Foundations of Data Science

Exercise 11.10 Compute an approximation to the scaling function that comes from the
dilation equation
φ(x) = 1 + √ 3
φ(2x) + 3 + √ 3
φ(2x − 1) + 3 − √ 3
φ(2x − 2) + 1√ 3
φ(2x − 3).
4
4
4
4
Exercise 11.11 Consider f(x) to consist of the semi circle (x − 1 2 )2 + y 2 = 1 and y ≥ 0
4
for 0 ≤ x ≤ 1 and 0 otherwise.
1. Using precision j = 4 find the coefficients for the scale functions and the wavelets
for D 4 defined by the dilation equation
φ(x) = 1 + √ 3
4
φ(2x) + 3 + √ 3
4
φ(2x − 1) + 3 − √ 3
4
2. Graph the approximation to the semi circle for precision j = 4.
Exercise 11.12 What is the set of all solutions to the dilation equation
φ(2x − 2) + 1√ 3
φ(2x − 3)
4
φ(x) = 1 + √ 3
φ(2x) + 3 + √ 3
φ(2x − 1) + 3 − √ 3
φ(2x − 2) + 1√ 3
φ(2x − 3)
4
4
4
4
Exercise 11.13 Prove that if scale functions defined by a dilation equation are orthogonal,
then the sum of the even coefficients must equal the sum of the odd coefficients in the
dilation equation. That is, ∑ c 2k = ∑ c 2k+1 .
k
k
function
= wavelets
acc=32; %accuracy of computation
phit=[1:acc zeros(1,3*acc)];
c1=(1+3^0.5)/4; c2=(3+3^0.5)/4; c3=(3-3^0.5)/4; c4=(1-3^0.5)/4;
for i=1:10
temp=(phit(1:2:4*acc)+phit(2:2:4*acc))/2;
phi2t=[temp zeros(1,3*acc)];
phi2tshift1=[ zeros(1,acc) temp zeros(1,2*acc)];
phi2tshift2=[ zeros(1,2*acc) temp zeros(1,acc)];
phi2tshift3=[ zeros(1,3*acc) temp ];
phit=c1*phi2t+c2*phi2tshift1+c3*phi2tshift2+c4*phi2tshift3;
plot(phit)
figure(gcf)
pause
end plot(phit) figure(gcf) end
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