Geometric Computer Vision

Notes for CMSC733 Classical and Deep Learning Approaches for Geometric Computer Vision taught by Prof. Yiannis Aloimonos.

• Course webpage

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

See Wikipedia: 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

\(\displaystyle \begin{bmatrix} u \\ v \\ w \end{bmatrix} = \begin{bmatrix} f & 0 & 0 & 0\\ 0 & f & 0 & 0\\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x_s \\ y _s \\ z_s \\ 1 \end{bmatrix} \)