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

2. 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

3. Artifact of wavelet(con’t) DWT

4. Artifact of wavelet(con’t) originalLL band

5. Artifact of wavelet

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

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

8. |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

9. 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)

10. Hölder regularity α can be determined from Wavelet coefficient

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

12. 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

13. High Pass: Y(2n+1) = X(2n+1) - floor( (X(2n) + X(2n+2)) / 2 ) Example of DWT: Lifting-based (5, 3) filter 171172164165142996355691711645563 012345678 3 174 12-3-113 168 144 60 64 Low Pass: Y(2n) = X(2n) + floor( (X(2n-1) + X(2n+1) + 2) / 4 ) Extension Run… 2 2 1414 1414 1 2 2 8 1 4 3 4 1 4 8

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

15. Modifying the Uniform Quantizer to Constrain τ= (β-1)*q/2 -15 -10 0 10 ex: τ=5 q=20 149 15

16. 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

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

18. 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

19. Experimental Results and Comparisons(con’t)

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