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Announcements

Announcements. Take home quiz given out Thursday 10/23 Due 10/30. Office hours tomorrow at 1:30-2:30. I’ll be around a lot tomorrow, so you can also drop by or send email. Web page updated. Reminder: Fourier Transform. P=(x,y) means P = x(1,0)+y(0,1) Similarly: .

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Announcements

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  1. Announcements • Take home quiz given out Thursday 10/23 • Due 10/30. • Office hours tomorrow at 1:30-2:30. I’ll be around a lot tomorrow, so you can also drop by or send email. • Web page updated.

  2. Reminder: Fourier Transform • P=(x,y) means P = x(1,0)+y(0,1) • Similarly: Note, I’m showing non-standard basis, these are from basis using complex functions.

  3. Orthonormal Basis • ||(1,0)||=||(0,1)||=1 • (1,0).(0,1)=0 • Similarly we use normal basis elements eg: • While, eg:

  4. Convolution Imagine that we generate a point in f by centering h over the corresponding point in g, then multiplying g and h together, and integrating.

  5. Problem with Fourier Transform Images contain different objects, of different sizes, with different types of textures. So interesting features are spatially localized, occur at different scales, and different frequencies.

  6. So, what’s the right representation? • Fourier transform good for accurate rep. of entire image. • We want a representation in which interesting events are captured in one or few elements. • This means spatially localized. • At the same time frequency information needed. • Better rep. for discontinuities.

  7. Localize in a space and frequency Space Frequency In general, we can localize in space or frequency but not both (Heisenberg).

  8. But we can trade-off • Example: Gabor function. • Gaussian times sine or cosine. • Using convolution theorem, fourier transform is a shifted Gaussian. • Model for neurons in visual cortex.

  9. Wavelet Transforms • The mother wavelet, y, is a smooth function: • Zero avg. • ||y|| = 1 • Centered at t = 0. • Wavelet atom is translated and dilated version: We convolve the signal with these wavelets.

  10. Redundancy/Orthogonality • If we apply these at all scales and translations, we represent a 1D signal in 2D. So they are redundant. • And they are not orthogonal. • Of course some discretization is needed.

  11. Wavelet Basis • Subsample wavelets so that an orthogonal set is chosen that spans the space of functions. • Sample scales by factor of 2 • Sample translates so functions orthogonal.

  12. Example, Haar wavelet Translate by multiples of 1, scale by factors of 2. We can verify that these are orthonormal.

  13. Every function representable by Haar wavelets • Function must have zero mean. • Add DC component to basis. • Divide function in half. Haar allows us to represent each half by its average. • Each half is a zero mean function. • Repeat recursively. • In limit, each tiny part of function is represented by constant function at its mean.

  14. Pros and cons of basis • Orthonormal basis good for compression. • Represents signals with independent components. • Output sensitive to translation. • Alternative; apply wavelets at many more translations.

  15. Efficient Transforms • Straightforward transform is to multiply function by each basis function and integrate. • Fast fourier transform reduces cost of fourier transform. • Subband coding efficient wavelet transform.

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