Gaussian Smoothing:

• The Gaussian filters of different sizes have been extensively used in

shape and image smoothing
• Marr and Hildreth,  Witkin, Lindeberg,
• Koenderink [Koenderink:Structure:84,Koenderink:etal:Dynamic:Shape] showed that the Gaussian scale-space can be modeled by the heat diffusion partial differential equation

where u represents the image, x, y  the spatial dimensions, and t specifies the scale parameter

• There are two major shortcomings of Gaussian smoothing:
• Gaussian smoothing shrinks  shapes and dislocates boundaries when moving from finer to coarser scales
• Gaussian smoothing blurs important image features.
• Examples:

• The above image in animation (830K)

• The above image in animation (950K)

Nonlinear diffusion filtering:

• Main idea: reduce smoothing at the edges to preserve the contrast information and the location of the object boundaries.
• Perona and Malik
where I(x,y) is the image and

• this approach smooths regions of low brightness gradient while regions of high gradients are not smoothed.
• Similar approaches by Catte et. al., Whitaker and Pizer
• see "Review of Nonlinear Diffusion Filtering"  by J. Weickert, in First International Conference, Scale-Space for an excellent review

• However, these anisotropic diffusion approaches are contrast-driven and

smooth globally salient but low contrast image features

Nonlinear Geometry Driven Diffusion:

• In these approaches,  the grey-level image is smoothed by the curvature
• Specifically,  grey-level intensity or range information can be used as

a evolving surface, 

This is  also known as the mean curvature flow.
• Evans and Spruck,  Alvarez et al, Kimia and Siddiqi , and others

• Examples:

      
 
• The above smoothing in animation (850K)

• The above smoothing in animation (800K)