Geometric Computer Vision
Notes for CMSC733 Classical and Deep Learning Approaches for Geometric Computer Vision taught by Prof. Yiannis Aloimonos.
Convolution and Correlation
See Convolutional neural network.
Traditionally, fixed filters are used instead of learned filters.
Edge Detection
Two ways to detect edges:
- Difference operators
- Models
Image Gradients
- Angle is given by \(\displaystyle \theta = \arctan(\frac{\partial f}{\partial y}, \frac{\partial f}{\partial x})\)
- Edge strength is given by \(\displaystyle \left\Vert (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}) \right\Vert\)
Sobel operator is another way to approximate derivatives:
\(\displaystyle
s_x =
\frac{1}{8}
\begin{bmatrix}
-1 & 0 & 1\\
-2 & 0 & 2\\
-1 & 0 & 1
\end{bmatrix}
\) and
\(\displaystyle
s_y =
\frac{1}{8}
\begin{bmatrix}
1 & 2 & 1\\
0 & 0 & 0\\
-1 & -2 & -1
\end{bmatrix}
\)
You can smooth a function by convolving with a Gaussian kernel.
- Laplacian of Gaussian
- Edges are zero crossings of the Laplacian of Gaussian convolved with the signal.
Effect of \(\displaystyle \sigma\) Gaussian kernel size:
- Large sigma detects large scale edges.
- Small sigma detects fine features.
- Scale Space
- With larger sigma, the first derivative peaks (i.e. zero crossings) can move.
- Close-by peaks can also merge as the scale increases.
- An edge will never split.
Subtraction
- Create a smoothed image by convolving with a Gaussian
- Subtract the smoothed image from the original image.
Finding lines in an image
Option 1: Search for line everywhere.
Option 2: Use Hough transform voting.
Hough Transform
Duality between lines in image space and points in Hough space.
Equation for a line in \(\displaystyle d = x \cos \theta + y \sin \theta\).
for all pixels (x,y) on an edge:
for all (d, theta):
if d = x*cos(theta) + y*sin(theta):
H(d, theta) += 1
d, theta = argmax(H)
- Hough transform handles noise better than least squares.
- Each pixel votes for a line in the Hough space. The line in the image space is the intersection of lines in the Hough space.
- Extensions
- Use image gradient.
- Give more votes for stronger edges
- Change sampling to give more/less resolution
- Same procedure with circles, squares, or other shapes.
- Hough transform for curves
Works with any curve that can be written in a parametric form.
Finding corners
\(\displaystyle C = \begin{bmatrix} \sum I_x^2 & \sum I_x I_y\\ \sum I_x I_y & \sum I_y^2 \end{bmatrix} \)
Consider \(\displaystyle C = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix} \)
Theoretical model of an eye
- Pick a point in space and the light rays passing through it.
- Pinhole cameras
- Abstractly, a box with a small hole in it.
Homography
Cross-ratio
Solving for homographies
Given 4 correspondences, you can solve for a homography.
Point and line duality
Points on the image correspond to lines/rays in 3D space.
The cross product of these correspond to a plane.
Calibration
Central Projection
\(\displaystyle \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} f & 0 & 0 & 0\\ 0 & f & 0 & 0\\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_s \\ y _s \\ z_s 1 \end{bmatrix} \)