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