1 / 18

CS448f: Image Processing For Photography and Vision

CS448f: Image Processing For Photography and Vision. Wavelets Continued. Last Time:. Last time we saw the Daubechies filter satisfied the following: fully orthogonal four taps as smooth as possible wavelet filter a simple modification of the scaling filter

scott
Download Presentation

CS448f: Image Processing For Photography and Vision

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CS448f: Image Processing For Photography and Vision Wavelets Continued

  2. Last Time: • Last time we saw the Daubechies filter satisfied the following: • fully orthogonal • four taps • as smooth as possible • wavelet filter a simple modification of the scaling filter • Why did we care about our wavelet basis functions being orthogonal?

  3. Orthogonality • Why did we care about our wavelet basis functions being orthogonal? • Easy to invert • Orthogonal transforms preserve distance • We can probably relax this requirement, provided we get something that’s still easy to invert

  4. Lifting • Let’s construct our filters using the following sequence: • Divide the inputs into evens and odds • Add some function of the odds to the evens • Add some function of the evens to the odds • Repeat as long as you like • Eventually the evens form a coarse layer and the odds form a fine layer • This is easy to invert

  5. Forward Transform Evens Coarse + Input Filter 1 Filter 2 Odds Fine +

  6. Inverse Transform Evens Coarse - Output Filter 1 Filter 2 Odds Fine -

  7. What makes a good fine layer? • Average value is 0 • So filter 1 should probably be something that enforces that.

  8. What makes a good coarse layer? • Why is subsampling bad? • Some pixels in the input count more than others • Each pixel in the input should count equally • E.g. Averaging down • Something should sum up to 1

  9. Let’s track the linear transform

  10. Divide the rows into even and odd

  11. Add a filter of the even rows to the odd rows: [-½ 0 -½]

  12. The odd rows are now a fine layer

  13. Add a filter of the odd rows to the even rows: [ ¼ 0 ¼ ]

  14. Add a filter of the odd rows to the even rows: [ ¼ 0 ¼ ]

  15. Why did I pick ¼?

  16. In the coarse layer, each input pixel now counts equally (sum along columns is constant)

  17. Lifting Evens Coarse + Input Predict Update Odds Fine - • Using an interpolation for the predict filter gives an appropriate fine layer • The update filter can be computed from the predict filter

  18. Wavelets • A coarse/fine decomposition that is fast to compute and takes no more memory than the original • Better or worse than a Laplacian pyramid?

More Related