O(N) operations. Discrete Wavelet basis: no closed-form expression. Discrete Computations Discrete orthogonal basis { m } of CN . m [n] = 1 N e2i N nm f = m f, m m
a[m] = ⇥f, m ⇤ R coding transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
a[m] = ⇥f, m ⇤ R coding decoding q[m] Z transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z
R =0.2 bit/pixel f forward Dequantization: ˜ a[m] = sign(q[m]) |q[m] + 1 2 ⇥ T a[m] = ⇥f, m ⇤ R coding decoding q[m] Z ˜ a[m] dequantization transform backward fR = m IT ˜ a[m] m transform Entropic coding: use statistical redundancy (many 0’s). Quantization: q[m] = sign(a[m]) |a[m]| T ⇥ Z ˜ a[m] T T 2T 2T a[m] Quantized q[m] bin T q[m] Z “Theorem:” ||f fM ||2 = O(M ) =⇥ ||f fR ||2 = O(log (R)R )
images are not smooth. But can be compressed e ciently. Shannon interpolation: if Supp( ˆ ˜ f) [ N , N ] 0,1,0,. . . h(t) = sin( t) t Sample and compress simultaneously? Pointwise Sampling and Smoothness ˜ f L2 f RN f[i] = ˜ f(i/N) JPEG-2k
f L2 f RN y RP micro mirrors array resolution CS hardware , ... f Operator K K CS Hardware Model CS is about designing hardware: input signals ˜ f L2(R2). , ,
dim(ker(K)) = N P is huge. Prior information: f is sparse in a basis { m }m . J (f) = Card {m \ | f, m | > } is small. Inversion and Sparsity f Operator K f, m f
i.e. J0 (f) k If P C log(N/k)k then 1-CS reconstruction is exact. Extensions to: noisy observation y = Kf + approximate sparsity f = fk sparse + Theoretical Performance Guaranties
i.e. J0 (f) k If P C log(N/k)k then 1-CS reconstruction is exact. Extensions to: noisy observation y = Kf + approximate sparsity f = fk sparse + Research problem: optimal value of C ? for N/k = 4, C log(N/k) 5. “CS is 5 less e cient than JPEG-2k” Theoretical Performance Guaranties