A restarted iterative homotopy analysis method for two nonlinear models from image processing

Abstract

Total variation (TV) minimization-based nonlinear models have been proven to be very useful and successful in image processing. A lot of effort has been devoted to overcome the nonlinearity of the model and at the same time to obtain fast numerical schemes. In this paper, we propose a restarted iterative homotopy analysis method (HAM) to improve the computational efficiency for the TV models and will show by experiments that this method demonstrates great potential for recovering the noise and with great speed in both image denoising and image segmentation models. The method modifies the existing HAM and makes it suitable to potentially solve other nonlinear partial differential equations arising from image processing models. In our examples, we will demonstrate the validity of a restarted HAM and that this method is efficient and robust even for images with large ratios of noise and with much less CPU time than other methods.

Keywords:

• image processing
• total variation
• denoising
• image segmentation
• discrete homotopy analysis method
• finite difference scheme
• series solution

2010 AMS Subject Classifications::

• 62H35
• 65N22
• 68U10
• 35A15
• 65C20
• 74G65
• 74G75
• 35C10

Notes

1. The operator in Equation (11) is a general case of the linear operator applied by Liao 24

2. In this paper, we consider the most common additive noise such as random Gaussian noise with mean 0 and standard deviation σ and uniform distributed noise

3. Without no need of solving a system.

4. f′₀(φ₀) has been calculated by the chain rule differentiation formula and c′₁ and c′₂ in Equations (44) and (45) has been differentiated with respect to q.