Random Walks for Mesh Denoising

1
Random Walks for Mesh Denoising

• Xianfang Sun
• Paul L. Rosin
• Ralph R. Martin
• Frank C. Langbein
• Cardiff University
• UK

2
Outline

• Introduction
• Random Walks
• Normal Filtering
• Vertex Position Updating
• Experimental Results
• Conclusion

3
Introduction

• Mesh models generated by 3D scanner always
  contain noise. It is necessary to remove the
  noise from the meshes.
• We want to distinguish mesh denoising, mesh
  smoothing, and mesh fairing.
• Mesh denoising remove noises, feature-preserving
• Mesh smoothing remove high-frequency information
  
• Mesh fairing smoothing, aesthetically pleasing
  surface

4
Introduction (cont.)

• Mesh denoising can be performed in one step or
  two steps
• One-step directly move vertex
• Two-step first adjust face normals, then based
  on new normals to move vertex
• When the result of a signal pass of the vertex
  displacement is not good, iteration is necessary.
• Two schemes of iterative two-step method
• (Step 1Step 2)n
• (Step 1) n1(Step 2)n2
• Where
• Step 1 face normal filtering
• Step 2 vertex position updating

5
Introduction (cont.)

• Our algorithm is an iterative two-step method
• We use random walks for face normal filtering
• and conjugate-gradient descent method for
  vertex position updating

Face normal filtering (Iterate n1 times)
Vertex position updating (Iterate n2 times)
6
Markov Chains andRandom Walks

• Random walks is closely related to Markov Chain.
• Markov Chains a sequence of random variables
• with the property that
  given the present state, the future state is
  conditionally independent of any earlier state.
• Random Walks
• A special Markov Chain with sparse transition
  probability matrix.

7
Markov Chains and Random Walks (cont.)

• Notation
• Initial probability distribution
• nth step probability distribution
• n-step transition probability matrix
• (i,j)th element of
• kth step transition probability matrix
• (i,j)th element of

8
Normal Filtering

• Motivation
• If the probabilities of stepping from one
  triangle to another depend on how similar their
  normals are, and we average normals according to
  the final probabilities of random walks, we will
  give greater weights to triangles with similar
  normals and less weights to ones that are not so
  similar.
• Similar ideas were used by Smolka and
  Wojciechowski 2001 for image denoising.
• This idea may be used in other mesh processing
  problems.

9
Normal Filtering

• Normal updating formula
• and are the current and updated normal
  of the face i, respectively F is the face set
  is the probability of going from face i to
  face j after n steps of random walks.
• depends on k 1, , n and n,
  where is the probability from face i to
  face j at the kth step.

10
Normal Filtering (cont.)

• Choice of
• It is a decreasing function of
• is the 1-ring neighbourhood of the
  face i.
• Choice of n
• When non-iterative scheme is chosen, n must be
  large enough to guarantee good results,
• When iterative scheme is chosen, n can be
  small.

11
Normal Filtering (cont.)

• Two computational schemes
• In each iteration, We compute
  sequentially from
• i 1 to number-of-faces
• Batch scheme always use the same old
  obtained
• in the last iteration
• Progressive scheme use the newly updated
• obtained in the current iteration, once the
  new value is
• available.

12
Normal Filtering (cont.)

• Computational Tip
• For the case of ngt1, because will become
  non-sparse, as n grows, the computational cost
  will grows quickly, and additional memory will be
  required to store the whole matrix , we
  propose
• Not to compute , and then use to
  compute ,
• but to update normals sequentially by
• and

13
Normal Filtering (cont.)

• Adaptive parameter adjustment
• Because choosing a suitable parameter value
  affects the quality of the results, we need to
  dynamically adjust the parameter. We consider to
  minimise the cost function
• where is the initial noisy normal of
  face i. And we use stochastic gradient-descent
  algorithm to update the parameter

14
Normal Filtering (cont.)

• Feature-preserving property
• The feature is related to the face normals
• The updated normal is weighted average of its
  neighbouring normals
• The weight function is a decreasing function of
  the normal difference
• Adaptive adjustment of the parameter further
  improves the feature-preserving property.

15
Vertex Position Updating

• Orthogonality between the normal and the three
  edges of each face on the mesh
• Minimise the error function
• Solution by conjugate gradient descent algorithm.

16
Experiment Results Choice of Parameters
17
Experimental Results Adaptive Parameter

• original noisy
  ß8, NA (non-adaptive)
• ß8, A ß5, NA
  ß5, A (adaptive)

18
Experimental Results Quality Comparison

• original noisy
  BF (bilateral)
• MF (median) FF (fuzzy)
  RF (random walks)

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Experimental Results Quality Comparison (cont.)

• original noisy
  BF
• MF FF
  RF

20
Experimental Results Quality Comparison (cont.)

• original noisy
  BF
• MF FF
  RF

21
Experimental Results Quality Comparison (cont.)

• original noisy
  BF
• MF FF
  RF

22
Experimental Results Quality Comparison (cont.)

• BF
  MF
• original
• 
  FF RF

23
Experimental Results Quality Comparison (cont.)

• 
  BF MF
• original
• 
  FF RF

24
Experimental Results Quality Comparison (cont.)

• 
  BF MF
• original
• 
  FF RF

25
Experimental Results Timing Comparison
26
Conclusins

• Random walks approach is introduced into mesh
  denoising it may also be used in other mesh
  processing problems.
• Adaptively adjust parameter and progressively
  update face normals is the best implementation of
  our approach.
• It is a fast and efficient feature-preserving
  approach
• Our approach is as fast as the bilateral
  filtering (BF) approach, however, our approach
  preserves sharp edges better than the BF
  approach.
• Compared to the fuzzy vector median filtering
  (FF) approach, our approach is over ten times
  faster, yet produces a final surface quality
  similar to or better than that approach.

27
Thank You!

• Questions?