Equi-chordal and equi-isoclinic tight fusion frames (ECTFFs and EITFFs) are both types of optimal packings of subspaces in Euclidean spaces. In the special case where these subspaces are one-dimensional, ECTFFs and EITFFs both correspond to types of optimal packings of lines known as equiangular tight frames. In this brief note, we review some of the fundamental ideas and results concerning ECTFFs and EITFFs.
An equiangular tight frame (ETF) is an M×N matrix which has orthogonal equal norm rows, equal norm columns, and the inner products of all pairs of columns have the same modulus. ETFs arise in numerous applications, including compressed sensing. They also seem to be rare: despite over a decade of active research by the community, only a few construction methods have been discovered. In this article we introduce a new construction of ETFs which uses a particular set of combinatorial designs called quasi-symmetric designs. For ETFs whose entries are contained in {+1;-1}, called real constant amplitude ETFs (RCAETFs), we see that this construction is reversible, giving new quasi-symmetric designs from the known constructions RCAETFs.
An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. They arise in numerous applications. It is well known that real ETFs are equivalent to a certain subclass of strongly regular graphs. In this note, we give some alternative techniques for understanding this equivalence. In a later document, we will use these techniques to further generalize this theory.
Chromotomography is a form of hyperspectral imaging that utilizes a spinning diffractive element to resolve a rapidly
evolving scene. The system captures both spatial dimensions and the spectral dimension at the same time. Advanced
algorithms take the recorded dispersed images and use them to construct the data cube in which each reconstructed
image is the recorded scene at a specific wavelength. A simulation tool has been developed which uses Zemax to
accurately trace rays through real or proposed optical systems. The simulation is used here to explore the limitations of
tomographic reconstruction in both idealized and aberrated imaging systems. Results of the study show the accuracy of
reconstructed images depends upon the content of the original target scene, the number of projections measured, and the
angle through which the prism is rotated. For cases studied here, 20 projections are sufficient to achieve image quality
99.51% of the max value. Reconstructed image quality degrades with aberrations, but no worse than equivalent
conventional imagers.
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.
We present the current state of our work on a mathematical framework for identification and delineation of
histopathology images-local histograms and occlusion models. Local histograms are histograms computed over
defined spatial neighborhoods whose purpose is to characterize an image locally. This unit of description is
augmented by our occlusion models that describe a methodology for image formation. In the context of this
image formation model, the power of local histograms with respect to appropriate families of images will be shown
through various proved statements about expected performance. We conclude by presenting a preliminary study
to demonstrate the power of the framework in the context of histopathology image classification tasks that, while
differing greatly in application, both originate from what is considered an appropriate class of images for this
framework.
The state of the art in compressed sensing uses sensing matrices which satisfy the restricted isometry property
(RIP). Unfortunately, the known deterministic RIP constructions fall short of the random constructions, which
are only valid with high probability. In this paper, we consider certain deterministic constructions and compare
different proof techniques that demonstrate RIP in the deterministic setting.
In information fusion, one is often confronted with the following problem: given a preexisting set of measurements
about an unknown quantity, what new measurements should one collect in order to accomplish a given fusion
task with optimal accuracy and efficiency. We illustrate just how difficult this problem can become by considering
one of its more simple forms: when the unknown quantity is a vector in a Hilbert space, the task itself is vector
reconstruction, and the measurements are linear functionals, that is, inner products of the unknown vector with
given measurement vectors. Such reconstruction problems are the subject of frame theory. Here, we can measure
the quality of a given frame by the average reconstruction error induced by noisy measurements; the mean square
error is known to be the trace of the inverse of the frame operator. We discuss preliminary results which help
indicate how to add new vectors to a given frame in order to reduce this mean square error as much as possible.
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.
We propose an active mask segmentation framework that combines the advantages of statistical modeling,
smoothing, speed and flexibility offered by the traditional methods of region-growing, multiscale, multiresolution
and active contours respectively. At the crux of this framework is a paradigm shift from evolving
contours in the continuous domain to evolving multiple masks in the discrete domain. Thus, the active
mask framework is particularly suited to segment digital images. We demonstrate the use of the framework
in practice through the segmentation of punctate patterns in fluorescence microscope images. Experiments
reveal that statistical modeling helps the multiple masks converge from a random initial configuration to
a meaningful one. This obviates the need for an involved initialization procedure germane to most of the
traditional methods used to segment fluorescence microscope images. While we provide the mathematical
details of the functions used to segment fluorescence microscope images, this is only an instantiation of the
active mask framework. We suggest some other instantiations of the framework to segment different types
of images.
In recent years, the focus in biological science has shifted to understanding complex systems at the cellular and molecular levels, a task greatly facilitated by fluorescence microscopy. Segmentation, a fundamental yet difficult problem, is often the first processing step following acquisition. We have previously demonstrated that a stochastic active contour based algorithm together with the concept of topology preservation (TPSTACS) successfully segments single cells from multicell images. In this paper we demonstrate that TPSTACS successfully segments images from other imaging modalities such as DIC microscopy, MRI and fMRI. While this method is a viable alternative to hand segmentation, it is not yet ready to be used for high-throughput applications due to its large run time. Thus, we highlight some of the benefits of combining TPSTACS with the multiresolution approach for the segmentation of fluorescence microscope images. Here we propose a multiscale active contour (MSAC)
transformation framework for developing a family of modular algorithms for the segmentation of fluorescence microscope images in particular, and biomedical images in general. While this framework retains the flexibility and the high quality of the segmentation provided by active contour-based algorithms, it offers a boost in the
efficiency as well as a framework to compute new features that further enhance the segmentation.
Multiscale moment-based transformations, such as the multiscale Harris corner point detector, have been used in
image processing applications for several years. Typically, such transforms are used to identify objects-of-interest
in a given image, which, in turn, facilitate target tracking and registration. Though these transforms are usually applied to digitally sampled images, many of their properties were previously only known to hold for images over continuous domains. We prove that many of these properties indeed generalize to images over discrete domains. In particular, after introducing a mathematically well-behaved method for rotating an image over the two-dimensional integer lattice, we show that this rotation commutes with the moment-based transform in the expected manner.
Chirps arise in many signal processing applications, and have been extensively studied, especially in the case where chirps are regarded as functions of the real-line or of the integers. However, less attention has been paid to study of chirps over finite cyclic groups. We discuss the basic properties of such chirps, including a way in which they may be used to construct finite tight frames.
We give a physical interpretation for finite tight frames along the lines of Columb's Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an N-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a fundamental inequality that all finite tight frames must satisfy.
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