Linear inverse problems with nonnegativity constraints: Singularity of optimisers

Figure 1.  An example of map $v \mapsto d(1, v) $ which fulfills the assumptions above. In particular, the derivative tends to −∞ when v approaches zero, and the divergence is zero only at the point v = 1. Note that this is a plot of the function $v \mapsto d_\beta(1, v) $ (see subsubsection 2.3.1) for β = 1.2

Figure 2.  Example of shifts with $ P(x) : = 1 -x^4 $, $ a = -1, b = 1 $, $ h = \frac{2}{5} $, shown over the compact $ K = [0,3] $. The resulting functions satisfy Assumption (C), as per Corollary 4.2

Figure 3.  Two views of the functions described in subsection 4.2: here the two detectors are the line segment $ [u,v] $ with $ u = 1+2i $, $ v \sim -0.7 + 1.4i $, and the symmetric line segment $ [\bar u, \bar v] $ with respect to the real axis. We only plot the function on half of its domain since it is symmetric with respect to the real axis. As can be seen, the functions cannot be continuously extended up to the vertices

Figure 4.  Example of PET functions for the regular polygonk $ P_N $, with $ N = 12 $. Outside of the pale blue region, $ a_{7,8} $ vanishes. Outside of the pale green region (a trapezium whose diagonals intersect at the point denoted $ c = c_{0,4} $), the function $ a_{0,4} $ vanishes. At the point $ z $, $ a_{0,4}(z) $ is given by $ \frac{\alpha}{\pi} $

Figure 5.  Case $ t = 1 $. (A) Divergence along iterates. (B) Maximum of reconstruction along iterates. (C) Reconstruction after 100 iterates. (D) Reconstruction after 1000 iterates

Figure 6.  Case $ t = 10^{-1} $. (A) Divergence along iterates. (B) Maximum of reconstruction along iterates. (C) Reconstruction after $ 100 $ iterates. (D) Reconstruction after $ 1000 $ iterates

Figure 7.  The singular solutions of the problem (6) with two detectors, a₁ = 1, a₂(x) = x on the interval K = [0, 1]. There are three distinct regions outside the cone $A \mathcal{M}_{+} $, but only one which intersects the first quadrant $ y \in {\mathbb R}_+^2 $, where the optimal solution is a Dirac, given by ξδ₁ for some $ \xi \geq 0 $

Figure 8.  The solutions for $ y = (0,1) $ of the problem in Figure 7 with total variation regularisation. Each curve corresponds to a different regularisation parameter, labelled by its base-10 logarithm. We see that when the regularisation parameter goes to zero, the computed solution converges to the expected exact solution, which, we see from Figure 7, is µ = .5δ₁, depicted by a blue circle

Figure 9.  Relation between regularisation parameter and the reconstruction of the Shepp--Logan phantom $ \mu $. Here, the data is given by $ y\sim\max(\mathcal{N}(A\mu, \sigma^2),0) $, with $ \sigma = 10 $