel-based’ algorithms in this book: the bilinear conjugategradient and set-theoretic algorithms.
NLSE is suitable when there are only a few parameters that are required to
define a model. There are, of course, any number of practical problems that cannot
be simply defined in terms of a few variables. For example, an actual fatigue
crack may not be well-represented by a simple slot or EDM notch that could
be parameterized for NLSE. Actual cracks may emanate from a number of sites
(‘multisite damage’) or may simply meander through the structure, so that they are
not so easily parametrized. This suggests that we need to reconstruct the anomaly
voxel-by-voxel, which is what the set-theoretic and bilinear conjugate-gradient
algorithms are designed to do.
By ’voxel-based’, we mean that each voxel in the grid of the anomalous region
is to be reconstructed in order to determine the anomaly, rather than to model the
anomaly at the outset as a canonical structure that is defined by a few parameters
that are to be reconstructed. In this sense, voxel-based algorithms are an attempt to
eliminate the ’curse of dimensionality’ malady that is present in NLSE due to the
huge interpolation grid that is required for large problems [111].
We have studied two such voxel-based algorithms: the bilinear conjugategradient algorithm and the set-theoretic algorithm. The conjugate-gradient algorithm is rather well-known in the literature and reconstructs the grid in the usual
manner of considering the mutual interaction of each voxel on the other, much as
in solving the forward problem with VIC-3D ® . On the contrary, the set-theoretic
algorithm reconstructs each voxel independently of the other (after a first stage
of linear data processing), i.e., it is non-Markovian and yields estimation-theoretic
quality estimates of each voxel as well. Furthermore, the set-theoretic algorithm
introduces some interesting statistical notions that are relevant to the research theme
v