v2009.01.01 - Convex Optimization

332 CHAPTER 4. SEMIDEFINITE PROGRAMMING
vec −1 f
Figure 88: Aliasing of Shepp-Logan phantom in Figure 86 resulting from
k-space subsampling pattern in Figure 87. This image is real because binary
mask Φ is vertically and horizontally symmetric. It is remarkable that the
phantom can be reconstructed, by convex iteration, given only U 0 = vec −1 f .
Express an image-gradient estimate
⎡
U ∆
∇U =
∆ ⎢ U ∆ T
⎣ ∆ U
∆ T U
⎤
⎥
⎦ ∈ R4n×n (763)
that is a simple first-order difference of neighboring pixels (Figure 89) to
the right, left, above, and below. 4.49 ByA.1.1 no.25, its vectorization: for
Ψ i ∈ R n2 ×n 2
vec ∇U =
⎡
⎢
⎣
⎤
∆ T ⊗ I
∆ ⊗ I
I ⊗ ∆
I ⊗ ∆ T
⎥
⎦ vec U ∆ =
⎡
⎢
⎣
Ψ 1
Ψ T 1
Ψ 2
Ψ T 2
⎤
⎥
⎦ vec U ∆ = Ψ vec U ∈ R 4n2 (764)
where Ψ∈ R 4n2 ×n 2 . A total-variation minimization for reconstructing MRI
4.49 There is significant improvement in reconstruction quality by augmentation of a
normally two-dimensional image-gradient to a four-dimensional estimate per pixel by
inclusion of two polar directions. We find small improvement on real-life images, ≈1dB
empirically, by further augmentation with diagonally adjacent pixel differences.