2017/01/15 by Andrey Akinshin, Akinshin, Andrey, Gil Goldman +3
Medicine · Physics and Astronomy · #65H10 #65J22 #94A12 #Advanced MRI Techniques and Applications #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Laser-Matter Interactions and Applications #Medical Imaging Techniques and Applications
paper · pdf · doi:10.48550/arxiv.1701.04058
openalex publication_date 2017/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a reconstruction problem for ``spike-train'' signals F of an a\npriori known form F(x)=\∑j=1daj\δ\(x-xj\), from\ntheir moments mk(F)=\∫ xkF(x)dx. We assume that the moments mk(F),\nk=0,1,\…,2d-1, are known with an absolute error not exceeding \ε >\n0. This problem is essentially equivalent to solving the Prony system\n\∑j=1d ajxjk=mk(F), k=0,1,\…,2d-1. We study the ``geometry of\nerror amplification'' in reconstruction of F from mk(F), in situations\nwhere the nodes x1,\…,xd near-collide, i.e. form a cluster of size h\n\≪ 1. We show that in this case, error amplification is governed by certain\nalgebraic varieties in the parameter space of signals F, which we call the\n``Prony varieties''. Based on this we produce lower and upper bounds, of the\nsame order, on the worst case reconstruction error. In addition we derive\nseparate lower and upper bounds on the reconstruction of the amplitudes and the\nnodes. Finally we discuss how to use the geometry of the Prony varieties to\nimprove the reconstruction accuracy given additional a priori information.\n