Conditions For Exact Reconstruction Research Articles

A Hybrid FDK (H-FDK) Algorithm for Circular Cone-beam CT that Combines Variable Filtering and Conjugate Rays

Circular cone-beam computed tomography (CT) is a popular configuration in industrial non-destructive testing. Corresponding to the circular cone-beam CT, the Feldkamp, Davis, and Kress (FDK) algorithm is the most popular reconstruction method. However, in the case of large cone angle scanning, the slice artifacts of the reconstruction using the FDK algorithm are visible because the data collected with the circular orbit scan mode do not satisfy the completeness conditions for exact reconstruction. To solve the problem, this study proposed a hybrid FDK (H-FDK) algorithm using a method that combines variable filtering and conjugate rays. First, the shape and curvature of the detector were changed for the filtering paths to be variable. Then, the conjugate weighting function for backprojection was constructed. Subsequently, optimization experiments were conducted to find optimal parameters for the detector and conjugate weighting function. Finally, the comparative experimental results demonstrated that the slices reconstructed by the H-FDK algorithm produced a discernible quality improvement.

Minimization of the [formula omitted]-ratio sparsity with [formula omitted] for signal recovery

In this paper, we propose a general scale invariant approach for sparse signal recovery via the minimization of the q-ratio sparsity sq(z)=(∥z∥1/∥z∥q)qq−1 with q∈[0,∞]. The properties of the q-ratio sparsity measure are studied and illustrated with examples. For the proposed q-ratio sparsity minimization problem with 1<q≤∞, we establish a verifiable exact reconstruction condition and derive its concise error bounds in terms of q-ratio constrained minimal singular values (CMSV). From an algorithmic point of view, we recognize that the proposed problem belongs to the nonlinear fractional programming and investigate two kinds of methods for solving it including the parametric methods and the change of variable method. Numerical experiments are conducted to demonstrate the advantageous performance of the proposed approaches over the state-of-the-art sparse recovery methods.

Exact reconstruction condition for angle-limited computed tomography of chemiluminescence.

Computed tomography of chemiluminescence (CTC) is an effective technique for three-dimensional (3D) combustion diagnostics. It reconstructs the 3D concentrations of intermediate species or 3D images of flame topology by multiple chemiluminescence projections captured from different perspectives. In the previous studies of CTC systems, it was assumed that projections from arbitrary perspectives are available. However, for some practical applications, the range of view angles and the number of projections might be restricted due to the optical access limitation, greatly affecting the reconstruction quality. In this paper, the exact reconstruction condition for angle-limited computed tomography of chemiluminescence was studied based on Mojette transform theories, and it was demonstrated by numerical simulations and experiments. The studies indicate that the object tested within limited angles can be well reconstructed when the number of grids, the number of projections, and the sampling rate of projections satisfy the exact reconstruction condition. By increasing the sampling rate of projections, high-quality tomographic reconstruction can be achieved by a few projections in a small angle range. Although this technique is discussed under combustion diagnostics, it can also be used and adapted for other tomography methods.

High spatial resolution computed tomography of chemiluminescence with densely sampled parallel projections.

Computed tomography of chemiluminescence (CTC) is an effective tool for combustion diagnostics by using optical detectors to capture the projections of luminescence from multiple views and realizing the three-dimensional (3D) reconstruction by computed tomography (CT) theories. In the existing CTC, ordinary commodity lenses were employed in the system for imaging, the imaging effects complicate the projection model and the low sampling rate decreases the spatial resolution and reconstruction accuracy. In classical CT techniques, parallel projection based on 2D Radon transform is the simplest model, which has been widely used in CT applications. In this work, double telecentric lens is introduced in CTC to realize the acquisition of parallel projection with high sampling rate. Despite the parallel projection CT theories have been well studied, there are still a few theoretical and technological drawbacks need to be solved when utilizing double telecentric lens in CTC. Firstly, a simple method based on bilinear interpolation is studied to improve the calculation accuracy of the weight matrix. Secondly, the exact reconstruction condition for parallel projections is studied based on the discrete Radon transform theory. It establishes the theoretical relationship of the reconstruction quality and the sampling rate of the projections, the number of views, the range and the spatial resolution of the reconstructed region. In experiment, camera calibration technique for double telecentric lens is studied, and the results of which are used for projection correction. The tomographic reconstructions of an axisymmetric flame demonstrate the feasibility and accuracy of the studies.

Robustness of Sparse Recovery via &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$F$ &lt;/tex-math&gt;&lt;/inline-formula&gt;-Minimization: A Topological Viewpoint

A recent trend in compressed sensing is to consider non-convex optimization techniques for sparse recovery. The important case of $F$-minimization has become of particular interest, for which the exact reconstruction condition (ERC) in the noiseless setting can be precisely characterized by the null space property (NSP). However, little work has been done concerning its robust reconstruction condition (RRC) in the noisy setting. We look at the null space of the measurement matrix as a point on the Grassmann manifold, and then study the relation between the ERC and RRC sets, denoted as $\Omega_J$ and $\Omega_J^r$, respectively. It is shown that $\Omega_J^r$ is the interior of $\Omega_J$, from which a previous result of the equivalence of ERC and RRC for $\ell_p$-minimization follows easily as a special case. Moreover, when $F$ is non-decreasing, it is shown that $\overline{\Omega}_J\setminus\interior(\Omega_J)$ is a set of measure zero and of the first category. As a consequence, the probabilities of ERC and RRC are the same if the measurement matrix $\mathbf{A}$ is randomly generated according to a continuous distribution. Quantitatively, if the null space $\mathcal{N}(\bf A)$ lies in the "$d$-interior" of $\Omega_J$, then RRC will be satisfied with the robustness constant $C=\frac{2+2d}{d\sigma_{\min}(\mathbf{A}^{\top})}$; and conversely if RRC holds with $C=\frac{2-2d}{d\sigma_{\max}(\mathbf{A}^{\top})}$, then $\mathcal{N}(\bf A)$ must lie in $d$-interior of $\Omega_J$. We also present several rules for comparing the performances of different cost functions. Finally, these results are capitalized to derive achievable tradeoffs between the measurement rate and robustness with the aid of Gordon's escape through the mesh theorem or a connection between NSP and the restricted eigenvalue condition.

Non‐minimum phase switched systems: HOSM‐based fault detection and fault identification via Volterra integral equation

SUMMARYIn this paper, the problem of continuous and discrete state estimation for a class of linear switched systems with additive faults is studied. The class of systems under study can contain non‐minimum phase zeroes in some of their ‘operating modes’. The conditions for exact reconstruction of the discrete state are given using structural properties of the switched system. The state space is decomposed into the strongly observable part, the non‐strongly observable part, and the unobservable part, to analyze the effect of the unknown inputs. State observers based on high‐order sliding mode to exactly estimate the strongly observable part and Luenberger‐like observers to estimate the remaining parts are proposed. For the case when the exact estimation of the state cannot be achieved, the ultimate bounds on the estimation errors are provided. The proposed strategy includes a high‐order sliding‐mode‐based fault detection and a fault identification scheme via the solution of a Volterra integral equation. The feasibility of the proposed method is illustrated by simulations. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Two-Band Hybrid FIR–IIR Filters for Image Compression

Two-band analysis-synthesis filters or wavelet filters are used pervasively for compressing natural images. Both FIR and IIR filters have been studied in this context, the former being the most popular. In this paper, we examine the compression performance of these two-band filters in a dyadic wavelet decomposition and attempt to isolate features that contribute most directly to the performance gain. Then, employing the general exact reconstruction condition, hybrid FIR-IIR analysis-synthesis filters are designed to maximize compression performance for natural images. Experimental results are presented that compare performance with the popular biorthogonal filters in terms of peak SNR, subjective quality, and computational complexity.

Modified-CS: Modifying Compressive Sensing for Problems With Partially Known Support

We study the problem of reconstructing a sparse signal from a limited number of its linear projections when a part of its support is known, although the known part may contain some errors. The “known” part of the support, denoted T, may be available from prior knowledge. Alternatively, in a problem of recursively reconstructing time sequences of sparse spatial signals, one may use the support estimate from the previous time instant as the “known” part. The idea of our proposed solution (modified-CS) is to solve a convex relaxation of the following problem: find the signal that satisfies the data constraint and is sparsest outside of T. We obtain sufficient conditions for exact reconstruction using modified-CS. These are much weaker than those needed for compressive sensing (CS) when the sizes of the unknown part of the support and of errors in the known part are small compared to the support size. An important extension called regularized modified-CS (RegModCS) is developed which also uses prior signal estimate knowledge. Simulation comparisons for both sparse and compressible signals are shown.

A Feldkamp-type approximate algorithm for helical multislice CT using extended scanning helix

In this paper, a Feldkamp-type approximate algorithm is proposed for helical multislice Computed Tomography (CT) image reconstruction. For reconstruction of practically required planar transversal image slices, the algorithm proposes a new short scan helical trajectory which satisfies Tuy’s exact reconstruction condition. A planar slice reconstructed using this new trajectory has potential to be exactly reconstructed. This method improves the approximate helical reconstruction in terms of artifacts reduction and efficiency enhancement.

Exact Image Super-Resolution for Pure Translational Motion and Shift-Invariant Blur

In this work, a special case of image super-resolution problem where the only type of motion is global translational motion and the blurs are shift-invariant is investigated. The necessary conditions for exact reconstruction of the original image by using finite impulse-response reconstruction filters are developed. If the number of low-resolution images is larger than a threshold and the blur functions meet a certain property, then a reconstruction filter set for exact super-resolution can be generated. If the conditions are not fully satisfied, then the high-resolution image is reconstructed with some error.

Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals

We examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples. This sampling scheme, which has been studied previously, has an interesting optimality property that uniform sampling lacks: one can sample and reconstruct the class /spl Bscr/(/spl Fscr/) of multiband signals with spectral support /spl Fscr/, at rates arbitrarily close to the Landau (1969) minimum rate equal to the Lebesgue measure of /spl Fscr/, even when /spl Fscr/ does not tile R under translation. Using the conditions for exact reconstruction, we derive an explicit reconstruction formula. We compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band-limited to the "span of /spl Fscr/" (the smallest interval containing /spl Fscr/) which is a bigger class than the valid signals /spl Bscr/(/spl Fscr/), band-limited to /spl Fscr/. We also examine the performance of the reconstruction system when the input contains additive sample noise.

Theory and design of uniform DFT, parallel, quadrature mirror filter banks

In this paper, the theory of uniform DFT, parallel, quadrature mirror filter (QMF) banks is developed. The QMF equations, i.e., equations that need to be satisfied for exact reconstruction of the input signal, are derived. The concept of decimated filters is introduced, and structures for both analysis and synthesis banks are derived using this concept. The QMF equations, as well as closed-form expressions for the synthesis filters needed for exact reconstruction of the input signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(n)</tex> , are also derived using this concept. In general, the reconstructed. signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{x}(n)</tex> suffers from three errors: aliasing, amplitude distortion, and phase distortion. Conditions for exact reconstruction (i.e., all three distortions are zero, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\hat{x}(n)</tex> is equal to a delayed version of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(n))</tex> of the input signal are derived in terms of the decimated filters. Aliasing distortion can always be completely canceled. Once aliasing is canceled, it is possible to completely eliminate amplitude distortion (if suitable IIR filters are employed) and completely eliminate phase distortion (if suitable FIR filters are employed). However, complete elimination of all three errors is possible only with some simple, pathalogical stable filter transfer functions. In general, once aliasing is canceled, the other distortions can be minimized rather than completely eliminated. Algorithms for this are presented. The properties of FIR filter banks are then investigated. Several aspects of IIR filter banks are also studied using the same framework.

Exact reconstruction techniques for tree-structured subband coders

In recent years, tree-structured analysis/reconstruction systems have been extensively studied for use in subband coders for speech. In such systems, it is imperative that the individual channel signals be decimated in such a way that the number of samples coded and transmitted do not exceed the number of samples in the original speech signal. Under this constraint, the systems presented in the past have sought to remove the aliasing distortion while minimizing the overall analysis/reconstruction distortion. In this paper, it is shown that it is possible to design tree-structured analysis/reconstruction systems which meet the sampling rate condition and which result in exact reconstruction of the input signal. The conditions for exact reconstruction are developed and presented. Furthermore, it is shown that these conditions are not overly restrictive and high-quality frequency division may be performed in the analysis section. A filter design procedure is presented which allows high-quality filters to be easily designed.