CURVELETS USING RIDGELETS BY: RON GRINFELD י"ט/כסלו/תשע"ט
POINTS OF DISCUSSION INTRODUCTION POINT DISCONTINUITIES FAILURE OF WAVELETS ON EDGES MOTIVATION THE CURVELET TRANSFORM ANALYSIS SUMMARY EXAMPLES י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
INTRODUCTION Example in 2-D The function: י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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: ? י"ט/כסלו/תשע"ט
THE CURVELET TRANSFORM CvT CvT INCLUDES 4 STAGES: -SUB-BAND DECOMPOSITION -SMOOTH PARTITIONING -RENORMALIZATION -RIDGELET ANALYSIS י"ט/כסלו/תשע"ט
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: י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
SUB-BAND DECOMPOSITION THE CURVELET TRANSFORM CvT SUB-BAND DECOMPOSITION י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
SMOOTH PARTITIONING DEFINE THE GRID OF DYADIC SQUARES Assume w be a smooth windowing function with ‘main’ support of size 2-s2-s. For each square, wQ is a displacement of w localized near Q Multiplying s f with wQ (QQs) produces a smooth dissection of the function into ‘squares’ י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
SMOOTH PARTITIONING ENERGY PARTITION RECONSTRUCTION PARSERVAL RELATION י"ט/כסלו/תשע"ט
SMOOTH PARTITIONING י"ט/כסלו/תשע"ט
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: י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
RIDGELET TILLING י"ט/כסלו/תשע"ט
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: י"ט/כסלו/תשע"ט
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: ? י"ט/כסלו/תשע"ט
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: י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
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 י"ט/כסלו/תשע"ט
LESS THEN 5% OF THE COEFFICIENTS AND YET SO MUCH INFORMATION ABOUT THE EDGES DEFINING THE OBJECTS IN THE IMAGES י"ט/כסלו/תשע"ט
THE END י"ט/כסלו/תשע"ט