Briefing

Corners and Feature Points

Differentiation

Harris Feature Detector

Exercises

Setup

We will use scipy for a small part of the exercises, if you haven’t already, run ‘pip install scipy’.

We need an image to work with, you can either load an image from disk or capture a new image from the webcam with the below code.

# Imports
import cv2 as cv
import numpy as np

# Capture a single frame
vid = cv.VideoCapture(0)
image = None

# Capture frames until we click the space button, use that image
while True:
    _, frame = vid.read()

    # Display the resulting frame
    cv.imshow("frame", frame)

    # Click space to capture frame
    k = cv.waitKey(1)
    if k % 256 == 32:
        # SPACE pressed
        image = frame
        break

# After the loop release the cap object
vid.release()
# Destroy all the windows
cv.destroyAllWindows()

As we will be working with a lot of different images in this exercise, it is recommended to save it to disk, we can do that with

# Imports
from pathlib import Path

# First we create a path to an images directory
p = Path.cwd() / "images" # <--- current working directory + /images
if not p.is_dir():
    p.mkdir()

# Then we save the image to the directory with name "frame.jpg"
cv.imwrite(str(p / "frame.jpg"), image)

Now we convert the image to gray-scale

# Convert frame to grayscale
image_gray = cv.cvtColor(image, cv.COLOR_BGR2GRAY)

# Save gray to images
cv.imwrite(str((p / "gray.jpg")), image_gray)

Exercise 1

The first exercise is to implement a Sobel-filter. Recall from the theory that we need to implement two 3x3 kernels to convolve with the original image.

This can be done using scipy.signal.convolve2d ( https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.convolve2d.html )

(Note: For a larger challenge you can also try implementing your own algorithm for the convolve function using numpy)

Code answers from here on will be collapsed, we recommend that you try to implement them yourself before reading an answer.

  from scipy import signal

  sobel_x = np.array([[-1, 0, 1],
                      [-2, 0, 2],
                      [-1, 0, 1]])
  sobel_y = np.array([[-1, -2, -1],
                      [0, 0, 0],
                      [1, 2, 1]])

  grad_x = signal.convolve2d(image_gray, sobel_x, boundary='symm', mode='same')
  grad_y = signal.convolve2d(image_gray, sobel_y, boundary='symm', mode='same')

You should then show the images using cv.imshow or save using cv.imwrite, as we did earlier. Discuss the results.

You can compare the results of your implementation with the built in function cv.Sobel (the cv.CV_64F

grad_x_cv = cv.Sobel(image_gray, cv.CV_32F, 1, 0, ksize=3)  # gradient along x axis,
grad_y_cv = cv.Sobel(image_gray, cv.CV_32F, 0, 1, ksize=3)  # gradient along y axis,

Compute the magnitude and orientation of the derivatives using

magnitude = np.sqrt((grad_x ** 2) + (grad_y ** 2))
orientation = np.arctan2(grad_y, grad_x) * (180 / np.pi) % 180

Show/Save the images, and discuss.

Exercise 2

TODO: ?

Step 1: Calculate the covariance matrix of a 3x3 kernel of the derivatives Step 2: Calculate the eigenvals and egeinvecs of the cov matrix (This is not trivial, might be a bit much to expect as an exercise)

As this is not a trivial step, we will calculate the EigenVals and Vecs using cv.cornerEigenValsAndVecs, note that this function also calculates applies a sobel-filter so we will call this function using the grayscale image.

bsize = 3  # the size of neighbourhood considered for corner detection
ksize = 3  # Aperture parameter of the Sobel derivative used.
eigenv = cv.cornerEigenValsAndVecs(image_gray, bsize, ksize)

The cornerEigenValsAndVecs algorithm will

1. Calculate the derivatives dI/dx and dI/dy using the sobel filter (as we did in exercise 1)
2. For each pixel p, take a neighborhood of blockSize bsize*bsize S(p), Calculate the covariation matrix M of the derivatives over the neighborhood as:

\[M = \begin{bmatrix} \sum_{S(p)}(dI/dx)^2 & \sum_{S(p)}(dI/dx)(dI/dy) \\ \sum_{S(p)}(dI/dx)(dI/dy) & \sum_{S(p)}(dI/dy)^2 \end{bmatrix} \]

After this the eigenvectors and eigenvalues of M are calculated and returned.

This should result in a h*w*6 array with (λ1,λ2,x1,y1,x2,y2) where

• λ1,λ2 are the non-sorted eigenvalues of M
• x1,y1 are the eigenvectors corresponding to λ1
• x2,y2 are the eigenvectors corresponding to λ2

Debrief