时间自适应临界变指数向量扩散流及其在图像处理中的应用
On time adaptive critical variable exponent vectorial diffusion flows and their applications in image processing
摘要：
变指数空间在实际问题中有着有趣的应用。最近，有相当大的兴趣，利用变指数的基础上的变分和演化问题的成像应用。与变指数相关的偏微分方程的主要类型包括$p(\cdot)$ -Laplacian。在成像应用中，变量指数可以接近临界值$1$，这在证明解的存在性方面提出了独特的挑战，这在以前还没有被掌握。在这项工作中，我们开发了一些额外的功能框架来研究临界变指数的含时抛物流。具体来说，我们考虑有界向量部分变分（BVPV）空间和它的变指数对应。在变指数空间中证明了临界向量$p(t,x)$ -Laplacian流弱解的存在性.对于非时变指数的临界向量$p(x)$ -Laplacian流，我们得到了一个半群解.结果是新的，即使在标量的情况下。这是一个理论导向的草案，全文将提供详细的实验结果，彩色图像恢复使用各种例子的可变指数，并比较他们传统的偏微分方程为基础的图像处理程序。我们的研究结果将表明在一般的图像处理，特别是图像恢复的变指数拉普拉斯流的适用性。
英文摘要：
Variable exponent spaces have found interesting applications in real world problems. Recently, there have been considerable interest in utilizing variational and evolution problems based on variable exponents for imaging applications. The main classes of partial differential equations (PDEs) related to the variable exponents involve the $p(\cdot)$-Laplacian. In imaging applications, the variable exponent can approach the critical value $1$, and this poses unique challenges in proving existence of solutions, which have not been mastered earlier. In this work, we develop some additional functional framework to study the time-dependent parabolic flows with critical variable exponents. Specifically, we consider bounded vectorial partial variation (BVPV) space and its variable exponent counterpart. We prove the existence of weak solutions of critical vectorial $p(t,x)$-Laplacian flow in our variable exponent space. For non-time-dependent variable exponent based critical vectorial $p(x)$-Laplacian flow we obtain a semigroup solution. The results are new even in the scalar case. This is a theory-oriented draft, and the full paper will provide detailed experimental results on color image restoration using various example for the variable exponents and compare them traditional PDE based image processing procedures. Our results will indicate the applicability of variable exponent Laplacian flows in image processing in general and image restoration in particular.
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