69 lines
1.6 KiB
Markdown
69 lines
1.6 KiB
Markdown
# CSE559A Lecture 18
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## Continue on Harris Corner Detector
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Goal: Descriptor distinctiveness
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- We want to be able to reliably determine which point goes with which.
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- Must provide some invariance to geometric and photometric differences.
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Harris corner detector:
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> Other existing variants:
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> - Hessian & Harris: [Beaudet ‘78], [Harris ‘88]
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> - Laplacian, DoG: [Lindeberg ‘98], [Lowe 1999]
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> - Harris-/Hessian-Laplace: [Mikolajczyk & Schmid ‘01]
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> - Harris-/Hessian-Affine: [Mikolajczyk & Schmid ‘04]
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> - EBR and IBR: [Tuytelaars & Van Gool ‘04]
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> - MSER: [Matas ‘02]
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> - Salient Regions: [Kadir & Brady ‘01]
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> - Others…
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### Deriving a corner detection criterion
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- Basic idea: we should easily recognize the point by looking through a small window
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- Shifting a window in any direction should give a large change in intensity
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Corner is the point where the intensity changes in all directions.
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Criterion:
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Change in appearance of window $W$ for the shift $(u,v)$:
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$$
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E(u,v) = \sum_{x,y\in W} [I(x+u,y+v) - I(x,y)]^2
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$$
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First-order Taylor approximation for small shifts $(u,v)$:
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$$
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I(x+u,y+v) \approx I(x,y) + I_x u + I_y v
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$$
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plug into $E(u,v)$:
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$$
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\begin{aligned}
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E(u,v) &= \sum_{(x,y)\in W} [I(x+u,y+v) - I(x,y)]^2 \\
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&\approx \sum_{(x,y)\in W} [I(x,y) + I_x u + I_y v - I(x,y)]^2 \\
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&= \sum_{(x,y)\in W} [I_x u + I_y v]^2 \\
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&= \sum_{(x,y)\in W} [I_x^2 u^2 + 2 I_x I_y u v + I_y^2 v^2]
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\end{aligned}
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$$
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Consider the second moment matrix:
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$$
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M = \begin{bmatrix}
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I_x^2 & I_x I_y \\
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I_x I_y & I_y^2
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\end{bmatrix}=\begin{bmatrix}
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a & 0 \\
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0 & b
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\end{bmatrix}
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$$
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If either $a$ or $b$ is small, then the window is not a corner.
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