---
title: Corner Detection
categories: session
---


**Reading** Ma 4.3 and 4.A and Szelisky Chapter 3.2

**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.

**Briefing** [Corner Lecture]()

# Exercises

## Learning Objectives

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 use the same setup as in [Image Filters]().

## Exercise 1

Learning goal: 1D derivatives and 1D convolutions

**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?


<details>
  <summary>Hint (Click to expand)</summary>
  </br>
```python
# Load image im as usual and convert to greyscale.
row = im[50,:]   # Row number 50
n = row.size

import matplotlib.pyplot as plt
plt.plot(range(n),row,color="blue",label="Pixel Row")
plt.show()
```
</details>

**Part 2**
Convolve the row with a $[1/2,-1/2]$ kernel and visualize.<br>
Do the values make sense?

<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>
&nbsp;

<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>
&nbsp;
<details>
  <summary>Cross-correlation (Click to expand)</summary>
  </br>
  If you use cross-correlation instead of convolution, flip the kernel.
</details>
&nbsp;

## Exercise 2

Learning goals: 2D Derivatives

**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</summary>
  </br>
  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>

+ 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?

**Part 2**<br>
Show $I_x$ as an image.
You probably have negative pixel values, so you may have to 
scale the image.

+ 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.

**Part 3**<br>
Repeat Parts 1 and 2 with vertical derivation, i.e. use $G_y$ instead
of $G_x$.

$$G_y = 1/8 \begin{bmatrix}
        1 & 2 & 1 \\
        0 & 0 & 0 \\
        -1 & -2 & -1 \\
      \end{bmatrix}
$$

+ Compare the images.  What differences can you make out?

## Exercise 3

Learning Goal: find rotation invariant heuristics for edges 

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.

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.

### 3.1

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 $\mathbf{x}$.

Note that this is not a *pseudo-image*.  For each $\mathbf{x}$ we
have $2\times2$ matrix and not just a scalar.

### 3.2

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.

You can make matrices containing, for each $\mathbf{x}$ 

+ 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 $G(\mathbf{x})$

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?

## Exercise 4

Learning goals: Introduction to Harris corner detector

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.



**Part 1**<br>

Consider the grayscale image we have been working with, and the gradient magnitude from 3.3.<br>
Where do you expect to find corners?

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
  cx = cv.cornerHarris(img_gray, bsize, ksize, k)

  T = 0.1 # Threshold
  c_image = img

  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>
&nbsp;

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.