## 1 Introduction

The task of recovering a complex signal from phaseless magnitude measurements is called the phase retrieval problem. These types of problems appear in many applications such as optics [4, 5] and x-ray crystallography [6, 7]. Here, we are interested in phase retrieval problems arising from ptychography [8], an imaging technique involving a sample illuminated by a coherent and often localized probe of illumination. When the probe interacts with the sample, light is diffracted and a diffraction pattern is detected. The probe, or the sample, is then shifted laterally in space to illuminate a new area of the sample while ensuring there is sufficient overlap between each neighboring shift. The intensity of the diffraction pattern detected at position resulting from the shift of the probe along the sample takes the general form of

(1) |

where is the sample being imaged, is a mask which represents the probe’s incident illumination on (a portion of) the sample, denotes the Hadamard (pointwise) product, is a shift operator, and is a function that describes the diffraction of the probe radiation from the sample to the plane of the detector after possibly passing though, e.g, a lens.

Prior work in the computational mathematics community related to ptychographic imaging has primarily focused on far-field^{1}^{1}1Far-field versus near-field measurements are defined based on the Fresnel number of the imaging system. See, e.g., [9] for details. ptychography (FFP) in which

is the action of a discrete Fourier transform matrix (see, e.g.,

[10, 11, 12, 13, 14, 15]) in (1). Here, in contrast, we consider the less well studied setting of near-field ptychography (NFP) which describes situations where the masked sample is too close to the detector to be well described by the FFP model. See, e.g., [1, 2, 3] for such imaging applications as well as for more detailed related discussions. In all of these NFP applications the acquired measurements can again be written in the form of (1) where is now a convolution operator with a given Point Spread Function (PSF) .Let denote an unknown sample, be a known mask, and be a known PSF, respectively. For the remainder of this paper we will suppose we have noisy discretized NFP measurements of the form

(2) |

where is a circular shift operator , is an additive noise matrix, and

. Throughout this paper we will always index vectors and matrices modulo

unless otherwise stated.### 1.1 Results, Contributions, and Contents

Our main theorem guarantees the existence of a PSF and a locally supported mask with , , for which the measurements (2) can be inverted up to a global phase factor by a computationally efficient and noise robust algorithm. In particular, we prove the following result which we believe to be the first theoretical error guarantee for a recovery algorithm in the setting of NFP.

###### Theorem 1 (Inversion of NFP Measurements).

Choose such that divides . One can construct a PSF and a mask with such that Algorithm 1 below, when provided with input measurements (2

), will return an estimate

of satisfyingHere is an absolute constant^{2}^{2}2In this paper we will use to denote absolute constants which may change from line to line., and denotes the smallest magnitude of any entry in .

Looking at Theorem 1 we can see, e.g., that in the noiseless setting where the output of Algorithm 1 is guaranteed to match the measured signal up to a global phase factor whenever has no zeros.^{3}^{3}3Note that prior work on far-field ptychography assumed that itself was non-vanishing (see e.g. [11, 12]). However, requiring to not vanish is more easily verifiable in practice. Moreover, the method is also robust to small amounts of additive noise. The proof of Theorem 1 consists of two parts: First, in Section 3, we show that a specific PSF and mask choice results in NFP measurements (2) which are essentially equivalent to far-field ptychographic measurements (4) that are known to be robustly invertible by prior work [11, 12, 14]. This guarantees the existence of a PSF and mask which allow for the robust inversion of (2) up to a global phase. However, these prior works all prove error bounds on which scale quadratically in (see, e.g., Corollary 3 in [12] and Theorem 1 in [14]). This motivates the second part of the proof in Section 4, where we use the results of [16] to improve these results so that they only depend linearly on . We also note that the improved dependence on proven in Section 4 for the FFP methods previously analyzed in [11, 12, 14] may be of potential independent interest.

Theorem 1 applies to a specific pair (described precisely in Lemma 2). In order to be able to handle more general and , in Section 5, we also show that the NFP measurements (2) may be recovered via a Wirtinger Flow based algorithm, Algorithm 2. This approach is particularly useful in situations where the mask is non-compactly supported and, unlike Algorithm 1, it allows us to use fewer shifts. Similar to Algorithm 1, Algorithm 2 relies on the observation that the NFP measurements (2) are essentially equivalent to FFP measurements as shown in Section 3. In Section 6, we evaluate Algorithms 1 and Algorithm 2, numerically, both individually and in comparison to one another in the case of locally supported masks. Finally, in Section 7, we conclude with a brief discussion of future work.

## 2 Preliminaries: Prior Results for Far-Field Ptychography using Local Measurements

Notation | Definition | Notes |
---|---|---|

Zero indexing | ||

Vector circular indexing | ||

Matrix circular indexing | ||

Complex inner product | ||

Support | ||

Discrete Fourier transform matrix | ||

Discrete Fourier transform | ||

Discrete inverse Fourier transform | ||

Circular shift | ||

Reversal | ||

Circular convolution | ||

Hadamard (pointwise) product |

Our method, described in Algorithm 1, is based on relating the near-field ptychographic measurements (2) to far-field ptychographic measurements of the form

(3) |

where is a compactly supported mask. If we let , then these measurements can be written as

(4) |

where as above denotes a circular shift of length , i.e., . In [11], phase retrieval measurements of this form are studied when is supported in an interval of length for some . The fast phase retrieval (fpr) method used there relies on using a lifted linear system involving a block-circulant matrix to recover a portion of the autocorrelation matrix . Specifically, letting , the authors define a block-circulant matrix by

(5) |

where the matrices are defined entry-wise by

(6) |

Letting be a vector obtained by subsampling appropriate entries of , the authors show that, in the noiseless setting,

(7) |

(See Equation (9) of [11] for explicit details on the arrangement of the entries.) For properly chosen , the matrix is invertible, and therefore one may solve for by multiplying by , i.e., . Then, one may reshape to recover a matrix whose non-zero entries are estimates of the autocorrelation matrix . One may then obtain a vector which approximates

by angular synchronization procedure such as the eigenvector-based method which we will discuss in Section

2.1.In [11], it is shown that exponential masks defined by

(8) |

lead to a lifted linear system which is well-conditioned and thus to provable recovery guarantees for the method described above. In particular, we may obtain the following upper bound for the condition number of block-circulant matrix obtained when one sets .

###### Theorem 2 (Theorem 4 and Equation (33) in [11]).

The condition number of , the matrix obtained by setting in (6), may be bounded by

Furthermore, can be inverted in

-time and its smallest singular value

is bounded from below by .### 2.1 Angular Synchronization

Inverting as described in the previous subsection allows one to obtain a portion of the autocorrelation matrix . This motivates us to consider angular synchronization, the process of recovering a vector from (a portion of) its autocorrelation matrix (or an estimate ). One popular approach, which we discuss below, is based on upon first entry-wise normalizing this matrix and then taking the lead eigenvector. Specifically, we define a truncated autocorrelation matrix corresponding to the true signal by

(9) |

We also define a truncated autocorrelation matrix corresponding to our estimate, , given by

(10) |

The method from [11] is based upon first solving for and then solving for . If is a good approximation of , then the results proved in [17] show that will be a good approximation of .

Moving forward, prior works [11, 17] effectively decomposed into its phase and magnitude matrices by setting and if with otherwise. One may then write . Note that by construction, if is nonvanishing, then we have and whenever . Letting be the leading eigenvector of and letting be the main diagonal of , the output of the resulting algorithm was then .

###### Example 1.

Let . Then defined as in (10) is given by

If we write , then we may compute

One may verify that the lead eigenvector is and therefore

In Section 4, we will discuss another slightly more sophisticated way for estimating the phases based on Algorithm 3 of [18] which involves taking the smallest eigenvector of an appropriately weighted graph Laplacian. Indeed, this new angular synchronization approach is what ultimately allows for the NFP error bound in Theorem 1 to have improved dependence on signal dimension over prior FFP error bounds in [11, 12, 14].

## 3 Near from Far: Guaranteed Near-Field Ptychographic Recovery via Far-Field Results

In this section, we show how to relate the near-field ptychographic measurements (2) to the far-field ptychographic measurements (4). This will allow us to recover by using methods similar to those introduced in [11]. In order get nontrivial bounds, we will also need to prove the existence of an admissible PSF and mask pair, and , which lead to a well conditioned linear system in (7). In particular, we will present a PSF and mask pair such that the resulting block-circulant matrix, denoted , will have the same condition number as the matrix constructed from the masks defined in (8). Therefore, Theorem 2 will allow us to obtain convergence guarantees for Algorithm 1.

Here, we will set the measurement index set considered in (2) to be where and . The following lemma proves that we can rewrite NFP measurements from (2) as local FFP measurements of the form (4) as long as the mask has local support and the PSF is periodic. It will be based upon defining masks

(11) |

where is the reversal of about its first entry modulo , i.e., Since the masks have compact support, this will then yield a lifted set of linear measurements of the type considered in [11, 12, 14].

###### Lemma 1.

Let and recall the measurements

defined in (2). Suppose that divides , that is periodic, and that satisfies . Then, we may rearrange the measurements (2) into a matrix of FFP-type measurements

(12) |

where is defined as in (11). As a consequence, recovering is equivalent to inverting a block-circulant matrix as described in (5) – (7).

###### Proof.

###### Remark 1.

If we instead restrict the domain on our NFP measurements to then we may remove the assumption that is periodic. In particular, if one substitutes and for some , then one has . Thus, since , , and so we may use the same calculation as above without assuming that is periodic.

Next, in Lemma 2 below, we will show how to choose a mask and PSF such that defined as in (11) and defined as in (8) will only differ by a global phase for each . As a consequence, we obtain the desired result that the block-circulant matrix arising from our the NFP measurements (2) is essentially equivalent (up to a row permutation and global phase shift) to the well-conditioned lifted linear measurement operator considered in Theorem 2.

###### Lemma 2.

Let have entries given by

where . Then for all , satisfies

(13) |

where is defined as in (8). As a consequence, if we let and be the lifted linear measurement matrices as per (5) obtained by setting each in (6) equal to and , respectively, then we will have

(14) |

where is a block diagonal permutation matrix. Thus and have the same singular values and

where denotes the condition number of a matrix.

###### Proof.

Using the definition of the Hadamard product , the circulant shift operator and the reversal operator , we see that

Therefore, inserting the definitions of and above shows that for

For , we have . Thus (13) follows.

To prove (14), let and be the matrices obtained by using the mask in (5) and (6) and let and be the matrices obtained using instead. Then combining (13) and (6) implies that For example, when one may check

(15) | ||||

and one may perform similar computations in the other cases. Since each and have rows and the mapping is a bijection on we see that each may be obtained by permuting the rows of (and that the permutation does not depend on ). Therefore, there exists a block diagonal permutation matrix such that . Finally, the condition number bound for now follows from Theorem 2 and the fact that permuting the rows of a matrix does not change its condition number or any of its singular values.∎

Lemma 1 above demonstrates how to recast NFP problems involving locally supported masks and periodic PSFs as particular types of FFP problems. Then, Lemma 2 provides a particular PSF and mask combination for which the resulting FFP problem can be solved by inverting a well-conditioned linear system. Together they imply that, for properly chosen and , one may robustly invert the measurements given in (2) by first recasting the NFP data as modified FFP data and then using the BlockPR approach from [11, 12, 14].
This is the main idea behind Algorithm 1. However, this approach will lead to theoretical error bounds which scale *quadratically* in . To remedy this, the final step of Algorithm 1 uses an alternative angular synchronization method (which originally appeared in [18]) based on a weighted graph Laplacian as opposed to previous works which used methods based on, e.g., the methods outlined in Section 2.1. As we shall see in the next section, this will allow us to obtain bounds in Theorem 1
which depend linearly in rather than quadratically.

## 4 Error Analysis for Algorithm 1

In this section, we will prove our main result, Theorem 1, which provides accuracy and robustness guarantees for Algorithm 1. For , we write its entry as and let and so that we may decompose as

(16) |

The following lemma upper bounds the total estimation error it terms of its phase and magnitude errors. For a proof, please see Appendix A.

###### Lemma 3.

Let be decomposed as in (16), and similarly let be decomposed . Then, we have that

(17) |

In light of Lemma 3, to bound the total error of our algorithm, it suffices to consider the phase and magnitude errors separately. In order to bound , we may utilize the following lemma which is a restatement of Lemma 3 of [11].

###### Lemma 4 (Lemma 3 of [11]).

Let denote the smallest singular value of the lifted measurement matrix from line 1 of Algorithm 1. Then,

Having obtained Lemma 4, we are now able to prove the following theorem bounding the total estimation error.

###### Theorem 3.

Let and be the admissible PSF, mask pair defined in Lemma 2. Then, we have that

###### Proof.

###### Remark 2.

In order to bound , we will need a few additional definitions. As in (9), let denote the partial autocorrelation matrix corresponding to the true signal and as in (10), and let denote the partial autocorrelation matrix corresponding to , i.e., the matrix obtained in step 3 of Algorithm 1). Let be a weighted graph whose vertices are given by , whose edge set is taken to be the set of such that and , and whose weight matrix is defined entrywise by

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