Forward and backward filtering based on backward stochastic differential equations

Dariusz  Borkowski

doi:10.3934/ipi.2016002

Article Contents

2016, Volume 10, Issue 2: 305-325. Doi: 10.3934/ipi.2016002

This issue Previous Article Restoration of manifold-valued images by half-quadratic minimization Next Article On the detection of several obstacles in 2D Stokes flow: Topological sensitivity and combination with shape derivatives

Forward and backward filtering based on backward stochastic differential equations

Dariusz Borkowski^1,

1.
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń

Received: February 28, 2013

Revised: December 31, 2014

Published: April 2016

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Abstract

In this paper we explore the problem of reconstruction of blurred and noisy images. The idea presented here provides a new methodology based on advanced tools of stochastic analysis which can be successfully used to solve the inverse problem. In order to solve this problem we use backward stochastic differential equations. The reconstructed image is characterized by smoothing noisy pixels and at the same time enhancing and sharpening edges. Our experiments show that the new approach gives very good results and compares favourably with deterministic partial differential equation methods.

Keywords:

Inverse problem,
backward stochastic differential equations,
stochastic processes,
image denoising,
image reconstruction.

Mathematics Subject Classification: Primary: 68U10, 60H10; Secondary: 60H30.

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