# CSE559A Lecture 21

## Continue on optical flow

### The brightness constancy constraint

$$
I_x u(x,y) + I_y v(x,y) + I_t = 0
$$
Given the gradients $I_x, I_y$ and $I_t$, can we uniquely recover the motion $(u,v)$?

- Suppose $(u, v)$ satisfies the constraint: $\nabla I \cdot (u,v) + I_t = 0$
- Then $\nabla I \cdot (u+u', v+v') + I_t = 0$ for any $(u', v')$ s.t. $\nabla I \cdot (u', v') = 0$
- Interpretation: the component of the flow perpendicular to the gradient (i.e., parallel to the edge) cannot be recovered!

#### Aperture problem

- The brightness constancy constraint is only valid for a small patch in the image
- For a large motion, the patch may look very different

Consider the barber pole illusion

### Estimating optical flow (Lucas-Kanade method)

- Consider a small patch in the image
- Assume the motion is constant within the patch
- Then we can solve for the motion $(u, v)$ by minimizing the error:

$$
I_x u(x,y) + I_y v(x,y) + I_t = 0
$$

How to get more equations for a pixel?
Spatial coherence constraint:  assume the pixel’s neighbors have the same (𝑢,𝑣)
If we have 𝑛 pixels in the neighborhood, then we can set up a linear least squares system:

$$
\begin{bmatrix}
I_x(x_1, y_1) & I_y(x_1, y_1) \\
\vdots & \vdots \\
I_x(x_n, y_n) & I_y(x_n, y_n)
\end{bmatrix}
\begin{bmatrix}
u \\ v
\end{bmatrix} = -\begin{bmatrix}
I_t(x_1, y_1) \\ \vdots \\ I_t(x_n, y_n)
\end{bmatrix}
$$

#### Lucas-Kanade flow

Let $A=
\begin{bmatrix}
I_x(x_1, y_1) & I_y(x_1, y_1) \\
\vdots & \vdots \\
I_x(x_n, y_n) & I_y(x_n, y_n)
\end{bmatrix}$

$b = \begin{bmatrix}
I_t(x_1, y_1) \\ \vdots \\ I_t(x_n, y_n)
\end{bmatrix}$

$d = \begin{bmatrix}
u \\ v
\end{bmatrix}$

The solution is $d=(A^\top A)^{-1} A^\top b$

Lucas-Kanade flow: 

- Find $(u,v)$ minimizing $\sum_{i} (I(x_i+u,y_i+v,t)-I(x_i,y_i,t-1))^2$
- use Taylor approximation of $I(x_i+u,y_i+v,t)$ for small shifts $(u,v)$ to obtain closed-form solution

### Refinement for Lucas-Kanade

In some cases, the Lucas-Kanade method may not work well:
- The motion is large (larger than a pixel)
- A point does not move like its neighbors
- Brightness constancy does not hold

#### Iterative refinement (for large motion)

Iterative Lukas-Kanade Algorithm

1. Estimate velocity at each pixel by solving Lucas-Kanade equations
2. Warp It towards It+1 using the estimated flow field
   - use image warping techniques
3. Repeat until convergence

Iterative refinement is limited due to Aliasing

#### Coarse-to-fine refinement (for large motion)

- Estimate flow at a coarse level
- Refine the flow at a finer level
- Use the refined flow to warp the image
- Repeat until convergence

![Lucas Kanade coarse-to-fine refinement](https://notenextra.trance-0.com/CSE559A/Lucas_Kanade_coarse-to-fine_refinement.png)

#### Representing moving images with layers (for a point may not move like its neighbors)

- The image can be decomposed into a moving layer and a stationary layer
- The moving layer is the layer that moves
- The stationary layer is the layer that does not move

![Lucas Kanade refinement with layers](https://notenextra.trance-0.com/CSE559A/Lucas_Kanade_refinement_with_layers.png)

### SOTA models

#### 2009

Start with something similar to Lucas-Kanade

- gradient constancy
- energy minimization with smoothing term
- region matching
- keypoint matching (long-range)

#### 2015

Deep neural networks

- Use a deep neural network to represent the flow field
- Use synthetic data to train the network (floating chairs)

#### 2023

GMFlow

use Transformer to model the flow field

## Robust Fitting of parametric models

Challenges:

- Noise in the measured feature locations
- Extraneous data: clutter (outliers), multiple lines
- Missing data: occlusions

### Least squares fitting

Normal least squares fitting

$y=mx+b$ is not a good model for the data since there might be vertical lines

Instead we use total least squares

Line parametrization: $ax+by=d$

$(a,b)$ is the unit normal to the line (i.e., $a^2+b^2=1$)
$d$ is the distance between the line and the origin
Perpendicular distance between point $(x_i, y_i)$ and line $ax+by=d$ (assuming $a^2+b^2=1$):

$$
|ax_i + by_i - d|
$$

Objective function:

$$
E = \sum_{i=1}^n (ax_i + by_i - d)^2
$$

Solve for $d$ first: $d =a\bar{x}+b\bar{y}$
Plugging back in:

$$
E = \sum_{i=1}^n (a(x_i-\bar{x})+b(y_i-\bar{y}))^2 = \left\|\begin{bmatrix}x_1-\bar{x}&y_1-\bar{y}\\\vdots&\vdots\\x_n-\bar{x}&y_n-\bar{y}\end{bmatrix}\begin{pmatrix}a\\b\end{pmatrix}\right\|^2
$$

We want to find $N$ that minimizes $\|UN\|^2$ subject to $\|N\|^2= 1$
Solution is given by the eigenvector of $U^\top U$ associated with the smallest eigenvalue

Drawbacks:

- Sensitive to outliers

### Robust fitting

General approach: find model parameters 𝜃 that minimize

$$
\sum_{i} \rho_{\sigma}(r(x_i;\theta))
$$

$r(x_i;\theta)$: residual of $x_i$ w.r.t. model parameters $\theta$
$\rho_{\sigma}$: robust function with scale parameter $\sigma$, e.g., $\rho_{\sigma}(u)=\frac{u^2}{\sigma^2+u^2}$

Nonlinear optimization problem that must be solved iteratively

- Least squares solution can be used for initialization
- Scale of robust function should be chosen carefully

Drawbacks:

- Need to manually choose the robust function and scale parameter

### RANSAC

Voting schemes

Random sample consensus: very general framework for model fitting in the presence of outliers

Outline:

- Randomly choose a small initial subset of points
- Fit a model to that subset
- Find all inlier points that are "close" to the model and reject the rest as outliers
- Do this many times and choose the model with the most inliers

### Hough transform