the wave, by the hologram Fourier transform holograms in transparency in located in the back focal gram S forming VanderLugt is subject lens plane of a at the records which amplitudes tw[r]

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plate is a key clue to uncover over-speed vehicles or the ones involved in hit-and-run accidents. However, the snapshot of over-speed vehicle captured by surveillance camera is frequently blurred due to fast motion, which is even unrecognizable by human. Those observed plate images are usually in low resolution and suffer severe loss of edge information, which cast great challenge to existing blind de- blurring methods. For license plate **image** blurring caused by fast motion, the blur kernel can be viewed as linear uniform convolution and parametrically modelled with angle and length. In this paper, we propose a novel scheme based on sparse representation to identify the blur kernel. By analyzing the sparse representation coefficients of the recovered **image**, we determine the angle of the kernel based on the observation that the recovered **image** has the most sparse representation when the kernel angle corresponds to the genuine motion angle. Then, we estimate the length of the motion kernel with Radon **transform** in **Fourier** domain. Our scheme can well handle large motion blur even when the license plate is unrecognizable by human. We evaluate our approach on real-world images and compare with several popular state-of-the-art blind **image** de-blurring algorithms. Experimental results demonstrate the superiority of our proposed approach in terms of effectiveness and robustness.

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Here a 1D cepstrum, extension of 1D i.e. 2D is used for filtering process and **image** registration process. As logarithm operators are used the cepstral features are invariant to amplitude changes and as cepstral **transform** is carried out in **Fourier** domain it is invariant to translation shift. In frequency domain it is observed that uniform motion blur has a periodic pattern by zero crossing of sine function [2]. Cepstrum or Cepstral mainly works on quefrency. With the **image** in quefrency the **image** is turned by the angle found in previous module. Collapse the 2D into 1D cepstral by taking average of the columns.

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Multichannel Sampling Theorem Methods -This reconstruction method implementation is achieved in the spatial domain method, but the technique is fundamentally related to the frequency domain because of the shift property of the **Fourier** **Transform** that translates the source **image** [2]. In this method, which is actually implemented in spatial domain, there is some consideration of the Point Spread Function (PSF) which is fundamentally included in the frequency domain method. So, Ur and Gross [11] consider some linear degradation to include the effect of the blur point Spread Function and global translation which is considered as delay in the method. That Observation the operation of the blurring effect and translation that the commutative and assuming[2] the single blur effect of the common for all channel, this observation shows that the Super-Resolution problem is divided into some distinct process which involves the “merging” of the under-sampled signal into single-band limit function; this is the “de- blurring” of the merged signal. De-blurring and merging process are different from each other, but there is a possibility that the some closed solution is derived for the merged signal from the under sampled and degraded channel outputs.

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Q. Shan, J. Jia, and A. Agarwala [8] represented a new method to remove motion blur from a single **image**. The method computes a deblurred **image** **using** a probabilistic model bringing together blur kernel estimation and unblurred **image** restoration. An analysis is presented for the causes of common artifacts which are found in current **deblurring** methods, inspired by this analysis many new terms has been introduced within this probabilistic model. One of these three new terms include a spatially random distribution of noise in the **image**. This model helps to separate the faults that rise during **image** noise approximation and blur kernel valuation. The second method is a new smoothness constraint that is imposed on low contrast area of latent **image**. This is very effective in restraining ringing artifacts. And final method is an optimization algorithm that alternates between blur kernel estimation and deblurred **image** restoration until convergence. As a result of these steps, they are able to produce high quality deblurred results in low computation time.

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In this paper, we proposed the novel architecture on the **lensless** camera that can capture the front and back scene at once. The merit of the proposed camera is the capability of super-FOV imaging by a thin and compact optical hardware constructed only by **image** sensors. To realize this, we exploited the CIS, which is the CMOS **image** sensor whose pixels are randomly drilled into air holes. In the proposed camera, CISs are placed facing with each other, where the front sensor works as a coded aperture and the back one works as a sparse **image** sampler of the coded optical **image**. The captured sparse coded **image** was computationally decoded into the subject **image** by the CS-based **image**-reconstruction algorithm.

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MOTION blur is the result of the relative motion between the camera and the scene during the integration time of the **image**. It has also been used to obtain motion and scene 3D structure information. Motion blur has also been also used in computer graphics to create more realistic images which are pleasing to the eye. Several representations and models for motion blur in human and machine vision have been proposed. Very often, motion blur is simply an undesired effect. It has plagued photography since its early days and is still considered to be an effect that can significantly degrade **image** quality. Every motion blurred **image** tends to be uniquely blurred. This makes the problem of motion **deblurring** hard[1].When a photograph is taken of a fast moving object, motion blur can cause significant degradation of the **image** .This is caused by the movement of the object relative to the sensor in the camera during the time the shutter is open. Both the object moving and camera shake contribute to this blurring. **Image** **deblurring** is usually the first process that is used in the analysis of digital images. In any **image** denoising technique it is very important that the denoising process should not have any blurring effect on the **image**, and makes no changes on the preserving of images to **image** edges [2].

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The MATLAB function fft2 computes two-dimensional DFTs **using** a fast **Fourier** **transform** algorithm. Y = fft2(X) is equivalent to Y = fft(fft(X).').', that is, to computing the one- dimensional DFT of each column X followed by the one-

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This testing does not include FFTW (The "Fastest **Fourier** **Transform** in the West"), which is a very high performance FFT implementation for any size **transform**. The goal of this work was to create a simple, single file, small FFT implementation suitable for many projects. FFTW is over 40,000 lines of code. The result of this work is well under 400 lines of code, including an FFT, an inverse FFT, a table based version of each, a real to complex FFT (covered below), and its inverse, all in a single standalone source file. The routines were tested on various sizes of . For each size, the algorithm is run 100 times to load all items into cache and otherwise smooth out results. Then a loop of 100 applications of the FFT is timed for 50 passes. The min, max, and average of these 50 scores are kept for each and FFT type combination. Sizes for were chosen to be small, around those needed in audio processing.

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The ability to obtain spectral information from objects or scenes is very useful in various fields including remote sensing, chemistry, astronomy, and industrial quality control. The methods of **Fourier** **transform** spectroscopy (FTS) have been used for over thirty years. As resolution of optical positioning devices and computer processing speeds increase, so does the usefulness and importance of FTS. There are many different designs of spectrometers available, but I have chosen to base this project on the Michelson interferometer.

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Capturing satisfactory images under low light conditions **using** a hand-held camera can be a frustrating experience. Often the taken images are blurry or noisy. The brightness of the **image** can be increased in three ways are shutter, aperture and ISO settings. First, reducing the shutter speed (the reciprocal of the focal length of the lens, in the unit of seconds) as well as safe shutter speed. But with a safe shutter speed and camera shake will result a blurred **image**. Second, the aperture should be large. A large aperture will reduce the depth of field. Moreover, the range of apertures in a consumer-level camera is very limited. Third, the ISO range should be high. However, the high ISO **image** is very noisy due to the amplification of noise as the camera’s gain increases. For taking a sharp **image** in a dim lighting environment, the best settings are: safe shutter speed, the largest aperture, and the highest ISO. Even with this combination, the captured **image** may still be dark and very noisy. To avoid that, flash is **using** in the camera. But unfortunately the flash introduces artifacts such as shadows and secularities. On the other hand, the flash is not effective for distant objects.

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Abstract – The digital cameras provided a great service for users of different ages as it facilitated the process of capturing the images, and in spite of that user still needs to improve some of the images marred by the lack of clarity when taking an **image** because of the lack of proper lighting as it is in cloudy weather or bright or dark sites light , or take the **image** from a distance, leading to blurred **image** details. We Propose in this paper a new method for contrast enhancement gray images based on Fast Discrete Curvelet **Transform** via Unequally Spaced Fast **Fourier** **Transform** (FDCT-USFFT). This type of transforms returns a table of curvelet coefficients indexed by a scale parameters, an orientation, and a spatial location. The FDCT-USFFT coefficients can be modified in order to enhancement contrast in an **image**. Results show that the proposed technique gave very good results, in comparison to the histogram equalization and wavelet **transform** based contrast enhancement method.

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parameters obtained refer to local features dened at dierent spatial resolu- tions and these are determined from the spatial frequency vectors on dierent levels of the MFT. For each local feature in a given orientation, the corre- lation stastistics used in the estimation scheme provide an estimate of the linear phase increment in an orthogonal orientation within the relevant spa- tial frequency vector. However, the correlation statistics provide a measure of the energy variation over all orientations, given the linear phase model. It is therefore necessary to nd other ways to model the magnitude distribution. However, both of Calway's and Li's methods are based on the assumption that each block only contains a single feature. If not, it will be divided into 4 sub-blocks, and each of these will be re-estimated at the higher spatial res- olution until a single feature is found or the block is too small to analyse [8]. Here the spatial frequency block is divided into orientation segments instead of dividing the spatial block into sub-blocks. The estimated features are dis- played by constructing an **image** in which each feature is represented by a straight line within the spatial region referred to by the spatial frequency vector. This line is shown at the appropriate orientation and position with a certain length. The luminance value of each line is then set to the magnitude of the MFT coecients. Figure 9 shows the result of this approach. In order to ensure that interesting features with relatively low energies are not missed, a normalisation process is performed to increase their visibility.

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The fractional **Fourier** **transform** (FrFT) is the generalized formula for the **Fourier** **transform** that transforms a function into an intermediate domain between time and frequency. The signals with significant overlap in both the time and frequency domain may have little or no overlap in the fractional **Fourier** domain. The fractional **Fourier** **transform** of order a of an arbitrary function x (t ) , with an angle α , is defined as [4]:

Alternatively, a parametric model m ay be used for statistical modelling. The most common stochastic models of texture are the autoregressive [58], or more generally Markovian Random Field (M RF) m odels [25] [44]. Chellappa and Kashyap [18], suggested the use of 2-D autoregressive, noncausal models for textures. In this model, the gray level of a pixel is characterised as a linear combination of the gray levels at neighbouring pixels and an additive linear combination of some noise field values. The model parameters are estimated by approximating a maximum likelihood solution for various possible neighbourhoods. One of these models is chosen, **using** a Bayesian decision rule, to approximate the original noncausal model and then to synthesise images. Chen [19] proposed a maximum entropy power spectrum estimation method, which involves a parameteric way of estimating spectrum based on an autoregressive model. These methods are widely used but are of limited applicability to images because the causality restriction is significant: the autoregressive model is limited by its inherent directionality, which introduces a degree of anisotropy.

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Abstract: In this study, complex differential equations are solved by us- ing the **Fourier** **transform**. First, we separate the real and imaginary parts of the equation. Thus, from one unknown equation we obtain a system of two unknown equations. We obtain the **Fourier** transforms of real and imaginary parts of the solutions **using** the **Fourier** **transform**. Finally, we obtain the real and imaginary parts of the solution by **using** the inverse **Fourier** **transform**. AMS Subject Classification: 45E05, 30G20, 32A55, 30E20

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CONTENTS CHAPTER I PAGE TITLE 1 STATEMENT OF OBJECTIVES 1.1 Definition of the Problem 1.2 Purpose of the Investigation 1.3 II Method of Investigation 2 5 TYPES OF BLUR 2.1 Introduction 2[r]

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which is large enough to fully describe the spatial domain **image**. Once the **image** is transformed into the frequency domain, filters can be applied to the **image** by convolution. FFT turns the complicated convolution operations into simple multiplications. An inverse **transform** is then applied in the frequency domain to obtain the result of convolution. The resultant **image** contains all the important details. Applying filters to **image** in frequency domain is computationally faster than to do the same in the **image** domain. This will speed up the process and provide better result. These are the steps followed:

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This project presents the identification of voltage disturbance based on mathematical method of Wavelet **Transform** and Fast **Fourier** **Transform** (FFT). A simple program is developed based on these two methods **using** MATLAB 7.8 software. The data of voltage disturbance used is limited into voltage sag, voltage swell and harmonic. FFT is the extension of the Discrete **Fourier** **Transform** (DFT). To locate the start time, end time and magnitude of the voltage disturbance occur on the recorded data, the Wavelet **Transform** is applied.

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Fig 3(a) and 3(b) were captured with a five minute interval between images [3]. These images provide an example of a rapid change in daylight that causes difficulty with respect to change detection. Fig 3(c) shows change detected region **using** DFrFT [4]. Intensity changes due to the appearance of shadows can be observed in difference **image**. But with the help of discussed method we detect only significant changes i.e. objects removing insignificant changes, occurring due to shadowing. Through the gradient correlation processes, the appearances of people are detected, as indicated by red blocks. In this case, precision value for proposed method is increased by 50% than previous method, but recall value is same for both methods.

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