KEYWORDS: Fourier transforms, Image processing, Digital image processing, Wavelets, Monte Carlo methods, Image compression, Mathematics, MATLAB, Precision measurement, Transform theory
Shearlab is a Matlab toolbox for digital shearlet transformation of two-D (image) data we developed following
a rational design process. The Pseudo-Polar FFT fits very naturally with the continuum theory of the Shearlet
transform and allows us to translate Shearlet ideas naturally into a digital framework. However, there are
still windows and weights which must be chosen. We developed more than a dozen performance measures
quantifying precision of the reconstruction, tightness of the frame, directional and spatial localization and other
properties. Such quantitative performance metrics allow us to: (a) tune parameters and objectively improve our
implementation; and (b) compare different directional transform implementations. We present and interpret the
most important performance measures for our current implementation.
We used digital image processing and statistical clustering algorithms to segment and classify brush strokes in
master paintings based on two-dimensional space and
three-dimensional chromaticity coordinates. For works
executed in sparse overlapping brush strokes our algorithm identifies candidate clusters of brush strokes of the
top (most visible) layer and digitally removes them. Then, it applies modified inpainting algorithms based on
statistical structure of strokes to fill in or "inpaint" the remaining, partially hidden brush strokes. This processes
can be iterated, to reveal and fill in successively deeper (partially hidden) layers of brush strokes-a process we
call "de-picting." Of course, the reconstruction of strokes at each successively deeper layer is based on less and
less image data from the painting and requires cascading estimates and inpainting; as such our methods yield
poorer accuracy and fidelity for such deeper layers. Our current software is semi-automatic; the operator such
as a curator or art historian guides certain steps. Future versions of our software will be fully automatic, and
estimate more accurate statistical models of the brush strokes in the target painting yield better estimates of
hidden brush strokes. Our software tools may aid art scholars in characterizing the images of paintings as well
as the working methods of some master painters.
We develop a wavelet transform on the sphere, based on the spherical HEALPix coordinate system (Hierarchical
Equal Area iso-Latitude Pixelization). HEALPix is heavily used for astronomical data processing applications; it
is intrinsically multiscale and locally euclidean, hence appealing for building multiscale system. Furthermore, the
equal-area pixelization enables us to employ average-interpolating refinement, giving wavelets of local support.
HEALPix wavelets have numerous applications in geopotential modeling. A statistical analysis demonstrates
wavelet compressibility of the geopotential field and shows that geopotential wavelet coefficients have many
of the statistical properties that were previously observed with wavelet coefficients of natural images. The
HEALPix wavelet expansion allows to evaluate a gravimetric quantity over a local region far more rapidly than
the classic approach based on spherical harmonics. Our software tools are specifically tailored to demonstrate
these advantages.
Current digital imaging devices often enable the user to capture still
frames at a high spatial resolution, or a short video clip at a lower spatial resolution. With bandwidth limitations inherent to any sensor, there is clearly a tradeoff between spatial and temporal sampling rates, which can be studied, and which present-day sensors do not exploit. The fixed sampling rate that is normally used does not capture the scene according to its temporal and spatial content and artifacts such as aliasing and motion blur appear. Moreover, the available bandwidth on the camera transmission or memory is not optimally utilized. In this paper we outline a framework for an adaptive sensor where the spatial and temporal sampling rates are adapted to the scene. The sensor is adjusted to capture the scene with respect to its content. In the adaptation process, the spatial and temporal content of the video sequence are measured to evaluate the required sampling rate. We propose a robust, computationally
inexpensive, content measure that works in the spatio-temporal
domain as opposed to the traditional frequency domain methods. We
show that the measure is accurate and robust in the presence of noise and aliasing. The varying sampling rate stream captures the scene more efficiently and with fewer artifacts such that in a post-processing step an enhanced resolution sequence can be effectively composed or an overall lower bandwidth for the capture of the scene can be realized, with small distortion.
The present paper concerns the statistical analysis of limits to achievable resolution in a so-called "diffraction-limited" imaging system. The canonical case study is that of incoherent imaging of two closely-spaced sources of possibly unequal intensities. The objective is to study how far beyond the classical Rayleigh limit of resolution one can reach at a given signal to noise ratio. We consider the definition of resolution limit from a statistical point of view as the ability of the imaging system to distinguish two closely-located sources in presence of additive noise. This problem can be stated in a
hypothesis testing framework where the hypotheses consider whether
one or two point sources are present. In terms of signal detection/ estimation, this leads to composite detection/estimation problem where a deterministic signal with unknown parameters is being sought. To solve this problem, we use locally optimal statistical tests with respect to a desired range of (small) separations between the point sources. Specifically, we will derive explicit relationships between the minimum detectable distance between two point sources, and the required SNR. For a specific point spread function, the required SNR can be expressed as a function of probabilities of detection and false alarm and the distance between point sources.
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