---
title: Lecture: Corner Detection
categories: lecture
---

# Briefing

## Corners and Feature Points

![Universitetsområdet i Ålesund](Images/ntnuaes1.jpg)

![Universitetsområdet i Ålesund (ny vinkel)](Images/ntnuaes2.jpg)

+ What are distinctive points in the image?
+ Distinctive points can (to some extent) be matched in two different images.

## Corners in Mathematical Terms

+ Luminance (colour) is a function $I(x,y)$ in the co-ordinates $x$ and $y$
+ Corners are sharp changes in colour/luminance.
+ Sharp changes are large values in the derivates of $I$,
    + i.e. a large gradient $\nabla I(x,y)$

## Differentiation

+ Sampled signal $f[x]$.
    + The derivative is only defined on continuous functions $f(x)$.
+ Reconstruct the original signal.
    + Assume that it is bandwidth limited.
+ Consider the Discrete Fourier Transform
    + Gives a Frequency Domain representation
    + The signal represented as a sum of sine waves.
    + Nyquist tells us that we can reconstruct the signal perfectly
      if it is sampled at twice the highest non-zero frequency.
      (At least to samples per wave.)
+ Let $T$ be sampling period
    + $\omega_s=\frac{2\pi}{T}$ is the sampling frequency
+ Ideal reconstruction filter
    + Frequency domain $H(\omega)=1$ between $\pm\pi/T$
    + Time domain
      $$h(x)=\frac{\sin(\pi x/T)}{\pi x/T}$$
+ Apply filter
    + Multiply in frequency domain
    + Convolve in time domain
+ Reconstructed function: $f(x) = f[x]* h(x)$


## Harris Feature Detector

# Debrief