^{1}

^{2}

^{2}

^{1}

^{1}

^{2}

We apply the

The rapidly growing Si technology in semiconductor electronics [

Due to the principal loss in phase information the scattering data can be considered to be undersampled and algorithms tailored to this lack of information, like basis pursuit approaches [

The paper is organized as follows. In Section

Exposing crystals to nondestructive coherent X-ray radiation in

Following standard textbooks, for example, [

As the units of lengths and wave vectors are carried by the basis sets the scalar products read, with dimensionless factors

Restricting to a finite grid with ^{1}

Thus the scalar product separates into the

The bold parallelogram represents periodic boundary conditions with, for example,

Of course for the application the formulas are only needed up to

Carrying out the remaining summation over all equidistant bilayers in growth direction

On the contrary, setting the ^{2}

The scattering amplitudes in Fourier domain can be assigned with arbitrary phases. As this leaves the given intensity distribution

The equations for the Kalman filter usually apply to vectors over the field of real numbers

The stochastic behaviour of all the considered quantities is modelled by zero-mean Gaussian distributions with positive definite matrices

The main idea behind Kalman filtering is to invert the observation model (

An alternative form of the covariance cycle in the correction step above reads, in terms of the explicit so-called Kalman gain matrix

The main problem in CS is to minimize

Thus the

Due to vectors over the field

Involving the observation

By inserting (

Using the covariance cycle (^{3}

In the overdetermined case

If

By virtue of the same theorem the combination

So in each iteration step with adaptive

For an example of linear compressive sensing in the framework of Kalman filtering see [

Like the

With the biases

A properly chosen factor

To demonstrate the relative phase recovery from the observed intensity (

As in the original objective the number of bilayers is roughly known we use as initial values

A number of

For an exponential decaying lowering factor

Reconstructed moduli of all scatterers. In the phase domain the values corresponding to vanishing moduli are suppressed. Due to the symmetry of the sensing problem the phases are only reconstructed up to a global phase.

Adding a global phase (slider on the left) verifies that all relative phases are reconstructed correctly. The open rectangles mark the position of occupied lattice sites on the chain. In the original objective this would refer to bilayers and substrate in the nanowire.

To drive the

Note that the

In the nanowire the equidistant bilayers are at fixed positions in growth direction

The 2D reconstruction from (^{4}

The actual scatterers forming the

The real amplitudes are nicely reconstructed (a) after

Without parasitic scatterers the positions are correctly found (b) after

A random distribution of

We aimed to implement nonlinear compressive sensing for complex vectors

Because of the Jacobians building up the sensing matrix ^{1}) occupied linear grid covering all the approximately

As the reconstruction from quadratic constraints seems to depend only little on the explicit algorithm used for the

Vectors over the field

With

The possibility of multiplying matrices

Introducing the length of row or column vectors

Hermitian matrices

If

Thus all Hermitian matrices

The authors declare that they have no competing interests.

This work has been supported in part by the Deutsche Forschungsgemeinschaft under Grants Lo 455/20-1 and Pi 214/38-2.

In solid state physics the reciprocal vectors

However, in the hexagonal lattice there are three possible lateral shifts yielding additional phase factors in (

A completely factorized version of (

At least in linear CS with a constant sensing matrix the resolution of consecutive frequency bins is also to be considered in discrete Fourier transform (cf. [