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Parallel Algorithm for a Nonlocal Diffusion Model Applied to a 3D Input Domain
... This model can also be seen applied to image processing and other domains in [2,3,5,6]. In [7] we described a numerical approximation for a similar model applied to a 3D input domain but having fewer experimental results. ...
In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of interest, given that the numerical analysis usually takes place on large input domains having more than one dimension. After introducing the general reaction-diffusion model, we discuss a numerical approximation scheme for the diffusion term, based on a finite difference method. In the next sections we propose two algorithms to solve the numerical approximation scheme, focusing on finding a way to improve the time performance. While the first algorithm (sequential) is used as a baseline for performance measurement, the second algorithm (parallel) is implemented using two different memory-sharing parallelization technologies: Open Multi-Processing (OpenMP) and CUDA. All the results were obtained by using the model in image processing applications such as image restoration and segmentation.
The paper is concerned with a qualitative analysis for a nonlinear second-order parabolic problem, subject to non-homogeneous Cauchy–Stefan–Boltzmann boundary conditions, extending the types already studied. Under certain assumptions, we prove the existence, a priori estimates, regularity and uniqueness of a solution in the class Wp1,2(Q). Here we extend the results already proven by the authors for a nonlinearity of cubic type, making the present mathematical model to be more capable for describing the complexity of certain wide classes of real physical phenomena (phase separation and transition, for instance).
In this paper we discuss a nonlocal approximation to the classical heat equation with Neumann boundary conditions. We considerwt∊(x,t)=1∊N+2∫ΩJx-y∊(w∊(y,t)-w∊(x,t))dy+C1∊N∫∂ΩJx-y∊g(y,t)dSy,(x,t)∈Ω‾×(0,T),w(x,0)=u0(x),x∈Ω‾,
and we show that the corresponding solutions, w∊, converge to the classical solution of the local heat equation vt=Δv with Neumann boundary conditions, ∂v∂n(x,t)=g(x,t), and initial condition v(0)=u0, as the parameter ∊ goes to zero. The obtained convergence is in the weak star on L∞ topology.
A nonlocal quadratic functional of weighted differences is examined. The weights are based on image features and represent the affinity between different pixels in the image. By prescribing different formulas for the weights, one can generalize many local and nonlocal linear denoising algorithms, including the nonlocal means filter and the bilateral filter. In this framework one can easily show that continuous iterations of the generalized filter obey certain global characteristics and converge to a constant solution. The linear operator associated with the Euler-Lagrange equation of the functional is closely related to the graph Laplacian. We can thus interpret the steepest descent for minimizing the functional as a nonlocal diffusion process. This formulation allows a convenient framework for nonlocal variational minimizations, including variational denoising, Bregman iterations, and the recently proposed inverse scale space. It is also demonstrated how the steepest descent flow can be used for segmentation. Following kernel based methods in machine learning, the generalized diffusion process is used to propagate sporadic initial user’s information to the entire image. Unlike classical variational segmentation methods, the process is not explicitly based on a curve length energy and thus can cope well with highly nonconvex shapes and corners. Reasonable robustness to noise is still achieved.
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CUDA (compute unified device architecture) is a novel technology of general-purpose computing on the GPU, which makes users develop general GPU (graphics processing unit) programs easily. This paper analyzes the distinct features of CUDA GPU, summarizes the general program mode of CUDA. Furthermore, we implement several classical image processing algorithms by CUDA, such as histogram equalization, removing clouds, edge detection and DCT encode and decode etc., especially introduce the first two algorithms. If we donpsilat take the data transfer time in experiment between host memory and device memory into account, as the image size increase, histogram computation can get a more than 40x speedup, removing clouds can get an about 79x speedup, DCT can gain around 8x and edge detection more than 200x.
A new definition of scale-space is suggested, and a class of
algorithms used to realize a diffusion process is introduced. The
diffusion coefficient is chosen to vary spatially in such a way as to
encourage intraregion smoothing rather than interregion smoothing. It is
shown that the `no new maxima should be generated at coarse scales'
property of conventional scale space is preserved. As the region
boundaries in the approach remain sharp, a high-quality edge detector
which successfully exploits global information is obtained. Experimental
results are shown on a number of images. Parallel hardware
implementations are made feasible because the algorithm involves
elementary, local operations replicated over the image