Fast Multipole Method FMM to Approximate

1
Fast Multipole Method (FMM) to Approximate Thin
Plate Spline (TPS) Mapping
Ali Zandifar, Ser-nam Lim, Ramani Duraiswami,
Neil Gumerov and Larry S. Davis
Results
TPS Computer Vision Applications
Fast Multipole Method
Finger Print Matching
(a) Comparing speed of MLFMM and standard method
(b) Variation of speed with the maximum number of
data points per box at lowest level (c)Variation
of Error with the maximum number of data (d)
Actual error versus number of data points

• Introduced by Rokhlin Greengard 1987
• Called one of the most significant advances
  computing
• Trades accuracy for complexity in a principled
  way
• Speeds up matrix-vector products of the type

Image Restoration, Image Morphing, etc.
Near and far field expansions and translations
(a) (b)(d)(e) original images (c) morphed images
using TPS with MLFMM and 10 landmarks
S-expansion

• for M evaluation points and N source points
  , the complexity of standard method is O(MN)
• for a given precision of e the FMM achieves
  evaluation in O(MN)

(a) Original Image (b) Sub-sampled image (c)
Reconstructed image from subsampled image using
TPS with MLFMM (d) Reconstructed image from
sub-sampled image using bilinear interpolation
Four key stones of using FMM
SS Translation

• Performance of MLFMM approximation with
  truncated series outperforms the standard method
  while giving only small error
• Complexity of O(N log N) vs. O( ) of
  standard method while maintaining small error

• Near and far field expansion
• Factorization of the function
• Translation of one factorization to another
• Error bounds of truncation and translation
• Grouping

Applications of FMM
Contribution and Future works
SR Translation

• Astrophysics
• Fluid and Plasma Dynamics
• Electronics
• Acoustics
• Computer Graphics
• Computer Vision and Image Processing

• Introduce applications of FMM in Computer vision
  
• MLFMM for fast evaluation of TPS mapping
• RR translation and R expansion representation of
  
• Develop a suitable pre-conditioner to the
  solution of linear matrix model
• Develop FMM for the other nonlinear mapping
  schemes

Thin Plate Spline (TPS)
A thin-plate spline f(x,y) is a smooth function
which interpolates a surface that is fixed at the
control points. This surface is like a thin metal
plate, then this plate takes a shape in which it
is least bent.
References
RR Translation

• R.K. Beatson and G.N. Newsam, "Fast Evaluation
  of Radial Basis Function I", In Computer Math.
  Applications, Vol. 24, No. 12, pp 7-19, 1992
• Nail Gumerov and Ramani Duraiswami, CMSC
  878R/AMSC698R course notes on Advanced Topics in
  Numerical Analysis Fast Multipole Method,
  URLhttp//www.umiacs.umd.edu/ramani

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(xi, yi)), U(r)r2log(r)