CURVELETS USING RIDGELETS

1. CURVELETS USING RIDGELETS BY: RON GRINFELD

2. POINTS OF DISCUSSION INTRODUCTION POINT DISCONTINUITIES FAILURE OF WAVELETS ON EDGES MOTIVATION THE CURVELET TRANSFORM ANALYSIS SUMMARY EXAMPLES

3. INTRODUCTION EDGE DEFINITION PHYSICAL– ONE OBJECT OCCLUDES ANOTHER OBJECT GEOMETRIC– DISCONTINUITIESALONGCURVES IMAGE PROCESSING – LUMINANCE UNDERGOING STEP DISCONTINUITIES AT BOUNDRIES

4. INTRODUCTION POINT DISCONTINUITIES IDEAL REPRESENTATION: HOW TO DO THIS ? -ASK AN ORACLE -DETECT THE EDGES IS IT POSSIBLE ? -NOISY AND BLURRED DATA -RESOURCES TO AN ORACLE

5. WAVELETS AND POINT DISCONTINUITIES -WAVELETS USE DIADIC SCALING -EACH SCALING SQUARE SIZE: -TO GET A SCALE OF RATE 1/n ONE NEEDS TO PERFORM n STAGES OF THE WAVELET PYRAMID -AT EACH STEP ONLY A FEW WAVELETS (C) “FEEL” THE POINT DISCONTINUITY INTRODUCTION

6. INTRODUCTION CONCLUSION: WAVELETS NEED TO KEEP A FACTOR OF ONLY MORE DATA THEN THE IDEAL REPRESENTATION WHEN HANDLING POINT DISCONTINUITIES POINT DISCONTINUITIES ALONG STARIGHT LINES

7. INTRODUCTION Example in 2-D The function:

8. INTRODUCTION FALIURE OF WAVELETS ON EDGES is smooth away from a discontinuity along a curve Note that this defines a line (2-D) discontinuity (edge), and not a point discontinuity At stage j of the wavelet pyramid: squares, size: “feel” the discontinuity Along

9. INTRODUCITON wavelet coeffs needed, each size N’th largest coeff’s size Rate of approximation

10. MOTIVATION TO GET THE BEST RATE OF APPROXIMATION RATE OF APPROXIMATION: BY TAKING THE BEST m TERMS OF THE TRANSFORM WE WANT THE SMALLEST ERROR RATE Fourier: Wavelets: Curvelets: ?

11. THE CURVELET TRANSFORM CvT CvT INCLUDES 4 STAGES: -SUB-BAND DECOMPOSITION -SMOOTH PARTITIONING -RENORMALIZATION -RIDGELET ANALYSIS

12. THE CURVELET TRANSFORM CvT SUB-BAND DECOMPOSITION -THE IMAGE IS DEVIDED INTO s RESOLUTION LAYERS BY A BANK OF SUB-BAND FILTERS: - (ALSO CALLED 0) IS A LOW PASS FILTER AND DEALS WITH FREQUENCIES NEAR ||1 -DEFINED AS: 2s(x) = 24s(22sx)DEALS WITH FREQUENCIES NEAR ||[22s, 22s+2] -THUS EACH SUB-BAND CONTAINS WIDE DETAILS -THE SUB-BAND DECOMPOSITION IS APPLYING A CONVOLUTION OPERATOR:

13. THE CURVELET TRANSFORM CvT SUB-BAND DECOMPOSITION USING THE WAVELET TRANSFORM TO APPROXIMATE SUB-BAND DECOMPOSITION -Using wavelet transform, f is decomposed into S0, D1, D2, D3, … -P0 f is partially constructed from S0 and D1, and may include also D2 and D3 -sf is constructed from D2s and D2s+1

14. THE CURVELET TRANSFORM CvT SUB-BAND DECOMPOSITION

15. THE CURVELET TRANSFORM CvT P0 f IS “SMOOTH” (NO 2-D EDGES) AND THUS CAN BE REPRESENTED USING WAVELETS ( ) BUT WHAT ABOUT THE DISCONTINUITIES ALONG THE CURVES REPRESENTED IN THE LAYERS sf ? NEXT STEP: SMOOTH PARTITIONING DIVIDING THE LAYERS INTO SQUARES IN A SPEACIAL WAY

16. SMOOTH PARTITIONING THE TRICK: THE SUB-BAND FILTERING CAUSED THE EDGES IN LAYER S TO BE WIDE WE WILL SEE A WAY TO DIVIDE THE LAYER INTO SIZE SQUARES, IN A SMART WAY THAT AVOIDS DAMAGING THE EDGES BY THE PARTITION THIS WILL RESOLVE IN THE FOLLOWING ASPECT RATIO OF THE EDGES: width  length2 AND WILL PRODUCE LONG,THIN AND DIRECTION ORIENTED EDGES, TO BE HANDLED BY RIDGELETS

17. SMOOTH PARTITIONING DEFINE THE GRID OF DYADIC SQUARES Assume w be a smooth windowing function with ‘main’ support of size 2-s2-s. For each square, wQ is a displacement of w localized near Q Multiplying sf with wQ (QQs) produces a smooth dissection of the function into ‘squares’

18. SMOOTH PARTITIONING The windowing function w is a nonnegative smooth function ENERGY PARTITION: The energy of certain pixel (x1,x2) is divided between all sampling windows of the grid

19. SMOOTH PARTITIONING ENERGY PARTITION RECONSTRUCTION PARSERVAL RELATION

20. SMOOTH PARTITIONING

21. RENORMALIZATION Renormalization is centering each dyadic square to the unit square [0,1][0,1] For each Q, the operator TQ is defined as: Each square is renormalized:

22. RIDGELETS – THE FINAL STAGE REMEMBER THE RATIO: width  length2 ? WE HAVE ACHIVED IT, AND NOW WE NEED A SET OF WAVELET BASED FUNCTIONS OF WHICH CONTAIN BOTH ANGULAR AND RADIAL LOCATIONS, AND CAN ENJOY THE BENEFIT OF THE RATIO width  length2 THESE FUNCTIONS ARE CALLED RIDGELETS

23. RIDGELETS The ridgelet element has a formula in the frequency domain:where is index to the ridge scale is the location, and are the angular scale and location of the periodic wavelets on the radon domain [-,  ) where j,kare Meyer wavelets for

24. RIDGELET TILLING

25. The energy of the input square sized: And scaled: is defined as: Coefficient’s amplitude N-th largest curvlet coeff. size Letting denote the N-th coeff’s amplitude We get:

26. REMEMBER OWER MOTIVATION? TO GET THE BEST RATE OF APPROXIMATION RATE OF APPROXIMATION: BY TAKING THE BEST m TERMS OF THE TRANSFORM WE WANT THE SMALLEST ERROR RATE Fourier: Wavelets: Curvelets: ?

27. REMEMBER OWER MOTIVATION? TO GET THE BEST RATE OF APPROXIMATION RATE OF APPROXIMATION: BY TAKING THE BEST m TERMS OF THE TRANSFORM WE WANT THE SMALLEST ERROR RATE Fourier: Wavelets: Curvelets:

28. LET US SUMMARIZE THE MAIN CONCEPTS OF THE CURVELET SYSTEM ANISOTROPIC SCALING – CREATING A RATIO OF width  length2 AND BY THAT, CONCENTRAITING THE EDGES, MAKING THEM THIN, STRAIGHT AND DIRECTED THE RIDGELET SYSTEM – A FORMULA CONSTRUCTED OF WAVELETS FITTED TO SCALE AND LOCATION IN BOTH  AND RADON DOMAINS, RECEIVING ANGLE SCALE AND LOCATION AS INPUT

29. SUMMARIZE THE TEXTURE (LUMINANCE) OF NATURAL AND SYNTHETIC IMAGES CAN BE DESCRIBED AS A COLLECTION OF CURVES AND POINTS THE CURVELET SYSTEM DIVIDES IMAGES TO POINTS AND LINES (APPROXIMATING CURVES) AND HANDLES THE POINTS BY WAVELETS AND THE LINES BY RIDGELETS

30. Wavelet partof the noisy image Ridgelets part of the image ILUSTRATING THE PRINCIPLE: image = points + lines (curves) sparse image representation can be achieved by: curvelets = wavelets(points) + ridgelets(lines) Originalimage

31. LESS THEN 5% OF THE COEFFICIENTS AND YET SO MUCH INFORMATION ABOUT THE EDGES DEFINING THE OBJECTS IN THE IMAGES

32. THE END