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Scalable Frames
(2012)
Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.
A basic task in signal analysis is to character-
ize data in a meaningful way for analysis and classification
purposes. Time-frequency transforms are powerful strategies
for signal decomposition, and important recent generalizations
have been achieved in the setting of frame theory. In parallel
recent developments, tools from algebraic topology, traditionally
developed in purely abstract settings, have provided new insights
in applications to data analysis. In this report, we investigate some
interactions of these tools, both theoretically and with numerical
experiments, in order to characterize signals and their frame
transforms. We explain basic concepts in persistent homology
as an important new subfield of computational topology, as well
as formulations of time-frequency analysis in frame theory. Our
objective is to use persistent homology for constructing topo-
logical signatures of signals in the context of frame theory. The
motivation is to design new classification and analysis methods by
combining the strength of frame theory as a fundamental signal
processing methodology, with persistent homology as a new tool
in data analysis.
A basic task in signal analysis is to character-
ize data in a meaningful way for analysis and classification
purposes. Time-frequency transforms are powerful strategies
for signal decomposition, and important recent generalizations
have been achieved in the setting of frame theory. In parallel
recent developments, tools from algebraic topology, traditionally
developed in purely abstract settings, have provided new insights
in applications to data analysis. In this report, we investigate some
interactions of these tools, both theoretically and with numerical
experiments, in order to characterize signals and their frame
transforms. We explain basic concepts in persistent homology
as an important new subfield of computational topology, as well
as formulations of time-frequency analysis in frame theory. Our
objective is to use persistent homology for constructing topo-
logical signatures of signals in the context of frame theory. The
motivation is to design new classification and analysis methods by
combining the strength of frame theory as a fundamental signal
processing methodology, with persistent homology as a new tool
in data analysis.
We study the exact recovery of signals from quantized frame coefficients. Here, the basis of the quantization
is hard thresholding, and we present a simple algorithm for the recovery of reconstructable signals. The set of
non-reconstructable signals is shown to be star-shaped and symmetric with respect to the origin. Moreover, we
provide a criterion on the frame for the boundedness of this set. In this case, we also give a priori bounds.