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title: Lecture: Corner Detection
categories: lecture
---
# Briefing
## Corners and Feature Points
+ 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
