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Smoothness-Constrained Quantization for Wavelet Image Compression W. Know Carey, Sheila S. Hemami, and Peter N. Heller IEEE TRANSACTIONS ON IMAGE PROCESSING

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Artifact of wavelet Ringing around edge spurious high-frequency in previously smooth areas =>fine scale wavelet coefficients in smooth areas of the image are increased during quantization

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Artifact of wavelet(con’t) DWT

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Artifact of wavelet(con’t) originalLL band

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Artifact of wavelet

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Hölder regularity α = n+r(0<=r<1) K is a constant α is larger =>function are smoother

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Hölder reqularity(con’t) function f(x) f(y) x y

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|f(x)-f(y)| < |f n (x)-f n (y)| |f1(x)-f1(y)| is lager than |f2(x)-f2(y)| => |f1 n (x)-f1 n (y)| growth faster than |f2 n (x)-f2 n (y)| when r(ex:0.1) and |y-x| fixed, n of f1 is smaller than f2 =>n is lager,function is smoother Hölder regularity(con’t) function f1function f2 f1(x) f2(y) x y f1(x) f2(y) x y

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If n of f1 and f2 are eqal, add r (ex:0.2) right-hand side is larger –Smooth function growth slowly => n of f2 is larger than f1 if n is equal,r is larger,function is smoother => Hölder regularity α is larger, function are smoother Hölder regularity(con’t)

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Hölder regularity α can be determined from Wavelet coefficient

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Smooth constrained quantization Quantizer will change Hölder regularity Define α+δ is the Hölder regularity after quantization Whenδ >0 when δ <0

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Smooth constrained quantization(con’t) Change in regularity is not only affected single wavelet coefficient To resolve artifact we discuss before, just discuss δ<0 When |ω k,l | is small, Quantization error affects δ larger

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High Pass: Y(2n+1) = X(2n+1) - floor( (X(2n) + X(2n+2)) / 2 ) Example of DWT: Lifting-based (5, 3) filter Low Pass: Y(2n) = X(2n) + floor( (X(2n-1) + X(2n+1) + 2) / 4 ) Extension Run…

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Modifying the Uniform Quantizer to Constrain Smoothness Original quantizer q q -2/q 2/q

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Modifying the Uniform Quantizer to Constrain τ= (β-1)*q/ ex: τ=5 q=

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Modifying the Uniform Quantizer to Constrain Smoothness Modify quantizer q β*q/2 q 2/q+τ=2/q+ (β-1)*q/2 = β*q/2

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Experimental Results and Comparisons Compare with deadzone –Deadzone ： β is selected by encoder and transmitted to the decoder

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Experimental Results and Comparisons(con’t) β = 1 : the same as the original quantization β = 2 : the reconstructed signal is prevent the coefficient from increasing 1<=β<=2

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Experimental Results and Comparisons(con’t)

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Conclusion Smoothness-constrained quantizer limits the amount by analyzing changes in local regularity can decrease

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