Semiclassical Sampling and Linear Inverse Problems Sampling meets - PowerPoint PPT Presentation

Semiclassical Sampling and Linear Inverse Problems Sampling meets Microlocal Analysis Plamen Stefanov IAS 2019

The classical Shannon sampling theorem says that if 𝑔�𝑦� has a Fourier � supported in the box �𝐶, 𝐶 � (i.e., it is band‐limited), then transform 𝑔 𝑔�𝑦� is uniquely and stably determined by its samples 𝑔 𝑡𝑙 , 𝑙 ∈ 𝒂 � if the sampling rate 𝑡 satisfies 0 � 𝑡 � 𝜌/𝐶 . More precisely, � 𝑔 𝑦 � � 𝑔 𝑡𝑙 𝜓 𝜌 𝑡 𝑦 � 𝑡𝑙 , 𝜓 𝑦 � � sinc�𝑦 � � �∈𝒂 � � and (unitarity) � 𝑔 � � 𝑡 � � 𝑔 𝑡𝑙 � . �∈𝒂 � � . Then the samples The proof is simple. Think of 𝑔 as the inverse FT of 𝑔 � , more precisely of its periodic 𝑔�𝑡𝑙� are the Fourier coefficients of 𝑔 extension over the lattice 𝐶𝒂 � .

Graph of sinc The red curve is the function reconstructed, indistinguishable from the original. The blue curve is a cubic spline interpolation. Note that the reconstruction has a global character. Here 𝑡 � 0.95 ∗ Nyquist .

In the frequency domain, things look like this: Nyquist condition satisfied (no aliasing). Since �𝐶, 𝐶 ⊂ �𝜌/𝑡, 𝜌/𝑡� , 𝟐 ���/�,�/�� 𝜓 � � 𝑔 � 𝜌 𝜌 �𝐶 𝐶 � 3𝜌 3𝜌 𝑡 𝑡 𝑡 𝑡 � � ; then the samples are its Fourier Take the 2𝜌/𝑡 ‐periodic extension 𝑔 ��� of 𝑔 coefficients: ��� 𝜊 � 𝑡 � � 𝑔 𝑡𝑙 𝑓 ���� � 𝑔 . � � back: Multiply that extension by 𝟐 ��/�,�/� 𝜊 to get 𝑔 � 𝜊 � 𝟐 ��/�,�/� 𝜊 𝑡 � � 𝑔 𝑡𝑙 𝑓 ���� . 𝑔 � Take ℱ �� . The effect of the multiplication is the appearance of ℱ �� of 𝟐 ��,� 𝜊 in a convolution, which is the sinc function. If we have oversampling ( supp�𝑔� strictly in ��𝐶, 𝐶� ), we can choose 𝜓̂ smooth, hence 𝜓 will be in the Schwartz class unlike the sinc function.

Aliasing: If the Nyquist condition is not satisfied: 𝟐 ���,�� � 𝑔 3𝐶 �3𝐶 �𝐶 𝐶 The 2𝐶 ‐periodic extension has overlapping segments. � truncated plus other stuff, i.e., we If we restrict back to ��𝐶, 𝐶� , we get 𝑔 � modulo 2𝐶 . Its inverse FT is not 𝑔 anymore. get 𝑔 This creates aliasing. Frequencies get shifted. If we have a smooth 𝜓̂ instead of 𝟐 ���,�� , we get an FIO, actually (in the sense below).

ℱ Aliasing in 𝐒 � : ℱ Oversampled Undersampled The FT of the �| |𝑔 undersampled 𝑔 on 81x81 grid on 41x41 grid The original consist of two patterns: one higher frequency than the other. First, we sample and reconstruct properly. Next, we undersample the higher frequency pattern but still sample properly the lower frequency one. The reconstruction changes the direction and the frequency of the undersampled pattern. The Fourier transforms demonstrate the shifting (folding) of the frequencies.

We are interested in sampling the data g � 𝐵𝑔 with 𝐵 linear, 𝑕 given, 𝑔 �? We assume that 𝐵 is an FIO, which is true very often: in integral geometry, thermo‐ acoustic tomography, etc. � (the smallest detail of 𝑔 ), how (i) Sampling 𝑩𝒈 : Having an estimate of supp 𝑔 dense should we sample 𝐵𝑔 ? (ii) Resolution limit on 𝒈 , given the sampling rate of 𝑩𝒈 . (iii) Aliasing: if we undersample 𝐵𝑔 , what kind of aliasing we get for 𝑔 ? (iv) Averaged measurements/anti‐aliasing: if we blur the data (either because the detectors are not points and average already or because we want to avoid aliasing), what do we get for 𝑔 ? � knowing supp 𝑔 � and apply the To answer (i), one may say – estimate supp 𝐵𝑔 sampling theorem, we are done. The problem is that this is not straightforward. 𝐵𝑔 may not be even band limited if 𝑔 is.

We look at the problem as an asymptotic one. The “smallest detail” is a small parameter tending to 0. Rescale 𝜊 to 𝜊/ℎ (i.e., 𝜊 � 𝜃/ℎ ) with 0 � ℎ ≪ 1 . This makes it a semi‐classical problem! If 𝜊 � 𝐶/ℎ , then for the rescaled 𝜊 we have 𝜊 � 𝐶 . We want to work in the phase space, i.e., to account for the 𝑦 dependence as well. So band limited now means that 𝑔 � �𝑦� has a compact WF � 𝑔 . This brings us to the first problem we need to study: (v) Semi‐classical sampling. How to sample 𝑔 � �𝑦� with a compact WF � 𝑔 ? • Uniform sampling on rectangular or non‐rectangular lattices? • Non‐uniform sampling? • How many sampling points are enough?

(v) Semi‐classical sampling Theorem (semi‐classical sampling): � Ω w ith WF � 𝑔 ⊂ Ω � �𝐶, 𝐶 � . Then for 𝑡 � 𝜌/𝐶 , Let 𝑔 � ∈ 𝐷 � 𝜌 � 𝑃 𝒯 ℎ � , 𝑔 � 𝑦 � � 𝑔 𝑡ℎ𝑙 𝜓 𝑡ℎ 𝑦 � 𝑡ℎ𝑙 �∈𝒂 � where 𝜓 is a product of sinc functions. Parseval’s equality holds, too, up to 𝑃 ℎ � . Just a rescaled classical version, with error estimates. The condition on WF � 𝑔 is a condition on ℱ � 𝑔 modulo 𝑃 ℎ � . The step size is 𝑡ℎ with 𝑡 � 𝜌/𝐶 . As above, if WF � 𝑔 ⊂ Ω � ��𝐶, 𝐶� � (oversampling), 𝜓 can be made rapidly decreasing. We call the projection Σ � �𝑔� of WF � 𝑔 onto 𝜊 the frequency set of 𝑔 .

(v) Semi‐classical sampling 𝑔 � 𝑔 𝐶 Σ � �𝑔� �𝐶 𝐶 𝑡ℎ � 𝜌ℎ 𝐶 �𝐶 The sampling lattice The Fourier domain

(v) Semi‐classical sampling 𝑔 � 𝑔 𝐶 Σ � �𝑔� �𝐶/2 𝐶/2 𝑡ℎ � 𝜌ℎ 𝐶 �𝐶/2 �𝐶 The sampling lattice The Fourier domain The lattice could be rectangular

(v) Semi‐classical sampling 𝑔 � 𝑔 𝐶 Σ � �𝑔� ‐ 𝐶/2 𝐶/2 𝑡ℎ � 2𝜌ℎ 𝐶 �𝐶 𝐶 The sampling lattice The Fourier domain The previous sampling lattice (rescaled by 2) works for Σ � �𝑔� supported like this. The reason is that we can tile the plane with (non‐intersecting) shifts generated by the vectors �𝐶/2,0� and �0, 𝐶� . Then 𝜓̂ has to be supported there.

(v) Semi‐classical sampling One can use linear transformations. If shifts of Σ � �𝑔� under the translations 𝜊 ↦ 𝜊 � 2𝜌�𝑋 ∗ � �� do not intersect, where det 𝑋 � 0 , then we can sample over the lattice 𝑦 � � 𝑡ℎ𝑋𝑙 , 𝑙 ∈ 𝒂 � with 𝑡 � 1 . This is a periodic but not a rectangular lattice (a parallelogram one). One can reduce the number of samples by taking some partition of unity 𝜓 � and in each one, sampling according to Σ � �𝜓 � 𝑔� . This gets us closer to the important question of non‐uniform sampling . In the classical case, there is a well cited (780 google scholar citations) paper by L ANDAU in Acta, saying, among the rest, that if we can sample uniquely and 𝑀 � ‐ stably some 𝑔 over a possibly non‐uniform set of points then the density of that set must have a Weyl type of lower bound � . This links sampling to proportional to the Lebesgue measure of supp 𝑔 spectral theory.

(v) Semi‐classical sampling We have the following microlocal analog of this result. Theorem. Let �𝑦 � �ℎ�� , 𝑘 � 1, … 𝑂�ℎ� be a set of points in 𝑺 � . Let 𝐿 ⊂ 𝑈 ∗ 𝑺 � be a compact set. If 𝐷 � � � 𝑔 � � � 𝑃 ℎ � 𝑂 ℎ � 𝑔 � 𝑦 � ℎ ��� for every 𝑔 � with WF � 𝑔 ⊂ 𝐿 , then 𝑂 ℎ � 2𝜌ℎ �� Vol�𝐿 ��� ��1 � 𝑝 ℎ � . The main novelty here, besides the semi‐classical setting, is that we relate the number of points needed to the phase volume of 𝑔 � , i.e., to �� (multiplied by Vol�supp 𝑔� ). Vol WF � 𝑔 instead of Vol�supp 𝑔 The proof uses the spectral asymptotics (D IMASSI & S JÖSTRAND ) for the smoothened version of the semiclassical Ψ DO 𝟐 � �𝑦, 𝜊� . In the uniform cases above, we have equality.

(i) Sampling FIOs (i) How to sample 𝐵𝑔 when 𝐵 is an FIO and WF � 𝑔 ⊂ Ω � � 𝜊 � 𝐶� (for example)? In other words, we know an a priori bound of the smallest detail 𝑔 has. How to sample 𝐵𝑔 ? We need to fit WF � 𝐵𝑔 in the smallest product Ω′ � 𝐿 with 𝐿 a box or a parallelogram or a domain satisfying the tiling property w.r.t. some lattice. Then we sample on the reciprocal one. Now, if 𝐵 is a semi‐classical FIO or if WF � 𝐵𝑔 was the classical WF set, then WF � 𝐵𝑔 is related by WF � 𝑔 by the canonical relation 𝐷 of 𝐵 . It turns out that, aside from the zero section, this is also true in this case WF � 𝐵𝑔 ∖ 0 ⊂ 𝐷 ∘ WF � 𝑔 ∖ 0. Then the sampling of 𝐵𝑔 is determined by the geometry of C ∘ Ω � 𝜊 � 𝐶 .