توصيفگر ها :
مشتقات مرتبه كسري , كاهش نويز تصوير , تانسور ساختاري , ضمني جهت متناوب , پايداري
چكيده فارسي :
كاهش نويز تصوير از جمله مسايل مورد توجه و كاربردي در پردازش تصوير است. برخي از روش هاي كاهش نويز تصوير با بهره گيري از معادلات با مشتقات كسري انجام مي‌شود. در اين پايان نامه به بررسي و تحليل يك مدل كاهش نويز تصوير بر اساس مشتقات مرتبه كسري و تانسور ساختاري پرداخته ايم. به همين منظور، ابتدا با اثبات وجود و يكتايي جواب در حل اين مدل مبتني بر مشتقات مرتبه كسري، يك روش عددي ضمني جهت متناوب براي حل آن به كار برده ايم و سپس پايداري و همگرايي اين روش عددي را اثبات مي كنيم. علاوه براين، با چندين آزمايش عددي و مقايسه آن با ساير مدل هاي كاهش نويز تصوير نشان مي دهيم، اين مدل توانسته است نتايج مدل هاي پيشين را بهبود بخشد.
چكيده انگليسي :
Digital images have many applications in satellite television, Magnetic resonance imaging, Tomography, and research and technology fields such as geographic information systems and astronomy. In general, data sets collected by image sensors are contaminated with noise that can damage image data. Therefore, image denoising is the first essential step before analyzing image data. In this thesis, we study and analyze an image denoising model based on fractional order derivatives and structural tensor. First, we outline the general concepts required for this dissertation. We review the concepts of LP, Hilbert, and Banach spaces, and after introducing Sobolov spaces, we study equations with fractional derivatives. Then, we introduce an image denoising model with the help of a structure tensor. In the presented model, since the structure tensor can provide more local structural information and direction information of the gradient, the eigenvalues of the structure tensor of an image are used to construct the norm parameter. Through theoretical analyses and experiments, we can confirm that the proposed norm parameter has the following adaptive characteristics: (1) In the flat region of the image, the proposed norm parameter tends to 2, so which has the characteristic of isotropic diffusion. (2) At the edge region of the image, the norm parameter tends to 1, and which spreads only along the tangential direction of the image edge. Then, using a nonlinear PDE equation, image denoising is investigated. For this purpose, we model the problem and after obtaining the PDE problem and checking the existence and uniqueness of the solution, we solve the PDE problem with the alternative direction implicit(ADI) method. We also examine the stability and convergence of the ADI method. Finally, we present an image denoising algorithm using the ADI method to solve the corresponding PDE problem. In the algorithm, it is expected to compute the inverse of an N1 × N2 matrix. This costs too much computer memory and time because N1 × N2 is so large. ADI scheme divides the 2D problem into several 1D problems. This keeps the stability of the algorithm and saves much computer memory. This scheme can be extended to 3D image denoising problems. In 3D cases, the advantage of this scheme can be even more obvious. This is the main motivation to design the numerical scheme. In this thesis, the signal-to-noise ratio (SNR) of the reconstructed image and the peak signal-to-noise ratio (PSNR) are used as indicators for quantitative comparison. Then, using MATLAB and with the help of an algorithm, we reduced the image noise. After that, we perform three numerical tests to show the efficiency of the denoising algorithm presented. In these three tests, Test 1 and Test 2 are performed to show the advantage of the tensor structure function. Test 3 is performed to show the efficiency of the presented algorithm.
