Fast Gaussian Filtering Algorithm Using Splines Kentaro Imajo (M2, - - PowerPoint PPT Presentation

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@imos http://imoz.jp/ /23 Fast Gaussian Filtering Algorithm Using Splines Kentaro Imajo (M2, Kyoto University) November 12, International Conference on Pattern Recognition 2012 1 /23 Contents 1. Background and Goal 2. Approximation of

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@imos http://imoz.jp/

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Fast Gaussian Filtering Algorithm Using Splines

Kentaro Imajo (M2, Kyoto University)

November 12, International Conference on Pattern Recognition 2012

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Contents

  • 1. Background and Goal
  • 2. Approximation of Gaussian Function
  • 3. 1D Convolution
  • 4. 2D Convolution
  • 5. Outline of Algorithm
  • 6. Experiments
  • 7. Conclusions

November 12, International Conference on Pattern Recognition 2012

Chapter 0. Contents 2

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Background and Goal

Chapter 1 of 7

November 12, International Conference on Pattern Recognition 2012

Chapter 1. Background and Goal 3

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Background

Many algorithms such as SIFT need to compute Gaussian filter.

Chapter 1. Background and Goal

November 12, International Conference on Pattern Recognition 2012

＊ =

2D Gaussian Func.

Figure Image Blurred by 2D Gaussian Filter

convolution

4

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Gaussian Filter

Gaussian Filter is computed by convolutions with 2D Gaussian function. 2D Gaussian Function: where σ is spread.

Chapter 1. Background and Goal

November 12, International Conference on Pattern Recognition 2012

1 √ 2πσ exp

  • −x2 + y2

2σ2

  • Circle

5

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Propose an algorithm to compute one Gaussian-filtered pixel in constant time to area size n (∝ σ2). Naïve method requires O(n) time.

Goal

Chapter 1. Background and Goal

November 12, International Conference on Pattern Recognition 2012

×

Area size is n Target Pixel

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Preceding Methods

  • Naïve method takes O(n) time.
  • FFT method is fast for all pixels,

but it cannot compute pixels apart.

  • Down-sampling method has

large errors. Within 3% error in constant time

Chapter 1. Background and Goal

November 12, International Conference on Pattern Recognition 2012

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Approximation of Gaussian Function

Chapter 2 of 7

Chapter 2. Approximation of Gaussian Function

November 12, International Conference on Pattern Recognition 2012

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Spline

Approximate Gaussian function with a spline, which is written as: where a, b, n are parameters, and (･)+ is max(･, 0). * Coordinates b are control points.

Chapter 2. Approximation of Gaussian Function

November 12, International Conference on Pattern Recognition 2012

˜ ψ(x) =

  • i

ai (x − bi)n

+

9

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  • 15
  • 12.5
  • 10
  • 7.5
  • 5
  • 2.5

2.5 5 7.5 10 12.5 15 80 160 240

˜ ψ(x) = 3(x + 11)2

+ 11(x + 3)2 +

+11(x 3)2

+ 3(x 11)2 +,

  • 2.657 102 exp
  • x2

5.27202

  • .

Approximation

Chapter 2. Approximation of Gaussian Function

November 12, International Conference on Pattern Recognition 2012

10

× × × ×

－

Approximation

－

Gaussian func.

×

Control points

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Evaluation

Approximation error is about 2%. (2D error is about 3%.) Higher order improves more: Approximation error of 4th order approximation is about 0.3%.

Chapter 2. Approximation of Gaussian Function

November 12, International Conference on Pattern Recognition 2012

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˜ ψ(x) = 70(x + 22)4

+ − 624(x + 11)4 + + 1331(x + 4)4 +

−1331(x − 4)4

+ + 624(x − 11)4 + − 70(x − 22)4 +.

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1D Convolution

Chapter 3 of 7

Chapter 3. 1D Convolution

November 12, International Conference on Pattern Recognition 2012

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Convolution

Convolution of a signal and a spline. Key Idea: Splines get discrete by differentiation.

Chapter 3. 1D Convolution

November 12, International Conference on Pattern Recognition 2012

13

  • 12
  • 8
  • 4
4 8 12
  • 240
  • 160
  • 80
80 160 240
  • 12
  • 8
  • 4
4 8 12
  • 25
25
  • 12
  • 8
  • 4
4 8 12
  • 10
  • 5
5 10
  • 12
  • 8
  • 4
4 8 12
  • 12
  • 8
  • 4
4 8 12

Differentiate

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Transformation

Transform convolution into summation.

Chapter 3. 1D Convolution

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( ˜ ψ ∗ I)(x) =

  • ∆x∈Z

˜ ψ(∆x)I(x − ∆x) =

  • ∆x∈Z

m

  • i=0

ai(∆x − bi)n

+

  • I(x − ∆x)

=

m

  • i=0

aiJ(x − bi), J(x) =

  • ∆x∈Z

∆xn

+I(x − ∆x)

Pre-computed Several calculations

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2D Convolution

Chapter 4 of 7

Chapter 4. 2D Convolution

November 12, International Conference on Pattern Recognition 2012

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2D Gaussian func. can be decomposed: Approximation can be written as: where

( ˜ ψ ∗ I)(x, y) =

m

  • i=1

ai

m

  • j=1

ajJ(x − bi, y − bi)

2D Convolution

Chapter 4. 2D Convolution

November 12, International Conference on Pattern Recognition 2012

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exp

  • −x2 + y2

2σ2

  • = exp
  • − x2

2σ2

  • exp
  • − y2

2σ2

  • 20 multiplications and 16 additions

J(x, y) =

(∆x,∆y)∈Z2

+ ∆xn∆ynI(x − ∆x, y − ∆y)

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Evaluation

Chapter 4. 2D Convolution

November 12, International Conference on Pattern Recognition 2012

17

  • 15

15 x

  • 15

15 y

× × × × × × × × × × × × × × × ×

· · · Gaussian — Approximation × Control point

Figure Elevation and control points

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Outline of Algorithm

Chapter 5 of 7

Chapter 5. Outline of Algorithm

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Outline of Algorithm

  • 1. Pre-compute J once an image

in linear time to the area of the image.

  • 2. Compute a Gaussian filtered value

in constant time (tens operations) for any combinations a, b.

Chapter 5. Outline of Algorithm

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( ˜ ψ ∗ I)(x, y) =

m

  • i=1

ai

m

  • j=1

ajJ(x − bi, y − bi) J(x, y) =

(∆x,∆y)∈Z2

+ ∆xn∆ynI(x − ∆x, y − ∆y)

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Experiments

Chapter 6 of 7

Chapter 6. Experiments

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Experiments

Chapter 6. Experiments

November 12, International Conference on Pattern Recognition 2012

21 100 101 102 103 104 # of application pixels of the filter 10−8 10−7 10−6 10−5 10−4 Computational time (sec.) — Proposed method - - - Na¨ ıve method

Figure Computational time for one pixel on average

For 70+ pixels

  • ur algorithm is faster

Diameter of 9 has 69 pixels

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Conclusions

Final Chapter

Chapter 7. Conclusions

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Conclusions

With pre-computing once an image, the proposed algorithm computes any size of the Gaussian filter

  • in constant time to size,
  • faster than naïve for 70+ pixels,
  • within 3% error.

Chapter 7. Conclusions

November 12, International Conference on Pattern Recognition 2012

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