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Register a new userDetection Theory for Union of Subspaces
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Added by
Waheed Bajwa
Publication date
2017
fields
Electronic Engineering
and research's language is
English
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The focus of this paper is on detection theory for union of subspaces (UoS). To this end, generalized likelihood ratio tests (GLRTs) are presented for detection of signals conforming to the UoS model and detection of the corresponding active subspace. One of the main contributions of this paper is bounds on the performances of these GLRTs in terms of geometry of subspaces under various assumptions on the observation noise. The insights obtained through geometrical interpretation of the GLRTs are also validated through extensive numerical experiments on both synthetic and real-world data.
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Calcified plaque in the aorta and pelvic arteries is associated with coronary artery calcification and is a strong predictor of heart attack. Current calcified plaque detection models show poor generalizability to different domains (ie. pre-contrast vs. post-contrast CT scans). Many recent works have shown how cross domain object detection can be improved using an image translation model which translates between domains using a single shared latent space. However, while current image translation models do a good job preserving global/intermediate level structures they often have trouble preserving tiny structures. In medical imaging applications, preserving small structures is important since these structures can carry information which is highly relevant for disease diagnosis. Recent works on image reconstruction show that complex real-world images are better reconstructed using a union of subspaces approach. Since small image patches are used to train the image translation model, it makes sense to enforce that each patch be represented by a linear combination of subspaces which may correspond to the different parts of the body present in that patch. Motivated by this, we propose an image translation network using a shared union of subspaces constraint and show our approach preserves subtle structures (plaques) better than the conventional method. We further applied our method to a cross domain plaque detection task and show significant improvement compared to the state-of-the art method.
We consider the phase retrieval problem for signals that belong to a union of subspaces. We assume that amplitude measurements of the signal of length $n$ are observed after passing it through a random $m times n$ measurement matrix. We also assume that the signal belongs to the span of a single $d$-dimensional subspace out of $R$ subspaces, where $dll n$. We assume the knowledge of all possible subspaces, but the true subspace of the signal is unknown. We present an algorithm that jointly estimates the phase of the measurements and the subspace support of the signal. We discuss theoretical guarantees on the recovery of signals and present simulation results to demonstrate the empirical performance of our proposed algorithm. Our main result suggests that if properly initialized, then $O(d+log R)$ random measurements are sufficient for phase retrieval if the unknown signal belongs to the union of $R$ low-dimensional subspaces.
We introduce Xampling, a unified framework for signal acquisition and processing of signals in a union of subspaces. The main functions of this framework are two. Analog compression that narrows down the input bandwidth prior to sampling with commercial devices. A nonlinear algorithm then detects the input subspace prior to conventional signal processing. A representative union model of spectrally-sparse signals serves as a test-case to study these Xampling functions. We adopt three metrics for the choice of analog compression: robustness to model mismatch, required hardware accuracy and software complexities. We conduct a comprehensive comparison between two sub-Nyquist acquisition strategies for spectrally-sparse signals, the random demodulator and the modulated wideband converter (MWC), in terms of these metrics and draw operative conclusions regarding the choice of analog compression. We then address lowrate signal processing and develop an algorithm for that purpose that enables convenient signal processing at sub-Nyquist rates from samples obtained by the MWC. We conclude by showing that a variety of other sampling approaches for different union classes fit nicely into our framework.
Traditional sampling theories consider the problem of reconstructing an unknown signal $x$ from a series of samples. A prevalent assumption which often guarantees recovery from the given measurements is that $x$ lies in a known subspace. Recently, there has been growing interest in nonlinear but structured signal models, in which $x$ lies in a union of subspaces. In this paper we develop a general framework for robust and efficient recovery of such signals from a given set of samples. More specifically, we treat the case in which $x$ lies in a sum of $k$ subspaces, chosen from a larger set of $m$ possibilities. The samples are modelled as inner products with an arbitrary set of sampling functions. To derive an efficient and robust recovery algorithm, we show that our problem can be formulated as that of recovering a block-sparse vector whose non-zero elements appear in fixed blocks. We then propose a mixed $ell_2/ell_1$ program for block sparse recovery. Our main result is an equivalence condition under which the proposed convex algorithm is guaranteed to recover the original signal. This result relies on the notion of block restricted isometry property (RIP), which is a generalization of the standard RIP used extensively in the context of compressed sensing. Based on RIP we also prove stability of our approach in the presence of noise and modelling errors. A special case of our framework is that of recovering multiple measurement vectors (MMV) that share a joint sparsity pattern. Adapting our results to this context leads to new MMV recovery methods as well as equivalence conditions under which the entire set can be determined efficiently.
In this letter, we study a few properties of Complex Conjugate Pair Sums (CCPSs) and Complex Conjugate Subspaces (CCSs). Initially, we consider an LTI system whose impulse response is one period data of CCPS. For a given input x(n), we prove that the output of this system is equivalent to computing the first order derivative of x(n). Further, with some constraints on the impulse response, the system output is also equivalent to the second order derivative. With this, we show that a fine edge detection in an image can be achieved using CCPSs as impulse response over Ramanujan Sums (RSs). Later computation of projection for CCS is studied. Here the projection matrix has a circulant structure, which makes the computation of projections easier. Finally, we prove that CCS is shift-invariant and closed under the operation of circular cross-correlation.
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