Linear Shift Variant Research Articles

Deep-learning-augmented computational miniature mesoscope.

Fluorescence microscopy is essential to study biological structures and dynamics. However, existing systems suffer from a trade-off between field of view (FOV), resolution, and system complexity, and thus cannot fulfill the emerging need for miniaturized platforms providing micron-scale resolution across centimeter-scale FOVs. To overcome this challenge, we developed a computational miniature mesoscope (CM2) that exploits a computational imaging strategy to enable single-shot, 3D high-resolution imaging across a wide FOV in a miniaturized platform. Here, we present CM2 V2, which significantly advances both the hardware and computation. We complement the 3 × 3 microlens array with a hybrid emission filter that improves the imaging contrast by 5×, and design a 3D-printed free-form collimator for the LED illuminator that improves the excitation efficiency by 3×. To enable high-resolution reconstruction across a large volume, we develop an accurate and efficient 3D linear shift-variant (LSV) model to characterize spatially varying aberrations. We then train a multimodule deep learning model called CM2Net, using only the 3D-LSV simulator. We quantify the detection performance and localization accuracy of CM2Net to reconstruct fluorescent emitters under different conditions in simulation. We then show that CM2Net generalizes well to experiments and achieves accurate 3D reconstruction across a ~7-mm FOV and 800-μm depth, and provides ~6-μm lateral and ~25-μm axial resolution. This provides an ~8× better axial resolution and ~1400× faster speed compared to the previous model-based algorithm. We anticipate this simple, low-cost computational miniature imaging system will be useful for many large-scale 3D fluorescence imaging applications.

Efficient shift-variant image restoration using deformable filtering (Part I)

In this study, we propose using the least squares optimal deformable filtering approximation as an efficient tool for linear shift variant (SV) filtering, in the context of restoring SV-degraded images. Based on this technique we propose a new formalism for linear SV operators, from which an efficient way to implement the transposed SV-filtering is derived. We also provide a method for implementing an approximation of the regularized inversion of a SV-matrix, under the assumption of having smoothly spatially varying kernels, and enough regularization. Finally, we applied these techniques to implement a SV-version of a recent successful sparsity-based image deconvolution method. A high performance (high speed, high visual quality and low mean squared error, MSE) is demonstrated through several simulation experiments (one of them based on the Hubble telescope PSFs), by comparison to two state-of-the-art methods.

Iterative PSF Estimation and Its Application to Shift Invariant and Variant Blur Reduction

Among image restoration approaches, image deconvolution has been considered a powerful solution. In image deconvolution, a point spread function (PSF), which describes the blur of the image, needs to be determined. Therefore, in this paper, we propose an iterative PSF estimation algorithm which is able to estimate an accurate PSF. In real-world motion-blurred images, a simple parametric model of the PSF fails when a camera moves in an arbitrary direction with an inconsistent speed during an exposure time. Moreover, the PSF normally changes with spatial location. In order to accurately estimate the complex PSF of a real motion blurred image, we iteratively update the PSF by using a directional spreading operator. The directional spreading is applied to the PSF when it reduces the amount of the blur and the restoration artifacts. Then, to generalize the proposed technique to the linear shift variant (LSV) model, a piecewise invariant approach is adopted by the proposed image segmentation method. Experimental results show that the proposed method effectively estimates the PSF and restores the degraded images.

Approximating filtered scale-variant signals

We develop theorems that place limits on the point-wise approximation of the responses of filters, both linear shift invariant (LSI) and linear shift variant (LSV), to input signals and images that are LSV in the following sense: they can be expressed as the outputs of systems with LSV impulse responses, where the shift variance is with respect to the filter scale of a single-prototype fillter. The approximations take the form of LSI approximations to the responses. We develop tight bounds on the approximation errors expressed in terms of filter durations and derivative (Sobolev) norms. Finally, we find application of the developed theory to defoveation of images, deblurring of shift-variant blurs, and shift-variant edge detection.

One-dimensional linear shift- variant digital filtering using two-dimensional techniques

This correspondence investigates some problems associated with the implementation of one-dimensional (1-D) linear shift-variant (LSV) digital filters. By mapping 1-D input and output sequences into two-dimensional (2-D) sequences, we may characterize the input-output relationship of a 1-D LSV digital filter as a 2-D convolution. In this way, a 1-D LSV digital filter may be approximately implemented as a 2-D recursive difference equation. The advantages and disadvantages of this implementation technique are discussed.

On Linear Shift-Variant Digital Signal Processing

The present paper reviews recent developments in the area of linear shift-variant (LSV) digital filters. The discussion encompasses the problem of determining filter characteristics in both time and frequency domains. In addition, important properties of LSV digital filters are explored. Based on these properties, synthesis and implementation techniques of LSV digital filters are examined.

Frequency-domain considerations of LSV digital filters

The present paper develops a framework for the analysis and synthesis of linear shift-variant (LSV) digital filters in the frequency domain. First, LSV digital filters are theoretically modeled by the successive use of linear shift-invariant (LSI) filters. On the basis of the model, we present an interpretation of shift-variant spectral modification or filtering. Further, shift-variant digital filtering is discussed in relation to the notions of the short-time spectrum and the generalized frequency function. In addition, we propose an efficient implementation procedure which reduces the number of filter coefficients and the amount of computation. The effectiveness of LSV digital filters in processing time-varying signals is demonstrated by experimental verification.

Shift variant image degradation and restoration using SVD

Restoration of images degraded by a linear shift variant (LSV) point spread function has been considered. Since the well-developed techniques of Fourier analysis are not directly applicable a numerical solution of the Fredholm integral equation describing the degradation has been attempted. The integral equation is first digitized to get a set of algebraic equations. The degradation matrix is factored using singular value decomposition (SVD). Presence of errors in the degraded image (due to detector noise) sometimes makes the solution obtained by inversion of the degrading matrix almost impossible. A major contribution to the noise in the solution thus obtained is due to the components of the error vector along the coordinate axes corresponding to smaller singular values. The effect of these errors has been minimized by estimating the copponents of the image along these coordinate axes from known statistical properties of images.