1. CURVELETS USING RIDGELETS
BY: RON GRINFELD
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2. POINTS OF DISCUSSION INTRODUCTION POINT DISCONTINUITIES
FAILURE OF WAVELETS ON EDGES
MOTIVATION
THE CURVELET TRANSFORM
ANALYSIS
SUMMARY
EXAMPLES
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3. EDGE DEFINITION PHYSICAL – ONE OBJECT OCCLUDES ANOTHER OBJECT
INTRODUCTION
EDGE DEFINITION
PHYSICAL – ONE OBJECT OCCLUDES ANOTHER OBJECT
GEOMETRIC – DISCONTINUITIES ALONG CURVES
IMAGE PROCESSING – LUMINANCE UNDERGOING STEP DISCONTINUITIES AT BOUNDRIES
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4. POINT DISCONTINUITIES
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
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5. WAVELETS AND POINT DISCONTINUITIES
INTRODUCTION
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
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6. POINT DISCONTINUITIES ALONG STARIGHT LINES
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
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7. INTRODUCTION
Example in 2-D
The function:
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8. FALIURE OF WAVELETS ON EDGES
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
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9. wavelet coeffs needed, each size N’th largest coeff’s size
INTRODUCITON
wavelet coeffs needed, each size
N’th largest coeff’s size
Rate of approximation
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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: ?
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11. THE CURVELET TRANSFORM CvT
CvT INCLUDES 4 STAGES:
-SUB-BAND DECOMPOSITION
-SMOOTH PARTITIONING
-RENORMALIZATION
-RIDGELET ANALYSIS
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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:
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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
-s f is constructed from D2s and D2s+1
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14. SUB-BAND DECOMPOSITION
THE CURVELET TRANSFORM CvT
SUB-BAND DECOMPOSITION
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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 s f ?
NEXT STEP: SMOOTH PARTITIONING
DIVIDING THE LAYERS INTO SQUARES IN A SPEACIAL WAY
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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
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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 s f with wQ (QQs) produces a smooth dissection of the function into ‘squares’
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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
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19. SMOOTH PARTITIONING ENERGY PARTITION RECONSTRUCTION PARSERVAL RELATION
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20. SMOOTH PARTITIONING
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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:
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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
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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,k are Meyer wavelets for 
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24. RIDGELET TILLING
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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:
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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: ?
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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:
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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
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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
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30. ILUSTRATING THE PRINCIPLE: image = points + lines (curves) sparse image representation can be achieved by: curvelets = wavelets(points) + ridgelets(lines)
Original image
Ridgelets part of the image
Wavelet part of the noisy image
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31. LESS THEN 5% OF THE COEFFICIENTS AND YET SO MUCH INFORMATION ABOUT
THE EDGES DEFINING
THE OBJECTS IN THE
IMAGES
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32. THE END
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