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---
title: Corner Detection
categories: session
---
# Briefing
## Corners and Feature Points
**Reading** Ma 4.3 and 4.A and Szelisky Chapter 3.2
## Differentiation
**Warning** Ma starts Chapter 4 by discussing *tracking*,
which means that motion as a function of time is considered as well
as the image as a function of spatial co-ordinates.
This is a lot of concepts and quantities to process at the same time.
We will instead start by discussing features in a still image.
When we have a good idea of what features are and how they behave,
we shall introduce motion.
## Harris Feature Detector
**Briefing** [Corner Lecture]()
# Exercises
## Setup
## Learning Objectives
We will use scipy for a small part of the exercises,
if you haven't already, run 'pip install scipy'.
1. What makes a feature in visual terms?
2. What makes a feature in mathematical terms?
3. How do we differentiate a sampled signal?
4. How does the Harris corner detector work?
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.
We use the same setup as in [Image Filters]().
```python
# Imports
import cv2 as cv
import numpy as np
## Exercise 1
# Capture a single frame
vid = cv.VideoCapture(0)
image = None
Learning goal: 1D derivatives and 1D convolutions
# Capture frames until we click the space button, use that image
while True:
_, frame = vid.read()
**Part 1**<br>
Extract one row from the grayscale image, and visualize it (e.g. with matplotlib.pyplot).<br>
Does the values correspond to what you would expect from the row?
# 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
<details>
<summary>Hint (Click to expand)</summary>
</br>
```python
# 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()
# Load image im as usual and convert to greyscale.
row = im[50,:] # Row number 50
n = row.size
# Then we save the image to the directory with name "frame.jpg"
cv.imwrite(str(p / "frame.jpg"), image)
import matplotlib.pyplot as plt
plt.plot(range(n),row,color="blue",label="Pixel Row")
plt.show()
```
Now we convert the image to gray-scale
```python
# Convert frame to grayscale
image_gray = cv.cvtColor(image, cv.COLOR_BGR2GRAY)
</details>
# Save gray to images
cv.imwrite(str((p / "gray.jpg")), image_gray)
```
**Part 2**
Convolve the row with a $[1/2,-1/2]$ kernel and visualize.<br>
Do the values make sense?
## Exercise 1
<details>
<summary>Manual with Numpy (Click to expand)</summary>
</br>
To implement convolution manually using a loop over a numpy array, you can
create a new 1D array with `np.zeros(<shape>)` and fill it in by
iterating over the row and the kernel.
Remember that the resulting array should be smaller than the original.
</details>
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.
<details>
<summary>Using the API (Click to expand)</summary>
</br>
If you are not able to get a result using numpy, use scipy.signal
(can also be used to compare your result):
```python
from scipy import signal
row_d = signal.convolve(row, kernel)
# or
# row_d = signal.correlate(row, cv.flip(kernel, -1))
```
</details>
<details>
<summary>Cross-correlation (Click to expand)</summary>
</br>
If you use cross-correlation instead of convolution, flip the kernel.
</details>
This can be done using scipy.signal.convolve2d ( https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.convolve2d.html )
## Exercise 2
(Note: For a larger challenge you can also try implementing your own algorithm for the convolve function using numpy)
Learning goals: 2D Derivatives
Code answers from here on will be collapsed, we recommend that you try to
implement them yourself before reading an answer.
**Part 1**<br>
Apply the sobel operator
$$G_x = 1/8 \begin{bmatrix}
1 & 0 & -1 \\
2 & 0 & -2 \\
1 & 0 & -1 \\
\end{bmatrix}
$$
to the entire grayscale image
using `scipy.signal.convolve2d`, `scipy.signal.correlate2d` or `cv.filter2d`.
Either show the image or write it to file with `cv.imwrite`.
This should give you the derivative $I_x$ of the image $I$ with
respect to $x$.
<details>
<summary>Hint (Click to expand)</summary>
<summary>Hint</summary>
</br>
Use `scipy.signal.convolve2d(<image>, <filter>, boundary='symm', mode='same')`
1. Look at the exercises from last Friday. We apply the Sobel filter just like a blurring
filter.
2. You can define the matrix $G_x$ as given above using the `numpy.array()` function.
</details>
<details>
<summary>Solution (Click to expand)</summary>
+ How does this relate to the 1D derivative you did in exercise 1.2?
+ What are the minimum and maximum values of the $I_x$ matrix?
```python
from scipy import signal
**Part 2**<br>
Show $I_x$ as an image.
You probably have negative pixel values, so you may have to
scale the image.
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]])
+ Try to take the absolute values of the luminence values.
+ Try to scale the luminences into the $0\ldots255$ range,
e.g. by adding $255$ and dividing by two.
+ What does the different visualisations tell you?
+ You may scale further to use the full $0\ldots255$ range and thus
increas contrast.
grad_x = signal.convolve2d(image_gray, sobel_x, boundary='symm', mode='same')
grad_y = signal.convolve2d(image_gray, sobel_y, boundary='symm', mode='same')
```
</details>
**Part 3**<br>
Repeat Parts 1 and 2 with vertical derivation, i.e. use $G_y$ instead
of $G_x$.
You should then show the images using cv.imshow or save using cv.imwrite, as we did earlier.
Discuss the results.
$$G_y = 1/8 \begin{bmatrix}
1 & 2 & 1 \\
0 & 0 & 0 \\
-1 & -2 & -1 \\
\end{bmatrix}
$$
You can compare the results of your implementation with the built in function
`cv.Sobel`.
+ Compare the images. What differences can you make out?
```python
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,
```
## Exercise 3
Compute the magnitude and orientation of the derivatives using
```python
magnitude = np.sqrt((grad_x ** 2) + (grad_y ** 2))
orientation = np.arctan2(grad_y, grad_x) * (180 / np.pi) % 180
```
Learning Goal: find rotation invariant heuristics for edges
Show/Save the images, and discuss.
In Exercise 2, we calculated $I_x$ and $I_y$ which give a lot of
edge information. Now we want to aggregate this information over
a Window.
## Exercise 2
Note that $I_x$ and $I_y$ are matrices with the same dimensions as
the original image. The index of an entry in these matrices will
be denoted $\mathbf{x}$ below, and we are going to make more
*pseudo-images* with the same dimensions.
TODO: ?
### 3.1
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.
For every point $\mathbf{x}$ we calculate the matrix
$$G(\mathbf{x}) = \begin{bmatrix}
\sum I_x^2 & \sum I_xI_y \\
\sum I_xI_y & \sum I_y^2
\end{bmatrix},$$
where the summations are made over a window, say a $5\times5$ window,
around $\mathbf{x}$.
```python
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)
```
Note that this is not a *pseudo-image*. For each $\mathbf{x}$ we
have $2\times2$ matrix and not just a scalar.
The cornerEigenValsAndVecs algorithm will
### 3.2
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 $S(p)$ of blockSize `bsize*bsize`,
Calculate the covariation matrix $M$ of the derivatives over the neighborhood as:
For each pixel position $\mathbf{x}$ calculate the eigenvectors/-values
of $G(\mathbf{x})$.
We know that large eigenvalues indicate features, and we want
to visualise this information.
Several variants are possible, and you may need only one or two
to get the picture.
$$M = \begin{bmatrix}
\sum_{S(p)}(dI/dx)^2 & \sum_{S(p)}(dI/dx)(dI/dy) \\
\sum_{S(p)}(dI/dx)(dI/dy) & \sum_{S(p)}(dI/dy)^2
\end{bmatrix}
$$
You can make matrices containing, for each $\mathbf{x}$
After this the eigenvectors and eigenvalues of M are calculated and returned.
+ The maximum of the eigenvalues of $G(\mathbf{x})$
+ The minimum of the eigenvalues of $G(\mathbf{x})$
+ The sum of the eigenvalues of $G(\mathbf{x})$
+ The product of the eigenvalues of $G(\mathbf{x})$
This should result in a `h*w*6` array with values ($\lambda_1,\lambda_2,x_1,y_1,x_2,y_2$) where
1. Scale these matrices so that they can be interpreted as
grey scale images, and visualise them.
2. Compare these visulisations to the edge plots from Exercise 2.
3. What do you see?
* $\lambda_1$,$\lambda_2$ are the non-sorted eigenvalues of M
* $x_1,y_1$ are the eigenvectors corresponding to $\lambda_1$
* $x_2,y_2$ are the eigenvectors corresponding to $\lambda_2$
## Exercise 4
Analyze? TODO: How can students learn from this exercise?
Learning goals: Introduction to Harris corner detector
## Exercise 3
If you have completed Exercise 3, you have done 90% of the
implementation of the Harris detector.
The missing part is the heuristics
$$C(G)=\lambda_{1}\lambda_{2} - k\cdot (\lambda _{1}+\lambda _{2})^{2}
=\det(G)-k\cdot \mathrm {tr} (G)^{2}$$
and the threshold used.
You can choose if you want to complete the implementation
of your very own Harris detector or use OpenCV's implementation.
Recall from the theory that we can calculate the harris-criterion with:
$$\lambda_1 * \lambda_2 - k * (\lambda_1 + \lambda_1)^2$$
Create a new image of the same size as image_gray, and calculate the
harris criterion for all pixels.
<details>
<summary>Solution (Click to expand)</summary>
**Part 1**<br>
A simple (naive) solution, is to create an empty image and
iterate over all pixels like this:
Consider the grayscale image we have been working with, and the gradient magnitude from 3.3.<br>
Where do you expect to find corners?
(Note that this is a very computationally expensive solution,
and takes a long time to complete)
Apply opencv's built in harris-detector.<br>
E.g. `cv.cornerHarris(img_gray, block_size, kernel_size, k)` with `block_size = 2`, `kernel_size = 5` and `k = 0.06`.<br>
Here, block_size is the size of neighbourhood considered for corner detection, kernel_size is the size of the sobel derivative kernel, while k is the harris free parameter.
Make a copy of the original image (with colors) and make circles around any corners found by the harris-detector.<br>
Example code for drawing circles is added below.
<details>
<summary>Code</summary>
</br>
```python
# Create an empty image of size image_gray.shape
c_x = np.empty(image_gray.shape, dtype=np.float32)
cx = cv.cornerHarris(img_gray, bsize, ksize, k)
# Define k, e.g. 0.04
k = 0.04 # Harris free parameter
T = 0.1 # Threshold
c_image = img
# Naive (slow) method, can be vectorized
for i in range(image_gray.shape[0]):
for j in range(image_gray.shape[1]):
lambda_1 = harris_dst[i, j, 0]
lambda_2 = harris_dst[i, j, 1]
c_x[i, j] = lambda_1 * lambda_2 - k * pow((lambda_1 + lambda_2), 2)
for i in range(c_image.shape[0]):
for j in range(c_image.shape[1]):
if c_x[i, j] > T:
cv.circle(c_image, (j, i), 2, (0, 0, 255), 2)
```
</details>
# Debrief
Save/visualize the result, how does it compare with your expectations?
**Part 2**<br>
Adjust the threshold `T` when drawing circles, what does this do?
**Part 3**<br>
Adjust the kernel_size (must be positive and odd),
block_size and/or $k$, and observe how they change the result.
## Optional Exercises
### Laplacian of Gaussian
Test different variations of LoG filters on your test images.
You can construct Gaussians as we did in the last session on
[Image Filters]() and the Laplacian as suggested in the
[Corner Lecture]() today.
How does LoG compare to the techniques you ghave tested above?
### Building on Exercise 1
Repeat part 2 with an image column (instead of row) and the transpose of the kernel.
### Building on Exercise 2
Using the same method as in exercise 1.2 and 1.3, compute the gradient of all
rows and columns, and compute the magnitude (as in exercise 3.3), compare with the
magnitude from 3.3.
