Stationarity of the noise distribution is a common assumption in image processing. This assumption greatly simplifies denoising estimators and other model parameters and consequently assuming stationarity is often a matter of convenience rather than an accurate model of noise characteristics. The problematic nature of this assumption is exacerbated in real-world contexts, where noise is often highly non-stationary and can possess time- and space-varying characteristics. Regardless of model complexity, estimating the parameters of noise dis- tributions in digital images is a difficult task, and estimates are often based on heuristic assumptions. Recently, sparse Bayesian dictionary learning methods were shown to produce accurate estimates of the level of additive white Gaussian noise in images with minimal assumptions. We show that a similar model is capable of accu- rately modeling certain kinds of non-stationary noise processes, allowing for space-varying noise in images to be estimated, detected, and removed. We apply this modeling concept to several types of non-stationary noise and demonstrate the model’s effectiveness on real-world problems, including denoising and segmentation of images according to noise characteristics, which has applications in image forensics.
A fusion frame is a collection of subspaces in a Hilbert space, generalizing the idea of a frame for signal representation.
A tool to construct fusion frames is the spectral tetris algorithm, a flexible and elementary method
to construct unit norm frames with a given frame operator having all of its eigenvalues greater than or equal to
two. We discuss how spectral tetris can be used to construct fusion frames with prescribed eigenvalues for its
fusion frame operator and with prescribed dimensions for its subspaces.
Redundant systems such as frames are often used to represent a signal for error correction, denoising
and general robustness. In the digital domain quantization needs to be performed. Given
the redundancy, the distribution of quantization errors can be rather complex. In this paper we
study quantization error for a signal X in Rd represented by a frame using a lattice quantizer. We
characterize the asymptotic distribution of the quantization error as the cell size of the lattice goes
to zero. We apply these results to get the necessary and sufficient conditions for the White Noise
Hypothesis to hold asymptotically in the case of the pulse-code modulation scheme.
This is an abbreviated version of a paper that will appear elsewhere in a regular refereed journal.
Fusion frames are an emerging topic of frame theory, with applications to communications and distributed
processing. However, until recently, little was known about the existence of tight fusion frames, much less how
to construct them. We discuss a new method for constructing tight fusion frames which is akin to playing Tetris
with the spectrum of the frame operator. When combined with some easily obtained necessary conditions, these
Spectral Tetris constructions provide a near complete characterization of the existence of tight fusion frames.
Recent work has shown that the mathematics of fractal geometry can be used to provide a quantitative signature
for the drip paintings of Jackson Pollock. In this paper we discuss the calculation of a related quantity, the "entropy
dimension" and discuss the possibility of its use a measure or signature for Pollock's work. We furthermore
raise the question of the robustness or stability of the fractal measurements with respect to variables like mode
of capture, digital resolution, and digital representation and include the results of a small experiment in the step
of color layer extraction.
The β-encoder, introduced as an alternative to binary encoding in A/D conversion,
creates a quantization scheme robust with respect to quantizer imperfections by
the use of a β-expansion, where 1 < β < 2. In this paper we introduce a more general encoder
called the βα-encoder, that can offer more flexibility in design and robustness without
any significant drawback on the exponential rate of convergence of the obtained expansion.
Although an extra multiplication is introduced, it needs not be very accurate. Mathematically,
the βα-encoder gives rise to a dynamical system that is both very interesting and
challenging.
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