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pages/CSE559A/CSE559A_L18.md

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

pages/CSE559A/_meta.js

@@ -20,4 +20,5 @@ export default {
     CSE559A_L15: "Computer Vision (Lecture 15)",
     CSE559A_L16: "Computer Vision (Lecture 16)",
     CSE559A_L17: "Computer Vision (Lecture 17)",
+    CSE559A_L18: "Computer Vision (Lecture 18)",
 }