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Image Processing in SIGGRAPH 06. Speaker: Qianqian Hu Date: March 31, 2006. Outlines. Fast Median and Bilateral Filtering Ben Weiss ( Shell & Slate Software ) Hybrid Images

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image processing in siggraph 06

Image Processing in SIGGRAPH 06

Speaker: Qianqian Hu

Date: March 31, 2006

outlines
Outlines
  • Fast Median and Bilateral Filtering
    • Ben Weiss (Shell & Slate Software)
  • Hybrid Images
    • Aude Oliva (Massachusetts Institute of Technology, Department of Brain and Cognitive Sciences), Antonio Torralba (Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory), Philippe G. Schyns (University of Glasgow)
  • Image Deformation Using Moving Least Squares
    • Scott Schaefer (Texas A&M University), Travis McPhail, Joe Warren (Rice University)
  • Appearance-Space Texture Synthesis
    • Sylvain Lefebvre, Hugues Hoppe (Microsoft Research)
image deformation using moving least squares

Image Deformation using Moving Least Squares

Scott Schaefer Travis McPhail Joe Warren

Texas A&M University Rice University Rice Univeristy

previous work
Previous Work
  • Grid-based techniques:
      • Bivariate cubic splines[Sederberg and Parry, 1986, Lee et al, 1995]
      • Shepard’s interpolant[Beier and Neely, 1992]
  • Transformation-based technique:
      • Radial Basis Function[Bookstein, 1989]
  • Triangulation-based technique[Igarashi et al, 2005]
characters of the deformation function
Characters of the Deformation Function
  • Given a set of handles p, and corresponding new positions q. The deformation function f satisfies
      • Interpolation: f(pi)=qi
      • Smoothness: smooth deformations
      • Identity: q=p f(v) = v
moving least squares
Moving Least Squares
  • Given a point v in the image, the best affine transformation lv(x) minimizes

where

DF: f(v)= lv(v)

affine transformation
Affine Transformation

lv(x)=xM +T,

  • M :a linear transformation matrix (rotation and scaling)
  • T :a translation

Best affine transformation

where

affine deformations
Affine Deformations
  • Solution for matrix M
  • Solution for deformation function
result
Result

Non-uniform scaling and shear

similarity deformations
Similarity Deformations
  • Constraints: uniform scaling, i.e,
  • Define , where
  • Least squares problem

where

similarity deformations1
Similarity Deformations
  • Solution for matrix M
  • Solution for deformation function

where

result1
Result

uniformscaling

rigid deformations
Rigid Deformations
  • Constraint: no uniform scaling, i.e.,
  • Theoretical base
rigid deformations1
Rigid Deformations
  • Solution for matrix M
  • Solution for deformation function

where , and Aiis as in similarity deformations.

deformation with line segments
Deformation with Line Segments
  • Least squares problem
affine lines
Affine Lines
  • Line segments are expressed in matrix form
  • Least squares problem
solution
Solution
  • The deformation function

and

similarity lines
Similarity Lines
  • The deformation function
rigid lines
Rigid Lines
  • The deformation function

where

implementation
Implementation
  • Every pixel is replaced by a grid
  • Every resulting pixel is calculated using bilinear interpolation
contributions
Contributions
  • A simple closed-form solution
      • a linear system (2X2) at each point
      • No use of linear solver
      • Simple, and realtime
  • Handles:
      • points,
      • line segments.
  • As-rigid-as possible
shortcoming
Shortcoming
  • Lack of topological information
future work
Future Work
  • Adding topological information
  • Generalizing to 3D to deform surfaces
  • Handles can be any curves
fast median and bilateral filtering

Fast Median and Bilateral Filtering

Ben Weiss

Shell & Slate Software Corp.

contributions1
Contributions
  • Improving Runtime from O(r) to O(logr)
      • Scalable to arbitrary radius
      • Realtime
  • Fitting for any bit-depth
related work
Related Work
  • A variety of O(r) algorithms

Huang, T.S., 1981. Two-Dimensional Signal Processing II: Transforms and Median Filters.

No good performance for large filtering kernels.

  • A tree-based O(log2r) algorithm

Gil, J. and Werman, M., 1993. Computing 2-D Min, Median, and Max Filters.

Ill-suited for deep-pipelined, vector-capable modern processors.

  • A parallel algorithm with time complexity of O(log4r)

Ranka, S., and Sahni, S., 1989. Efficient Serial and Parallel Algorithms for Median Filtering.

even worser than linear for r<55.

median filtering
Median Filtering
  • A pixel value is replaced by the median of its neighbours. [Tukey, 1977]
advantages
Advantages
  • Reducing image noise
  • Preserving edges
  • Basic algorithm of many image-processing techniques
      • Rank-order filtering
      • Morphological processing

slowness!!!

basic o r algorithm
Basic O(r) Algorithm
  • Consider a r-radius median filter to an 8-bit image.
histogram and mean value
Histogram and Mean Value
  • Use a 256-element histogram
  • Mean value = the index v*such that

Integral =2r2+2r+1

fundamental idea
Fundamental idea
  • If multiple columns are processed at once, the aforementioned redundant calculations become sequential.
distributive histograms
Distributive Histograms
  • For disjoint image regions A and B:
higher depth median filtering
Higher-Depth Median Filtering
  • 16-bit and HDR(High dynamic range) images
  • Histogram exponentially with bit-depth
the ordinal transform
The Ordinal Transform

cardinal values consecutive ordinal values

comparison 2
Comparison(2)
  • For 8-bit data
      • 50 times faster than Photoshop
  • For 16-bit data
      • 20 times as fast as Photoshop

MedianDemo

bilateral filtering
Bilateral Filtering
  • A normalized convolution[Tomasi, 1998]
      • Spatial distance
      • Relative difference in intensity
linear intensity bilateral
Linear-Intensity Bilateral
  • A box spatial and triangular intensity filter
logarithmic intensity bilateral
Logarithmic-Intensity Bilateral
  • Durand, F. and Dorsey, J. 2002. Fast Bilateral Filtering for the Display of High Dynamic Range Images. SIG’02
comparison
Comparison

BilateralDemo