[Show abstract][Hide abstract] ABSTRACT: The Schur-Horn theorem is a classical result in matrix analysis which
characterizes the existence of positive semidefinite matrices with a given
diagonal and spectrum. In recent years, this theorem has been used to
characterize the existence of finite frames whose elements have given lengths
and whose frame operator has a given spectrum. We provide a new generalization
of the Schur-Horn theorem which characterizes the spectra of all possible
finite frame completions. That is, we characterize the spectra of the frame
operators of the finite frames obtained by adding new vectors of given lengths
to an existing frame. We then exploit this characterization to give a new and
simple algorithm for computing the optimal such completion.
[Show abstract][Hide abstract] ABSTRACT: The restricted isometry property (RIP) is an important matrix condition in
compressed sensing, but the best matrix constructions to date use randomness.
This paper leverages pseudorandom properties of the Legendre symbol to reduce
the number of random bits in an RIP matrix with Bernoulli entries. In this
regard, the Legendre symbol is not special---our main result naturally
generalizes to any small-bias sample space. We also conjecture that no random
bits are necessary for our Legendre symbol--based construction.
[Show abstract][Hide abstract] ABSTRACT: We propose a new mathematical and algorithmic framework for unsupervised image segmentation, which is a critical step in a wide variety of image processing applications. We have found that most existing segmentation methods are not successful on histopathology images, which prompted us to investigate segmentation of a broader class of images, namely those without clear edges between the regions to be segmented. We model these images as occlusions of random images, which we call textures, and show that local histograms are a useful tool for segmenting them. Based on our theoretical results, we describe a flexible segmentation framework that draws on existing work on nonnegative matrix factorization and image deconvolution. Results on synthetic texture mosaics and real histology images show the promise of the method.
[Show abstract][Hide abstract] ABSTRACT: In certain real-world applications, one needs to estimate the angular frequency of a spinning object. We consider the image processing problem of estimating this rate of rotation from a video of the object taken by a camera aligned with the axis of rotation. For many types of spinning objects, this problem can be addressed with existing techniques: simply register two consecutive video frames. We focus, however, on objects whose shape and intensity changes greatly from frame to frame, such as spinning plumes of plasma that emerge from a certain type of spacecraft thruster. To estimate the angular frequency of such objects, we introduce the Geometric Sum Transform (GST), a new rotation-based generalization of the discrete Fourier transform (DFT). Taking the GST of a given video produces a new sequence of images, the most coherent of which corresponds to the object’s true rate of rotation. After formally demonstrating this fact, we provide a fast algorithm for computing the GST which generalizes the decimation-in-frequency approach for performing a Fast Fourier Transform (FFT). We further show that computing a GST is, in fact, mathematically equivalent to computing a system of DFTs, provided one can decompose each video frame in terms of an eigenbasis of a rotation operator. We conclude with numerical experimentation.
Advances in Computational Mathematics 02/2014; · 1.47 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We propose a methodology for the design of features mimicking the visual cues used by pathologists when identifying tissues in hematoxylin and eosin (H&E)-stained samples.
[Show abstract][Hide abstract] ABSTRACT: In many applications, signals are measured according to a linear process, but
the phases of these measurements are often unreliable or not available. To
reconstruct the signal, one must perform a process known as phase retrieval.
This paper focuses on completely determining signals with as few intensity
measurements as possible, and on efficient phase retrieval algorithms from such
measurements. For the case of complex M-dimensional signals, we construct a
measurement ensemble of size 4M-4 which yields injective intensity
measurements; this is conjectured to be the smallest such ensemble. For the
case of real signals, we devise a theory of "almost" injective intensity
measurements, and we characterize such ensembles. Later, we show that phase
retrieval from M+1 almost injective intensity measurements is NP-hard,
indicating that computationally efficient phase retrieval must come at the
price of measurement redundancy.
Linear Algebra and its Applications 07/2013; 449. · 0.97 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: An equiangular tight frame (ETF) is a set of unit vectors in a Euclidean
space whose coherence is as small as possible, equaling the Welch bound. Also
known as Welch-bound-equality sequences, such frames arise in various
applications, such as waveform design and compressed sensing. At the moment,
there are only two known flexible methods for constructing ETFs: harmonic ETFs
are formed by carefully extracting rows from a discrete Fourier transform;
Steiner ETFs arise from a tensor-like combination of a combinatorial design and
a regular simplex. These two classes seem very different: the vectors in
harmonic ETFs have constant amplitude, whereas Steiner ETFs are extremely
sparse. We show that they are actually intimately connected: a large class of
Steiner ETFs can be unitarily transformed into constant-amplitude frames,
dubbed Kirkman ETFs. Moreover, we show that an important class of harmonic ETFs
is a subset of an important class of Kirkman ETFs. This connection informs the
discussion of both types of frames: some Steiner ETFs can be transformed into
constant-amplitude waveforms making them more useful in waveform design; some
harmonic ETFs have low spark, making them less desirable for compressed
sensing. We conclude by showing that real-valued constant-amplitude ETFs are
equivalent to binary codes that achieve the Grey-Rankin bound, and then
construct such codes using Kirkman ETFs.
IEEE Transactions on Information Theory 06/2013; · 2.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The local histogram transform of an image is a data cube that consists of the histograms of the pixel values that lie within a fixed neighborhood of any given pixel location. Such transforms are useful in image processing applications such as classification and segmentation, especially when dealing with textures that can be distinguished by the distributions of their pixel intensities and colors. We, in particular, use them to identify and delineate biological tissues found in histology images obtained via digital microscopy. In this paper, we introduce a mathematical formalism that rigorously justifies the use of local histograms for such purposes. We begin by discussing how local histograms can be computed as systems of convolutions. We then introduce probabilistic image models that can emulate textures one routinely encounters in histology images. These models are rooted in the concept of image occlusion. A simple model may, for example, generate textures by randomly speckling opaque blobs of one color on top of blobs of another. Under certain conditions, we show that, on average, the local histograms of such model-generated-textures are convex combinations of more basic distributions. We further provide several methods for creating models that meet these conditions; the textures generated by some of these models resemble those found in histology images. Taken together, these results suggest that histology textures can be analyzed by decomposing their local histograms into more basic components. We conclude with a proof-of-concept segmentation-and-classification algorithm based on these ideas, supported by numerical experimentation.
[Show abstract][Hide abstract] ABSTRACT: Digital fingerprinting is a framework for marking media files, such as images, music, or movies, with user-specific signatures to deter illegal distribution. Multiple users can collude to produce a forgery that can potentially overcome a fingerprinting system. This paper proposes an equiangular tight frame fingerprint design which is robust to such collusion attacks. We motivate this design by considering digital fingerprinting in terms of compressed sensing. The attack is modeled as linear averaging of multiple marked copies before adding a Gaussian noise vector. The content owner can then determine guilt by exploiting correlation between each user's fingerprint and the forged copy. The worst case error probability of this detection scheme is analyzed and bounded. Simulation results demonstrate that the average-case performance is similar to the performance of orthogonal and simplex fingerprint designs, while accommodating several times as many users.
IEEE Transactions on Information Theory 01/2013; 59(3):1855-1865. · 2.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In many areas of imaging science, it is difficult to measure the phase of
linear measurements. As such, one often wishes to reconstruct a signal from
intensity measurements, that is, perform phase retrieval. In this paper, we
provide a novel measurement design which is inspired by interferometry and
exploits certain properties of expander graphs. We also give an efficient phase
retrieval procedure, and use recent results in spectral graph theory to produce
a stable performance guarantee which rivals the guarantee for PhaseLift in
[Candes et al. 2011]. We use numerical simulations to illustrate the
performance of our phase retrieval procedure, and we compare reconstruction
error and runtime with a common alternating-projections-type procedure.
[Show abstract][Hide abstract] ABSTRACT: In the field of compressed sensing, a key problem remains open: to explicitly
construct matrices with the restricted isometry property (RIP) whose
performance rivals those generated using random matrix theory. In short, RIP
involves estimating the singular values of a combinatorially large number of
submatrices, seemingly requiring an enormous amount of computation in even
low-dimensional examples. In this paper, we consider a similar problem
involving submatrix singular value estimation, namely the problem of explicitly
constructing numerically erasure robust frames (NERFs). Such frames are the
latest invention in a long line of research concerning the design of linear
encoders that are robust against data loss. We begin by focusing on a subtle
difference between the definition of a NERF and that of an RIP matrix, one that
allows us to introduce a new computational trick for quickly estimating NERF
bounds. In short, we estimate these bounds by evaluating the frame analysis
operator at every point of an epsilon-net for the unit sphere. We then borrow
ideas from the theory of group frames to construct explicit frames and
epsilon-nets with such high degrees of symmetry that the requisite number of
operator evaluations is greatly reduced. We conclude with numerical results,
using these new ideas to quickly produce decent estimates of NERF bounds which
would otherwise take an eternity. Though the more important RIP problem remains
open, this work nevertheless demonstrates the feasibility of exploiting
symmetry to greatly reduce the computational burden of similar combinatorial
linear algebra problems.
[Show abstract][Hide abstract] ABSTRACT: In this paper we provide rigorous proof for the convergence of an iterative voting-based image segmentation algorithm called Active Masks. Active Masks (AM) was proposed to solve the challenging task of delineating punctate patterns of cells from fluorescence microscope images. Each iteration of AM consists of a linear convolution composed with a nonlinear thresholding; what makes this process special in our case is the presence of additive terms whose role is to "skew" the voting when prior information is available. In real-world implementation, the AM algorithm always converges to a fixed point. We study the behavior of AM rigorously and present a proof of this convergence. The key idea is to formulate AM as a generalized (parallel) majority cellular automaton, adapting proof techniques from discrete dynamical systems.
[Show abstract][Hide abstract] ABSTRACT: Given a channel with additive noise and adversarial erasures, the task is to
design a frame that allows for stable signal reconstruction from transmitted
frame coefficients. To meet these specifications, we introduce numerically
erasure-robust frames. We first consider a variety of constructions, including
random frames, equiangular tight frames and group frames. Later, we show that
arbitrarily large erasure rates necessarily induce numerical instability in
signal reconstruction. We conclude with a few observations, including some
implications for maximal equiangular tight frames and sparse frames.
Linear Algebra and its Applications. 02/2012; 437(6).
[Show abstract][Hide abstract] ABSTRACT: The restricted isometry property (RIP) is a well-known matrix condition that
provides state-of-the-art reconstruction guarantees for compressed sensing.
While random matrices are known to satisfy this property with high probability,
deterministic constructions have found less success. In this paper, we consider
various techniques for demonstrating RIP deterministically, some popular and
some novel, and we evaluate their performance. In evaluating some techniques,
we apply random matrix theory and inadvertently find a simple alternative proof
that certain random matrices are RIP. Later, we propose a particular class of
matrices as candidates for being RIP, namely, equiangular tight frames (ETFs).
Using the known correspondence between real ETFs and strongly regular graphs,
we investigate certain combinatorial implications of a real ETF being RIP.
Specifically, we give probabilistic intuition for a new bound on the clique
number of Paley graphs of prime order, and we conjecture that the corresponding
ETFs are RIP in a manner similar to random matrices.
[Show abstract][Hide abstract] ABSTRACT: Given a parametrized family of finite frames, we consider the optimization
problem of finding the member of this family whose coefficient space most
closely contains a given data vector. This nonlinear least squares problem
arises naturally in the context of a certain type of radar system. We derive
analytic expressions for the first and second partial derivatives of the
objective function in question, permitting this optimization problem to be
efficiently solved using Newton's method. We also consider how sensitive the
location of this minimizer is to noise in the data vector. We further provide
conditions under which one should expect the minimizer of this objective
function to be unique. We conclude by discussing a related
variational-calculus-based approach for solving this frame optimization problem
over an interval of time.
Numerical Functional Analysis and Optimization - NUMER FUNC ANAL OPTIMIZ. 02/2012; 33.
[Show abstract][Hide abstract] ABSTRACT: Finite unit norm tight frames provide Parseval-like decompositions of vectors in terms of redundant components of equal weight. They are known to be robust against additive noise and erasures, and as such, have great potential as encoding schemes. Unfortunately, up to this point, these frames have proven notoriously difficult to construct. Indeed, though the set of all unit norm tight frames, modulo rotations, is known to contain manifolds of nontrivial dimension, we have but a small finite number of known constructions of such frames. In this paper, we present a new iterative algorithm—gradient descent of the frame potential—for increasing the degree of tightness of any finite unit norm frame. The algorithm itself is easy to implement, and it preserves certain group structures present in the initial frame. In the special case where the number of frame elements is relatively prime to the dimension of the underlying space, we show that this algorithm converges to a unit norm tight frame at a linear rate, provided the initial unit norm frame is already sufficiently close to being tight. By slightly modifying this approach, we get a similar, but weaker, result in the non-relatively-prime case, providing an explicit answer to the Paulsen problem: “How close is a frame which is almost tight and almost unit norm to some unit norm tight frame?”
Applied and Computational Harmonic Analysis. 01/2012;
[Show abstract][Hide abstract] ABSTRACT: We present a method for identifying colitis in colon biopsies as an extension of our framework for the automated identification of tissues in histology images. Histology is a critical tool in both clinical and research applications, yet even mundane histological analysis, such as the screening of colon biopsies, must be carried out by highly-trained pathologists at a high cost per hour, indicating a niche for potential automation. To this end, we build upon our previous work by extending the histopathology vocabulary (a set of features based on visual cues used by pathologists) with new features driven by the colitis application. We use the multiple-instance learning framework to allow our pixel-level classifier to learn from image-level training labels. The new system achieves accuracy comparable to state-of-the-art biological image classifiers with fewer and more intuitive features.
Image Processing (ICIP), 2012 19th IEEE International Conference on; 01/2012
[Show abstract][Hide abstract] ABSTRACT: Digital fingerprinting is a framework for marking media files, such as
images, music, or movies, with user-specific signatures to deter illegal
distribution. Multiple users can collude to produce a forgery that can
potentially overcome a fingerprinting system. This paper proposes an
equiangular tight frame fingerprint design which is robust to such collusion
attacks. We motivate this design by considering digital fingerprinting in terms
of compressed sensing. The attack is modeled as linear averaging of multiple
marked copies before adding a Gaussian noise vector. The content owner can then
determine guilt by exploiting correlation between each user's fingerprint and
the forged copy. The worst-case error probability of this detection scheme is
analyzed and bounded. Simulation results demonstrate the average-case
performance is similar to the performance of orthogonal and simplex fingerprint
designs, while accommodating several times as many users.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: 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.