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pages/CSE559A/CSE559A_L17.md

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+# CSE559A Lecture 17
+
+## Local Features
+
+### Types of local features
+
+#### Edge
+
+Goal: Identify sudden changes in image intensity
+
+Generate edge map as human artists.
+
+An edge is a place of rapid change in the image intensity function.
+
+Take the absolute value of the first derivative of the image intensity function.
+
+For 2d functions, $\frac{\partial f}{\partial x}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$
+
+For discrete images data, $\frac{\partial f}{\partial x}\approx \frac{f(x+1)-f(x)}{1}$
+
+Run convolution with kernel $[1,0,-1]$ to get the first derivative in the x direction, without shifting. (generic kernel is $[1,-1]$)
+
+Prewitt operator:
+
+$$
+M_x=\begin{bmatrix}
+1 & 0 & -1 \\
+1 & 0 & -1 \\
+1 & 0 & -1 \\
+\end{bmatrix}
+\quad 
+M_y=\begin{bmatrix}
+1 & 1 & 1 \\
+0 & 0 & 0 \\
+-1 & -1 & -1 \\
+\end{bmatrix}
+$$
+Sobel operator:
+
+$$
+M_x=\begin{bmatrix}
+1 & 0 & -1 \\
+2 & 0 & -2 \\
+1 & 0 & -1 \\
+\end{bmatrix}
+\quad 
+M_y=\begin{bmatrix}
+1 & 2 & 1 \\
+0 & 0 & 0 \\
+-1 & -2 & -1 \\
+\end{bmatrix}
+$$
+Roberts operator:
+
+$$
+M_x=\begin{bmatrix}
+1 & 0 \\
+0 & -1 \\
+\end{bmatrix}
+\quad 
+M_y=\begin{bmatrix}
+0 & 1 \\
+-1 & 0 \\
+\end{bmatrix}
+$$
+
+Image gradient:
+
+$$
+\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)
+$$
+
+Gradient magnitude:
+
+$$
+||\nabla f|| = \sqrt{\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2}
+$$
+
+Gradient direction:
+
+$$
+\theta = \tan^{-1}\left(\frac{\frac{\partial f}{\partial y}}{\frac{\partial f}{\partial x}}\right)
+$$
+
+The gradient points in the direction of the most rapid increase in intensity.
+
+> Application: Gradient-domain image editing
+>
+> Goal: solve for pixel values in the target region to match gradients of the source region while keeping the rest of the image unchanged.
+>
+> [Poisson Image Editing](http://www.cs.virginia.edu/~connelly/class/2014/comp_photo/proj2/poisson.pdf)
+
+Noisy edge detection:
+
+When the intensity function is very noisy, we can use a Gaussian smoothing filter to reduce the noise before taking the gradient.
+
+Suppose pixels of the true image $f_{i,j}$ are corrupted by Gaussian noise $n_{i,j}$ with mean 0 and variance $\sigma^2$.
+Then the noisy image is $g_{i,j}=(f_{i,j}+n_{i,j})-(f_{i,j+1}+n_{i,j+1})\approx N(0,2\sigma^2)$
+
+To find edges, look for peaks in $\frac{d}{dx}(f\circ g)$ where $g$ is the Gaussian smoothing filter.
+
+or we can directly use the Derivative of Gaussian (DoG) filter:
+
+$$
+\frac{d}{dx}g(x,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}
+$$
+
+##### Separability of Gaussian filter
+
+A Gaussian filter is separable if it can be written as a product of two 1D filters.
+
+$$
+\frac{d}{dx}g(x,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}}
+\quad \frac{d}{dy}g(y,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{y^2}{2\sigma^2}}
+$$
+
+##### Separable Derivative of Gaussian (DoG) filter
+
+$$
+\frac{d}{dx}g(x,y)\propto -x\exp\left(-\frac{x^2+y^2}{2\sigma^2}\right)
+\quad \frac{d}{dy}g(x,y)\propto -y\exp\left(-\frac{x^2+y^2}{2\sigma^2}\right)
+$$
+
+##### Derivative of Gaussian: Scale
+
+Using Gaussian derivatives with different values of 𝜎 finds structures at different scales or frequencies
+
+(Take the hybrid image as an example)
+
+##### Canny edge detector
+
+1. Smooth the image with a Gaussian filter
+2. Compute the gradient magnitude and direction of the smoothed image
+3. Thresholding gradient magnitude
+4. Non-maxima suppression
+   - For each location `q` above the threshold, check that the gradient magnitude is higher than at adjacent points `p` and `r` in the direction of the gradient
+5. Thresholding the non-maxima suppressed gradient magnitude
+6. Hysteresis thresholding
+   - Use two thresholds: high and low
+   - Start with a seed edge pixel with a gradient magnitude greater than the high threshold
+   - Follow the gradient direction to find all connected pixels with a gradient magnitude greater than the low threshold
+
+##### Top-down segmentation
+
+Data-driven top-down segmentation:
+
+#### Interest point
+
+Key point matching:
+
+1. Find a set of distinctive keypoints in the image
+2. Define a region of interest around each keypoint
+3. Compute a local descriptor from the normalized region
+4. Match local descriptors between images
+
+Characteristic of good features:
+
+- Repeatability
+  - The same feature can be found in several images despite geometric and photometric transformations 
+- Saliency
+  - Each feature is distinctive
+- Compactness and efficiency
+  - Many fewer features than image pixels
+- Locality
+  - A feature occupies a relatively small area of the image; robust to clutter and occlusion
+
+##### Harris corner detector
+
+
+
+
+
+### Applications of local features
+
+#### Image alignment
+
+#### 3D reconstruction
+
+#### Motion tracking
+
+#### Robot navigation
+
+#### Indexing and database retrieval
+
+#### Object recognition
+
+
+

pages/CSE559A/_meta.js

@@ -19,4 +19,5 @@ export default {
     CSE559A_L14: "Computer Vision (Lecture 14)",
     CSE559A_L15: "Computer Vision (Lecture 15)",
     CSE559A_L16: "Computer Vision (Lecture 16)",
+    CSE559A_L17: "Computer Vision (Lecture 17)",
 }

pages/Math4121/Exam_reviews/Math4121_E2.md

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+# Math4121 Exam 2 Review
+
+Range: Chapter 2-4 of Bressoud's A Radical Approach to Lebesgue's Theory of Integration
+
+## Chapter 2
+
+## Chapter 3
+
+## Chapter 4

pages/Math4121/_meta.js

@@ -3,6 +3,7 @@ export default {
     "---":{
         type: 'separator'
     },
+    Exam_reviews: "Exam reviews",
     Math4121_L1: "Introduction to Lebesgue Integration (Lecture 1)",
     Math4121_L2: "Introduction to Lebesgue Integration (Lecture 2)",
     Math4121_L3: "Introduction to Lebesgue Integration (Lecture 3)",