Both the Kingsbury dual-tree and the subsequent Selesnick
double-density dual-tree complex wavelet transform
approximate an analytic function. The classification of the phase dependency across scales is largely unexplored
except by Romberg et al.. Here we characterize the sub-band dependency of the orientation of phase gradients
by applying the Helmholtz principle to bivariate histograms to locate meaningful modes. A further characterization
using the Earth Mover's Distance with the fundamental
Rudin-Osher-Meyer Banach space decomposition
into cartoon and texture elements is presented. Possible applications include image compression and invariant
descriptor selection for image matching.
Infrared sensors and advanced signal processing are used to detect small (or point) targets in highly cluttered and noisy environments. In this paper, a wavelet detection algorithm and tracking of small targets in clutter will be discussed. A new registration algorithm based on optical flow estimates with matched subspace detectors against small maneuverable targets is also discussed. Both detectors incorporate adaptive constant false alarm rate (CFAR) detection statistics. Simulation of the detection and tracking algorithms using an unclassified database with a helicopter target and platform for the video cameras is summarized.
Wavelet-based detection algorithms are developed for detection small targets in non-Gaussian clutter. A wavelet transform is applied to reduce spatial correlation of the clutter. An adaptive matched filter is applied in the wavelet transform domain which uses estimated covariance matrices derived from the wavelet coefficients. Two problems hinder the use of covariance estimates for background clutter removal: slow computational speed and induced false alarms resulting from nearly singular covariance estimates due to small sample sizes. The issue of speed is dealt with by evaluating the covariance matrices on a sparse grid followed by low order interpolation. To control the problem of bad covariance estimates we filter the grid generated covariance matrices to remove outliers using peer group averaging. This procedure removes the false alarm problem associated with nearly singular covariance estimates without degrading the overall performance of the clutter removal process.
A Daubechies' wavelet-based constant false alarm rate (CFAR) small-target detection algorithm is evaluated using measured and simulated infrared images. The wavelet-based detection algorithm is compared with the matched filter to establish relative performance curves. The adaptive CFAR detection statistics are derived from the lexicographically ordered image vectors using Efron's bootstrap method. The bootstrap employs repeated resampling to overcome the difficulties of modeling the post-transform detection statistics of the underlying clutter or fixed pattern noise. The performance of the detection algorithm is evaluated using a simulated Gaussian target with parametrically varying amplitude, size, and polarity. It is embedded in fixed pattern noise and measured images that will stress the detection algorithms.
A Daubechies wavelet-based bootstrap detection strategy based on the research of Carmona was applied to a set of test signals. The detector was a function of the d-scales. The adaptive detection statistics were derived using Efron's bootstrap methodology, which relieved us from having to make parametric assumptions about the underlying noise and offered a method of overcoming the constraints of modeling the detector statistics. The test set of signals used to evaluate the Daubechies/bootstrap pulse detector were generated with a Hewlett-Packard Fast Agile Signal Simulator (FASS). These video pulses, with varying signal-to-noise ratios (SNRs), included unmodulated, linear chirp, and Barker phase-code modulations baseband (IF) video pulses mixed with additive white Gaussian noise. Simulated examples illustrating the bootstrap methodology are presented, along with a complete set of constant false alarm rate (CFAR) detection statistics for the test signals. The CFAR curves clearly show that the wavelet bootstrap can adaptively detect transient pulses at low SNRs.
This paper presents results on an approach to optical flow estimation and image segmentation based on treating the flow of image level sets rather than individual points. This allows the accurate estimation of object velocity even from low quality video sequences and has the advantage of simplifying the analysis of classical ill-condition problems for optical flow estimation such as the aperutre effect. This procedure has been tailored to motion estimation for small to intermediate sized objects and can be applied to the problem of estimating human locomotion from image sequences. Under reasonable assumptions it is shown analytically that the condition number of the from image sequences. Under reasonable assumptions it is shown analytically that the condition number of the aggregate velocity equations from optical flow is related in a natural way to the curvature of the image level set at the point of velocity estimation. The provides a link with affine invariant image processing and opens the door to curvature based chaining methods for estimating the flow velocity of moving targets. Numberical examples are presented illustrating the advantages of this approach over competing methods.
Filtering a sequence of images for target detection is discussed using multiresolution wavelet techniques and nonlinear size limited band filters based on partial differential operators associated with curvature of image level sets. In the latter procedure, evolution of the image under the differential operator is analogous to morphological erosion using structure sets, while backwards time evolution corresponds to dilation. The composition of these operations is subtracted from the original image to give the size limited `tophat' operator. Application of this procedure is demonstrated using synthesized targets embedded in actual infrared camera data.
Optical flow is an estimate of the velocity field based on the change of intensity patterns in successive images, and is an important quantity in computational vision for dense images. Because of the aperture problem optical flow computations can be ill-posed. This problem is compounded by derivative estimation errors. This paper presents an aggregate velocity scheme that uses iterative velocity refinement along object edge contours obtained via the Mallat- Zhong-Hwang wavelet and chaining algorithms. By working with edge information and aggregate velocities we avoid the aperture problem; iterative refinement compensates for errors in the derivative estimation. Our approach assigns a common velocity to the edge points of an image. When combined with a constant brightness assumption this yields an overdetermined set of linear equations. Since the data vector and matrix coefficients of this linear system consist of temporal and spatial derivative estimates, respectively, and both are subject to errors, the overdetermined system is solved using a total least squares approach. The resulting velocity estimate is then subtracted from the image sequence and the velocity estimation procedure is repeated for the new image sequence. This approach is very fast and accurate for images that have nearly the same edge velocity vectors as is usually the case for distant objects. A convergence analysis is given for the special case of 1D convected flow and it is shown that spatial and/or temporal smoothing enhances the convergence.
Daubechies wavelets are applied to the fixed pattern noise measured by a 336-by-165 Honeywell uncooled microbolometer infrared sensor and by an Amber AE 4128 128-by-128 indium antimonide (InSb) staring array. The main hypothesis presented in this paper is that Daubechies wavelets are the proper filter for the noise, because they decorrelate the temporal pixel responses on both arrays. These results are an experimental verification of the basic wavelet research of Flandrin with significant additions by Tewfik and Kim.
Results are presented of a numerical survey of optical flow algorithms for tracking problems associated with infrared imaging in high-speed missiles. The algorithms tested include those associated with normal flow subject to global smoothness constraints, edge detection via zero crossings of the image after convolution with spatiotemporal filters, and windowed matching techniques. The tracking problems considered in this survey fall into two classes: acquisition, and tracking after acquisition. These classes can be further divided into near and far range problems, characterized by extended and point target images. Other parameters of interest model allowable target and sensors motion and the amount of background clutter. Each of the above methods for determining optical flow can be used in conjunction with a variety of image preprocessing techniques such as kernel smoothing (especially by Gaussian kernels). and evolution under affine invariant partial differential operators. These preprocessing methods can also by used in combination with other approaches such as temporal layering in which successive image are combined to produce images with streaks whose edges are predominantly parallel to the optical flow.
In this paper a review of some significant recent theoretical connections between fractional Brownian motion, wavelets, and a low-frequency spectrum 1/f-type noise of the form (omega) -(alpha ) 1 <EQ (alpha) <EQ 2 is presented. Fractional Brownian motion is a parsimonious model (it depends on two parameters) that links the covariance of the sample path of a random signal with its power spectrum. The wavelet transform of fractional Brownian motion has a correlation function and spectral distribution that is known. The applicability of the theory is illustrated using data from an Amber focal plane array by showing that the wavelet transform can decorrelate a 1/f-type fixed pattern noise spectrum in a predictable fashion.
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