update

pages/CSE559A/CSE559A_L20.md

@@ -62,3 +62,84 @@ For matching:
 ## Local feature matching
 
 ### Matching
+
+Simplest approach: Pick the nearest neighbor. Threshold on absolute distance
+
+Problem: Lots of self similarity in many photos
+
+Solution: Nearest neighbor with low ratio test
+
+![Comparison of keypoint detectors](https://notenextra.trance-0.com/CSE559A/Comparison_of_keypoint_detectors.png)
+ 
+## Deep Learning for Correspondence Estimation
+
+![Deep learning for correspondence estimation](https://notenextra.trance-0.com/CSE559A/Deep_learning_for_correspondence_estimation.png)
+
+## Optical Flow
+
+### Field
+
+Motion field: the projection of the 3D scene motion into the image
+Magnitude of vectors is determined by metric motion
+Only caused by motion
+
+Optical flow: the apparent motion of brightness patterns in the image
+Magnitude of vectors is measured in pixels
+Can be caused by lightning
+
+### Brightness constancy constraint, aperture problem
+
+Machine Learning Approach
+
+- Collect examples of inputs and outputs
+- Design a prediction model suitable for the task
+  - Invariances, Equivariances; Complexity; Input and Output shapes and semantics
+- Specify loss functions and train model
+- Limitations: Requires training the model; Requires a sufficiently complete training dataset; Must re-learn known facts; Higher computational complexity
+
+Optimization Approach
+
+- Define properties we expect to hold for a correct solution
+- Translate properties into a cost function
+- Derive an algorithm to solve for the cost function
+- Limitations: Often requires making overly simple assumptions on properties; Some tasks can’t be easily defined
+
+Given frames at times $t-1$ and $t$, estimate the apparent motion field $u(x,y)$ and $v(x,y)$ between them
+Brightness constancy constraint: projection of the same point looks the same in every frame
+
+$$
+I(x,y,t-1) = I(x+u(x,y),y+v(x,y),t)
+$$
+
+Additional assumptions:
+
+- Small motion: points do not move very far
+- Spatial coherence: points move like their neighbors
+
+Trick for solving:
+
+Brightness constancy constraint:
+
+$$
+I(x,y,t-1) = I(x+u(x,y),y+v(x,y),t)
+$$
+
+Linearize the right-hand side using Taylor expansion:
+
+$$
+I(x,y,t-1) \approx I(x,y,t) + I_x u(x,y) + I_y v(x,y)
+$$
+
+$$
+I_x u(x,y) + I_y v(x,y) + I(x,y,t) - I(x,y,t-1) = 0
+$$
+
+Hence,
+
+$$
+I_x u(x,y) + I_y v(x,y) + I_t = 0
+$$
+
+
+
+