B-Spline Channels

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B-Spline Channels Channel Smoothing

• Michael Felsberg
• Computer Vision LaboratoryLinköping
  UniversitySWEDEN

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General Idea of Channels

• Encode single value (linear or modular) in N-D
  coefficient vector (channel vector)
• Locality of encoding
• Similar values in same coefficients
• Dissimilar values in different coefficients
• Stability by smooth, monopolar basis functions
• Small changes of value lead to small changes of
  coefficients
• Non-negative coefficients

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Example for Single Value
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Example for Multiple Values
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Overview

• Encoding with quadratic B-splines
• Decoding strategies
• Relation to kernel-density estimation
• Relation to robust M-estimation
• Channel smoothing
• Applications

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Quadratic B-Splines
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B-Splines Encoding

• The value of the nth channel at x is obtained by
• Encoding in practice
• mround(f)
• cm-1(f-m-0.5)2 /2
• cm0.75-(f-m)2
• cm1(m-f-0.5)2 /2

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Example
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Linear Decoding

• Normalized convolution of the channel vector
• Choice of n by heuristics
• Largest denominator (3-box filter)
• Additional local maximum

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Quantization Effect
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Quadratic Decoding I

• Idea detect local maximum of B-spline
  interpolated channel vector
• Step 1 recursive filtering to obtain
  interpolation coefficients

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Quadratic Decoding II

• Step 2 detect zeros

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Quadratic Decoding III

• Step 3 compute energy
• Step 4 sort the decoded values according to
  their energy(the energy represents the
  confidence)

The decoded values must be shiftedand rescaled
to the original interval
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Quantization Effect
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Kernel Density Estimation I

• Given several realizations of a stochastic
  variable (samples of the pdf)
• Goal estimate pdf from samples
• Method convolve samples with a kernel function

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Kernel Density Estimation II

• Requirements for kernel function
• Non-negative
• Integrates to one
• Expectation of estimate

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Relation to C.R.

• Adding channel representation of several
  realizations corresponds to a sampled kernel
  density estimation
• Ideal interpolation with B-splines possible!

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L2 vs. Robust Optimization

• Outliers are critical for L2 optimization
• Idea of robust estimation
• error norm is saturated for outliers
• Influence function becomes zero for outliers

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Robust Error Norm
E
f - f0
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Robust Influence Function
E
f - f0
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Influence Function of C.R.
Obtained from lineardecoding
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Error Norm of C.R.
Obtained by integrating the influence function
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Channel Smoothing
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Channel Smoothing Example

• Discontinuity is preserved
• Constant and linear regions are correctly
  estimated

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Stochastic Signals

• Stochastic signal single realization of a
  stochastic process
• Ergodicity assumption
• averaging over several realizations at a single
  point
• can be replaced with
• averaging over a neighborhood of a single
  realization

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Ergodicity C.S.

• Ergodicity often not fulfilled for signals /
  features, but trivial for channels
• Ergodicity of channels implies that averaging of
  channels corresponds to (sampled) kernel density
  estimation

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Quantization Effect and C.S.
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Outlier Rejection in C.S.
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Applications

• Image denoising
• Infilling of information
• Orientation estimation
• Edge detection
• Corner detection
• Disparity estimation

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Image Denoising
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Infilling of Information
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Orientation Estimation
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Corner Detection
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Corner Detection
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Corner Detection
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Disparity Estimation
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Disparity Estimation
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Further Reading

• B-Spline Channel Smoothing for Robust
  EstimationFelsberg, M., Forssén, P.-E., Scharr,
  H. LiTH-ISY-R-2579January, 2004