Matthew Fickus

Air Force Institute of Technology, Patterson, California, United States

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Publications (64)82.39 Total impact

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    ABSTRACT: Hyperspectral data is commonly used by astronomers to discern the chemical composition of stars. Unfortunately, conventional hyperspectral platforms require long exposure times, which can hamper their use in applications like celestial navigation. We propose a compressed sensing platform that exploits the spatial sparsity of stars to quickly sample the hyperspectral data. We leverage certain combinatorial designs to devise coded apertures, and then we apply block orthogonal matching pursuit to quickly reconstruct the desired imagery.
    IEEE Signal Processing Letters 11/2015; 22(11):1829-1833. DOI:10.1109/LSP.2015.2433837 · 1.64 Impact Factor
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    Matthew Fickus · Dustin G. Mixon
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    ABSTRACT: A Grassmannian frame is a collection of unit vectors which are optimally incoherent. The most accessible (and perhaps most beautiful) of Grassmannian frames are equiangular tight frames (ETFs); indeed, there are infinite families of known ETFs, whereas only finitely many non-ETF Grassmannian frames are known to date. This paper surveys every known construction of ETFs and tabulates existence for sufficiently small dimensions.
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    Matthew Fickus · Justin D. Marks · Miriam J. Poteet
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    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.
    Applied and Computational Harmonic Analysis 08/2014; DOI:10.1016/j.acha.2015.03.004 · 3.00 Impact Factor
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    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. Background: H&E staining is the gold standard in clinical histology; it is cheap and universally used, producing a vast number of histopathological samples. While pathologists accurately and consistently identify tissues and their pathologies, it is a time-consuming and expensive task, establishing the need for automated algorithms for improved throughput and robustness. Methods: We use an iterative feedback process to design a histopathology vocabulary (HV), a concise set of features that mimic the visual cues used by pathologists, e.g. “cytoplasm color” or “nucleus density”. These features are based in histology and understood by both pathologists and engineers. We compare our HV to several generic texture-feature sets in a pixel-level classification algorithm. Results: Results on delineating and identifying tissues in teratoma tumor samples validate our expert knowledge-based approach. Conclusions: The HV can be an effective tool for identifying and delineating teratoma components from images of H&E-stained tissue samples.
    06/2014; 5:19. DOI:10.4103/2153-3539.135606
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    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.
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    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.
    IEEE Transactions on Image Processing 05/2014; 23(5):2033-46. DOI:10.1109/TIP.2014.2307475 · 3.11 Impact Factor
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    Lindsay N. Smith · Matthew Fickus
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    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; 40(1). DOI:10.1007/s10444-013-9296-1 · 1.56 Impact Factor
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    Matthew Fickus · Dustin G. Mixon · Aaron A. Nelson · Yang Wang
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    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. DOI:10.1016/j.laa.2014.02.011 · 0.98 Impact Factor
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    John Jasper · Dustin G. Mixon · Matthew Fickus
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    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; 60(1). DOI:10.1109/TIT.2013.2285565 · 2.65 Impact Factor
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    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.
    Applied and Computational Harmonic Analysis 05/2013; 34(3):469-487. DOI:10.1016/j.acha.2012.07.005 · 3.00 Impact Factor
  • D.G. Mixon · C.J. Quinn · Negar Kiyavash · Matthew Fickus
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    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 03/2013; 59(3):1855-1865. DOI:10.1109/TIT.2012.2229781 · 2.65 Impact Factor
  • Matthew Fickus · Dustin G. Mixon · Miriam J. Poteet
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    ABSTRACT: Broadly speaking, frame theory is the study of how to produce well-conditioned frame operators, often subject to nonlinear application-motivated restrictions on the frame vectors themselves. In this chapter, we focus on one particularly well-studied type of restriction, i.e., having frame vectors of prescribed lengths. We discuss two methods for iteratively constructing such frames. The first method, called spectral tetris, produces special examples of such frames, and only works in certain cases. The second method combines the idea behind spectral tetris with the classical theory of majorization; this method can build any such frame in terms of a sequence of interlacing spectra, called eigensteps.
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    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.
    SIAM Journal on Imaging Sciences 10/2012; 7(1). DOI:10.1137/12089939X · 2.87 Impact Factor
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    Matthew Fickus · John Jasper · Dustin G. Mixon · Jesse Peterson
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    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.
    Linear Algebra and its Applications 09/2012; 479. DOI:10.1016/j.laa.2015.04.004 · 0.98 Impact Factor
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    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.
    Applied and Computational Harmonic Analysis 09/2012; 33(2):300-308. DOI:10.1016/j.acha.2012.03.008 · 3.00 Impact Factor
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    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.
    Proceedings / ICIP ... International Conference on Image Processing 09/2012; 2012:2809-2812. DOI:10.1109/ICIP.2012.6467483
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    Matthew Fickus · Dustin G. Mixon
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    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). DOI:10.1016/j.laa.2012.04.034 · 0.98 Impact Factor
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    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.
    Journal of Fourier Analysis and Applications 02/2012; 19(6). DOI:10.1007/s00041-013-9293-2 · 1.08 Impact Factor
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    Matthew Fickus · Dustin G. Mixon
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    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 02/2012; 33. DOI:10.1080/01630563.2012.682130 · 0.54 Impact Factor
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    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

Publication Stats

759 Citations
82.39 Total Impact Points

Institutions

  • 2005–2015
    • Air Force Institute of Technology
      Patterson, California, United States
  • 2009–2012
    • Wright-Patterson Air Force Base
      Dayton, Ohio, United States
  • 2008–2010
    • Carnegie Mellon University
      • Department of Biomedical Engineering
      Pittsburgh, Pennsylvania, United States
  • 2003
    • Cornell University
      • Department of Mathematics
      Ithaca, New York, United States