The general model which is used for the observed blurred image, g x , y of a scene, f x , y , is described by a convolution integral:.

## Deconvolution of Image and Spectra | Linear Subspace | Hilbert Space

Earlier work in image features invariant with respect to blur is divided into three different categories. First category belongs to deriving blur invariant properties of the PSF. The second one is to estimate the PSF, and the last is to obtain the original image via deblurring processes. A frequency domain approach for blur invariant has been done to develop some features for object recognition [ 14 ].

Blur invariant in moment domain is proposed by Flusser and Suk,which is invariant to convolution of an image with an arbitrary symmetric PSF kernel [ 15 ]. Two recognition methods for motion blurred images have been developed based on a relation between the moments of the blurred and the original images [ 16 ]. A set of invariants are derived from Zernike moments which are simultaneously invariant to similarity transformation and to convolution with circularly symmetric PSF [ 17 ]. Dai et al. Yan et al. Khan et al. In this method, the iris is segmented from the acquired image of an eye using an edge detection algorithm.

Wavelet-domain blur invariants are proposed for 2D discrete signals which are invariant to centrally symmetric blurs [ 21 ].

Honarvar et al. The adaptive total variation TV minimization technique by Yoon et al.

Accurate sparse-projection image reconstruction via nonlocal TV regularization is proposed by Zhang et al. In this paper, the blur invariant features in Fourier domain are proposed. By establishing a relationship between the normalized Fourier transform of the blurred and the original images, it is possible to estimate the original image.

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In this section, we establish a new frequency blur invariant based on a Gaussian PSF for degraded images. Since an imaging system can be modeled as a 2D convolution in 1 , it is possible to transform this equation to the Fourier domain. For frequency analysis, we consider the imaging system in the presence and absence of noise, respectively. In the presence of noise, the degradation model can be expressed in the Fourier domain as.

The Wiener deconvolution method has widespread use in image deconvolution applications, as the frequency spectrum of most visual images is fairly well behaved and may be estimated easily [ 25 , 26 ]. The filter is described in the frequency domain:. Using this technique to find the best reconstruction of a noisy image can be compared with other algorithms such as Gaussian filtering. If noise n x , y is neglected, 2 can be reduced to. It is clear that 5 is a separable function in terms of x and y , and we can rewrite that as follows:.

Substituting 9 into 4 , we get. To obtain the frequency domain blur invariant, we set both frequencies u , v to 1,1 in 10 which leads to. In this paper, these features are obtained by normalizing the Fourier transform of the original and blurred images with their respective Fourier transforms, F 1,1 and G 1,1. In this section, we show an image deconvolution method based on the derivatives of the blurred image function which are defined in terms of differences.

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We begin with 1D version of 10 because the separable property of the Gaussian distributions will allow the easy 2D implementation of them. To deconvolve the original signal, we rewrite 10 in 1D form as. If we use the Taylor series expansion of the squared exponential function, it is possible to connect the degraded signal to its original form.

### Noise free case

Equation 13 leads to. By using the high order derivative property of the Fourier transform differentiation property , we have. Using 15 in 14 and taking the inverse Fourier transform yields. Equation 16 includes the even order derivative of the degraded signal that can be defined as a difference. For example, the definition of a second-order derivative as the difference is [ 28 ]. By generalizing the definition of a high order derivative as differences, we are able to approximate the continuous derivatives with discrete differences as.

Sustituting 18 into 16 yields the original signal in terms of the degraded signal as.

Similarly, for a 2D blurred image, the desired image deconvolution can be obtained from. Different numerical experiments are conducted in order to prove the validity and the efficiency of the proposed methods. The detailed description of these numerical experiments will be presented in this section. The performance for the proposed methods is evaluated based on the binary, gray-scale, and real images.

### 1. Introduction

This section is divided into two subsections. In the first subsection, the accuracy of the proposed blur invariant features in the Fourier domain is validated by using the frequency analysis of the blurred and the original images. The efficiency of the proposed image deconvolution algorithm based on the Gaussian PSF is carried out with different experiments in the second subsection. Results of nine numerical experiments are used to ensure the efficiency of the proposed image deconvolution method. Visit Our Stores. This landmark volume was the first published on deconvolution to provide both an overview of the field, and practical methods and results.

In the twelve years since the first book was published, major advances have taken place. For example, researchers have refined projections onto convex sets, improved understanding of other relaxation methods, applied principles of neural networks, and extended the most effective nonlinear methods to image processing.

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The significant advances in the years since the First Edition have created the need for this Second Edition, which addresses both the most recent and effective nonlinear constrained methods, and their practical application to a variety scientific and engineering fields.

Deconvolution of Images and Spectra embraces all the advantages of its predecessor by conveying a clear understanding while providing a selection of effective and practical techniques. The authors assume only a working knowledge ofcalculus, and emphasize practical applications over topics of purely theoretical interest, focusing on areas that have been pivotal to the evolution of the most effective methods.

This tutorial is essentially self-contained; readers will find it to be easy to understand and practical.

Figures are clear and the text is very readable We are happy to have this book and recommend it to anyone involved in data analysis or signal processing which involves deconvolution. Deconvolution is a technique in signal or image processing that is applied when data is difficult to read due to spreading and blurring of corrupt images and experimental results.

This landmark volume was first published to provide both an overview of the field, and practical methods and results.

Default of imfilter is correlation. How does deconvblind deconvolve? Ask Question. Asked 4 years, 1 month ago. Active 4 years, 1 month ago. Viewed times. Please help me! I can't find any information on this on the MathWorks site at all. See: "Acceleration of iterative image restoration algorithms, by D. Thanks so much! Your note on symmetry helped a lot I guess I am still mostly confused as to why Matlab's imfilter uses correlation rather than convolution when their functions deconvolve by convolution.

You really don't see much difference in terms of the spatial version because the operations are the same. BTW, if I have helped, please consider accepting my answer. That can be done by clicking on the checkmark icon at the top of my post, to the left below the up and down voting arrows. Good luck! Sign up or log in Sign up using Google. Sign up using Facebook.