Raster Grid Pathology and the Cure

RASTER GRID P A THOLOGY AND THE CURE ALBER T F ANNJIANG Abstract. Blind pt yc hograph y is a phase retriev al metho d using m ultiple co ded diffraction patterns from different, o v erlapping parts of the unkno wn extended ob ject illuminated with an unkno wn windo w function. The windo w function is also kno wn as the prob e in the optics literature. As suc h blind pt yc hograph y is an in v erse problem of sim ultaneous reco v ery of the ob ject and the window function given the intensities of the window ed F ourier transform and has a multi-scale set-up in which the prob e has an intermediate scale b etw een the pixel scale and the macro-scale of the extended ob ject. Uniqueness problem for blind ptyc hog- raph y is analyzed rigorously for the raster scan (of a constan t step size τ ) and its v ariants, in which another scale comes into pla y: the ov erlap b etw een adjacent blo cks (the shifted windo ws). The block phases are shown to form an arithmetic progression and the com- plete characterization of the raster scan am biguities is given, including: First, the p erio dic raster grid pathology of degrees of freedom prop ortional to τ 2 and, second, a non-p erio dic, arithmetically progressing phase shift from block to blo ck. Finally irregularly p erturb ed raster scans are sho wn to remov e all ambiguities other than the inheren t ambiguities of the scaling factor and the affine phase am biguit y under the minimum requirement of roughly 50% o v erlap ratio. 1. Introduction In the last decade, pt yc hography has made rapid technological adv ances and dev elop ed in to a p ow erful lensless coherent imaging metho d [18, 36, 40]. Pt ychograph y collects the diffrac- tion patterns from o v erlapping illuminations of v arious parts of the unkno wn ob ject using a lo calized coheren t source (the probe) [27, 30, 31], and builds on the adv ances in synthetic ap erture metho ds to extend phase retriev al to unlimited ob jects and enhance imaging res- olution [5, 19, 25, 26, 29]. Blind ptyc hography go es a step further and seeks to reconstruct b oth the unknown ob ject and the unknown prob e simultaneously [28, 35]. Mathematically , blind ptyc hography is an inv erse problem of sim ultaneous reco v ery of the ob ject and the windo w function (the prob e) giv en the in tensities of the window ed F ourier transform. In pt ychograph y , the windo w function has an intermediate scale betw een the pixel scale and the macro-scale of the extended ob ject. The performance of ptyc hography dep ends on factors such as the t yp e of illumination and the measuremen t scheme, including the amoun ts of o v erlap and prob e p ositions. F or example, the use of randomly structured illuminations can improv e ptyc hographic reconstruction o v er that with regular illuminations [3, 7, 8, 10, 11, 16, 21, 29, 32 – 34, 38, 39]. Exp erimen ts suggest an ov erlap ratio of at least 50%, t ypically 60-70% b etw een adjacen t illuminations for blind pt ychograph y [2, 22]. Optimizing the scan pattern can significantly impro ve the p erformance of ptyc hography and is an imp ortan t part of the exp erimental design. 1 In particular, empirical evidences rep eatedly p oin t to the pitfalls of the raster scan, which is exp erimen tally the easiest to implemen t [14]. Mathematically sp eaking, blind ptyc hograph y with raster scan seeks to recov er b oth the ob ject and the window function (the prob e) as unkno wns but only the 2D window ed F ourier intensities (co ded diffraction patterns) as the data. Raster scanning refers to the p ositions of the windo w function. The raster scan scheme is susceptible to p erio dic artifacts, kno wn as r aster grid p atholo gy , attributed to the regularit y and symmetry of the scan p ositions [35]. On the other hand, to the b est of our kno wledge, raster grid pathology has not b e precisely form ulated and analyzed. The purp ose of the present w ork is a complete analysis of raster grid pathology from the p ersp ective of in verse problems. Uniqueness of solution is funda- men tal to an y in v erse problem. The exceptions to uniqueness are called the ambiguities. W e identify the rater grid pathology rep orted in optics literature as p erio dic ambiguities of p erio d equal to the step size of the raster scan. Moreo ver, we will characterize all the am biguities inherent to the raster scan pt yc hograph y and prop ose a simple mo dification that can eliminate all the ambiguities except for those inherent to any blind ptyc hograph y . The first thing to note is that raster grid pathology only app ears in blind pt ychograph y but not in ptyc hography with a kno wn prob e. In the latter case, the only ambiguit y is a constan t phase factor which has no real significance (and will b e ignored) and the con vergence b eha viors of the raster scan ptyc hography with a kno wn prob e has b een rigorously established [3]. Second, there are tw o ambiguities inherent to an y blind ptyc hography: a scaling factor and an affine phase factor. T o give a precise description, we introduce some notation as follo ws. Let Z 2 n = J 0 , n − 1 K 2 b e the ob ject domain con taining the supp ort of the discrete ob ject f where J k , l K denotes the integers b etw een, and including, k ≤ l ∈ Z . Let M 00 := Z 2 m , m < n, b e the initial probe area which is also the support of the prob e µ 00 describing the illumination field. Here n is the global scale and m the in termediate scale of the set-up. Let T b e the set of all shifts, including (0 , 0), in v olved in the ptyc hographic measuremen t. Denote b y µ t the t -shifted prob e for all t ∈ T and M t the domain of µ t . Let f t the ob ject restricted to M t . W e write f = ∨ t f t and refer to each f t as a part of f . In pt ychograph y , the original ob ject is broken up in to a set of ov erlapping ob ject parts, each of whic h pro duces a µ t -co ded diffraction pattern (i.e. F ourier in tensity). The totality of the co ded diffraction patterns is called the pt ychographic measurement data. Let ν 00 (with t = (0 , 0)) and g = ∨ t g t b e an y pair of the prob e and the ob ject estimates pro ducing the same ptyc hography data as µ 00 and f , i.e. the diffraction pattern of ν t  g t is identical to that of µ t  f t where ν t is the t -shift of ν 00 and g t is the restriction of g to M t . F or conv enience, w e assume the v alue zero for µ t , f t , ν t , g t outside of M t and the perio dic b oundary condition on Z 2 n when µ t crosses ov er the b oundary of Z 2 n . Consider the prob e and ob ject estimates ν 00 ( n ) = µ 00 ( n ) exp( − i a − i w · n ) , n ∈ M 00 (1) g ( n ) = f ( n ) exp(i b + i w · n ) , n ∈ Z 2 n (2) 2 for any a, b ∈ R and w ∈ R 2 . F or any t , we ha v e the follo wing calculation ν t ( n ) = ν 00 ( n − t ) = µ 00 ( n − t ) exp( − i w · ( n − t )) exp( − i a ) = µ t ( n ) exp( − i w · ( n − t )) exp( − i a ) and hence for all n ∈ M t , t ∈ T ν t ( n ) g t ( n ) = µ t ( n ) f t ( n ) exp(i( b − a )) exp(i w · t ) . (3) Clearly , (3) implies that g and ν 00 pro duce the same pt ychographic data as f and µ 00 since for each t , ν t  g t is a constant phase factor times µ t  f t . In addition to the affine phase am biguity (1)-(2), another am biguit y , a scaling factor ( g = cf , ν 00 = c − 1 µ 00 , c > 0), is also inheren t to an y blind pt ychograph y as can easily b e c heck ed. W e refer to the scaling factor and the affine phase ambiguit y as the inherent ambiguities of blind pt ychograph y . Note that when the prob e is exactly kno wn ν 00 = µ 00 , neither ambiguit y can o ccur. A recent theory of uniqueness for blind pty c hograph y with random prob es [9] establishes that for general sampling schemes and with high probabilit y (in the selection of the random prob e), we hav e the relation ν t  g t = e i θ t µ t  f t , t ∈ T , (4) for some constan ts θ t ∈ R (called blo c k phases here) if g and ν t pro duce the same diffraction pattern as f and µ t for all t ∈ T . Here  denotes the comp onen t-wise (Hadamard) pro duct. The mask ed ob ject parts ψ t := µ t  f t are also kno wn as the exit waves in the scanning transmission electron microscopy literature. W e refer to (4) as the lo c al uniqueness of the exit wa v es which means unique determination of the exit wa ves up to the blo ck phases but not globally since θ t can dep end on t and v ary from blo ck to blo c k. Ho wev er, the blo ck phase profile is not arbitrary . F or example, blo c k phases for the raster scan and the p erturb ed raster scan alwa ys form an arithmetic progression (see b elow), p ossessing tw o degrees of freedom. Once the exit w av es ψ t are determined up to blo c k phases, (4) with θ t treated as parameters represen ts a bilinear system (in ν 00 and g ) of m 2 × |T | equations coupled through the ov erlap b et w een adjacen t blo cks. The total num b er of complex v ariables is n 2 + m 2 . In the case of raster scan with step size τ , |T | ≈ n 2 /τ 2 and m 2 |T | ≈ n 2 ( τ /m ) − 2 where the shift ratio τ /m is 1 min us the o verlap ratio ( m − τ ) /m . F or 50% o v erlap ratio and m < n , m 2 |T | ≈ 4 n 2 , a couple times larger than ( n 2 + m 2 ). This sp eaks of the p oten tial redundancy of information in (4) on dimension coun t. Y et this simplistic analysis is deceptive as we will see that due to degenerate coupling the raster scan has ambiguities of exactly τ 2 + 2 degrees of freedom in addition to the three degrees of freedom of the inherent am biguities discussed ab ov e. W e will tak e (4) as the starting p oin t of our analysis of raster scan ambiguities, first to c haracterize all the am biguities in the raster scan and, second, to sho w ho w to harness the nonlinear coupling in (4) b y more n uanced design of measurement schemes in whic h pixel- scale c hanges result in total eradication of am biguities other than the inherent ones through the intermediate-scale coupling. 3 (a) Ptyc hography set-up (b) raster scan pattern Figure 1. Simplified ptyc hographic setup sho wing a Cartesian grid used for the ov erlapping raster scan p ositions [24]. 1.1. Our contribution. W e first prov e that the blo c k phases of the raster scan of any step size τ < m alw ays ha v e an affine profile (Section 3, Theorem 3.1). W e then give a complete c haracterization of the raster scan am biguities (Theorem 4.3). Roughly sp eaking, there are tw o types of am biguities besides the inheren t am biguities (the scaling factor and the affine phase am biguit y (1)-(2)). First, there is the non-p erio dic, arithmetically progressing am biguity , inherited from the aforementioned affine blo ck phase profile, which v aries on the blo c k scale while the affine phase am biguit y v aries on the pixel scale. Second, there are τ -p erio dic am biguities of τ 2 degrees of freedom, whic h we identify as mathematical description of the raster grid pathology rep orted in the optics literature. The larger the step size the (m uc h) greater the degrees of am biguity which can not be remo ved without extra prior information. Finally w e demonstrate a simple mec hanism for eliminating all the other ambiguities than the scaling factor and the affine phase ambiguit y by slightly p erturbing the raster scan with the minim um o v erlap ratio roughly 50%, consistent with exp erimen tal findings in the optics literature (Section 5, Theorem 5.5). The optimal tradeoff b etw een the speed of data acquisition and the conv ergence rate of reconstruction lies in the balance betw een the a verage step size and the ov erlap size. The rest of the pap er is organized as follows. In Section 2, we giv e a detailed presentation of the raster scan. In Section 3, we pro v e that the blo c k phases hav e an affine profile. In Section 4, w e give a complete c haracterization of the raster scan am biguities. In Section 5 w e sho w that slightly perturb ed raster scan has no other am biguities than the scaling factor and the affine phase ambiguit y . In Section 6, w e give n umerical demonstrate of the p erturb ed raster scan. W e conclude with a few remarks in Section 7. 4 2. Raster scan The raster scan can b e formulated as the 2D lattice with the basis { v 1 , v 2 } T = { t kl ≡ k v 1 + l v 2 : k , l ∈ Z } , v 1 , v 2 ∈ Z 2 (5) acting on the ob ject domain Z 2 n . Instead of v 1 and v 2 w e can also tak e u 1 = ` 11 v 1 + ` 12 v 2 and u 2 = ` 21 v 1 + ` 22 v 2 for integers ` ij with ` 11 ` 22 − ` 12 ` 21 = ± 1. This ensures that v 1 and v 2 themselv es are integer linear com binations of u 1 , u 2 . Ev ery lattice basis defines a fundamen tal parallelogram, whic h determines the lattice. There are five 2D lattice t yp es, called p erio d lattices, as given b y the crystallographic restriction theorem. In contrast, there are 14 lattice t yp es in 3D, called Brav ais lattices [4]. W e will fo cus on the simplest raster scan corresp onding to the squar e lattic e with v 1 = ( τ , 0) , v 2 = (0 , τ ) of step size τ ∈ N . Our results can easily b e extended to other lattice sc hemes. Under the p erio dic b oundary condition the raster scan with the step size τ = n/q , q ∈ N , T consists of t kl = τ ( k , l ), with k , l ∈ { 0 , 1 , · · · , q − 1 } . The p erio dic b oundary condition means that for k = q − 1 or l = q − 1 the shifted prob e is wrapp ed around in to the other end of the ob ject domain. Denote the t kl -shifted prob es and blocks b y µ kl and M kl , resp ectiv ely . Lik ewise, denote b y f kl the ob ject restricted to the shifted domain M kl . Dep ending on whether τ ≤ m/ 2 (the under-shifting case) or τ > m/ 2 (the o v er-shifting case), w e hav e t wo types of sc hemes. F or the former case, all pixels of the the ob ject participate in an equal n um b er of diffraction patterns. F or the latter case, how ever, 4( m − τ ) 2 pixels participate in four, 4(2 τ − m )( m − τ ) pixels participate in t w o and (2 τ − m ) 2 pixels participate in only one diffraction pattern, resulting in uneven co verage of the ob ject. 2.1. The under-shifting sc heme τ ≤ m/ 2 . F or simplicit y of presentation we consider the case of τ = m/p for some integer p ≥ 2 (i.e. pn = q m ). As noted ab ov e, all pixels of the the ob ject participate in the same n um b er (i.e. 2 p ) of diffraction patterns. The b orderline case τ = m/ 2 (dubb ed the minimalist scheme in [3]) corresp onds to p = 2. W e partition the cyclical t kl -shifted prob e µ kl and the corresp onding domain in to equal-sized square blo cks as µ kl =      µ kl 00 µ kl 10 · · · µ kl p − 1 , 0 µ kl 01 µ kl 11 · · · µ kl p − 1 , 1 . . . . . . . . . . . . µ kl 0 ,p − 1 µ kl 1 ,p − 1 · · · µ kl p − 1 ,p − 1      , µ kl ij ∈ C m/p × m/p (6) M kl =      M kl 00 M kl 10 · · · M kl p − 1 , 0 M kl 01 M kl 11 · · · M kl p − 1 , 1 . . . . . . . . . . . . M kl 0 ,p − 1 M kl 1 ,p − 1 · · · M kl p − 1 ,p − 1      , M kl ij ∈ Z m/p × m/p (7) 5 under the p erio dic b oundary condition µ q − 1 − i,k j,l = µ 0 k j − i − 1 ,l , µ k,q − 1 − i l,j = µ k 0 l,j − i − 1 , (8) M q − 1 − i,k j,l = M 0 k j − i − 1 ,l , M k,q − 1 − i l,j = M k 0 l,j − i − 1 (9) for all 0 ≤ i ≤ j − 1 ≤ p − 2 , k = 1 , . . . , q − 1 , l = 1 , . . . , p − 1 . Accordingly , w e divide the ob ject f in to q 2 non-o verlapping square blo c ks f =   f 00 . . . f q − 1 , 0 . . . . . . . . . f 0 ,q − 1 . . . f q − 1 ,q − 1   , f ij ∈ C m/p × m/p . (10) 2.2. The o v er-shifting sc heme τ > m/ 2 . Because of unev en co verage of the ob ject do- main, the ov er-shifting case is more complicated. W e divide the shifted prob e µ kl and its domain as µ kl =   µ kl 00 µ kl 10 µ kl 20 µ kl 01 µ kl 11 µ kl 21 µ kl 02 µ kl 12 µ kl 22   ∈ C m × m (11) M kl =   M kl 00 M kl 10 M kl 20 M kl 01 M kl 11 M kl 21 M kl 02 M kl 12 M kl 22   ∈ Z m × m (12) under the p erio dic b oundary condition M q − 1 ,k 2 j = M 0 k 0 j , M k,q − 1 i 2 = M k 0 i 0 (13) µ q − 1 ,k 2 j = µ 0 k 0 j , µ k,q − 1 i 2 = µ k 0 i 0 , (14) for all k = 1 , . . . , q − 1 and i, j = 0 , 1 , 2, where q is the n umber of shifts in each direction. Note that the sizes of these blo c ks are not equal: the four corner blo c ks are ( m − τ ) × ( m − τ ), the cen ter blo ck is (2 τ − m ) × (2 τ − m ) and the rest are either (2 τ − m ) × ( m − τ ) or ( m − τ ) × (2 τ − m ). As a result, the corresp onding partition of f also has unequally sized blo c ks. W e write f = q − 1 _ k,l =0 f kl , f kl =   f kl 00 f kl 10 f kl 20 f kl 01 f kl 11 f kl 21 f kl 02 f kl 12 f kl 22   ∈ C m × m (15) where, for i, j = 0 , 1 , 2 , k , l = 0 , · · · , q − 1, f kl 2 j = f k +1 ,l 0 j , f k,l i 2 = f k,l +1 i 0 . 3. Affine block phases Let S b e an y cyclic subgroup of T generated by v , i.e. S := { t j = j v : j = 0 , . . . , s − 1 } , of order s , i.e. s v = 0 mod n . F or ease of notation, denote b y µ k , f k , ν k , g k and M k for the resp ectiv e t k -shifted quantities. 6 Theorem 3.1. As in (4) , supp ose that ν k ( n ) g k ( n ) = e i θ k µ k ( n ) f k ( n ) (16) for al l n ∈ M k and k = 0 , . . . , s − 1 . If, for al l k = 0 , . . . , s − 1 , M k ∩ M k +1 ∩ supp( f ) ∩ (supp( f ) + v ) 6 = ∅ , (17) then the se quenc e { θ 0 , θ 1 , . . . , θ s − 1 } is an arithmetic pr o gr ession wher e ∆ θ = θ k − θ k − 1 is an inte ger multiple of 2 π /s . Remark 3.2. If f has a ful l supp ort, i.e. supp( f ) = Z 2 n , then (17) holds for any step size τ < m (i.e. p ositive overlap). Pr o of. Rewriting (16) in the form ν k +1 ( n ) g k +1 ( n ) = e i θ k +1 µ k +1 ( n ) f k +1 ( n ) (18) and substituting (16) into (18) for n ∈ M k ∩ M k +1 , we ha v e e i θ k f k ( n ) µ k ( n ) /ν k ( n ) = e i θ k +1 µ k +1 ( n ) /ν k +1 ( n ) f k +1 ( n ) and hence for all n ∈ M k ∩ M k +1 ∩ supp( f ), e i θ k µ k ( n ) /ν k ( n ) = e i θ k +1 µ k +1 ( n ) /ν k +1 ( n ) . (19) F or all j = 0 , . . . , s − 1 , substituting ν j ( n ) = ν j +1 ( n + v ) , µ j ( n ) = µ j +1 ( n + v ) , (20) in to (19), w e hav e that for n ∈ M k ∩ M k +1 ∩ supp( f ) e i θ k µ k +1 ( n + v ) /ν k +1 ( n + v ) = e i θ k +1 µ k +2 ( n + v ) /ν k +2 ( n + v ) , or equiv alently e i θ k µ k +1 ( n ) /ν k +1 ( n ) = e i θ k +1 µ k +2 ( n ) /ν k +2 ( n ) , (21) ∀ n ∈ M k +1 ∩ M k +2 ∩ (supp( f ) + v ) On the other hand, (19) also implies e i θ k +1 µ k +1 ( n ) /ν k +1 ( n ) = e i θ k +2 µ k +2 ( n ) /ν k +2 ( n ) , (22) ∀ n ∈ M k +1 ∩ M k +2 ∩ supp( f ) . Hence, if M k ∩ M k +1 ∩ supp( f ) ∩ (supp( f ) + v ) 6 = ∅ then (22) and (21) imply that e i θ k +1 e − i θ k = e i θ k e − i θ k − 1 , ∀ k = 0 , . . . , s − 1 (23) and hence ∆ θ = θ k − θ k − 1 is independent of k . In other w ords, { θ 0 , θ 1 , θ 2 . . . } is an arithmetic progression. Moreo ver, the p erio dic b oundary condition and the fact that s v = 0 mod 2 π imply that s ∆ θ is an in teger multiple of 2 π .  7 Applying Theorem 3.1 to the tw o-generator group T of the raster scan we ha ve the follo wing result. Corollary 3.3. F or the ful l r aster sc an T , the blo ck phases have the pr ofile θ kl = θ 00 + r · ( k , l ) , k , l = 0 , . . . , q − 1 , (24) for some θ 00 ∈ R and r = ( r 1 , r 2 ) wher e r 1 and r 2 ar e inte ger multiples of 2 π /q . 4. Raster scan ambiguities In this section we giv e a complete c haracterization of the raster scan am biguities other than the scaling factor and the affine phase ambiguit y (1)-(2), including the arithmetically progressing phase factor inherited from the block phases and the raster grid pathology whic h has a τ -p erio dic structure of τ × τ degrees of freedom. W e will use the notation in Section 2. Before we state the general result. Let us consider tw o simple examples to illustrate each t yp e of ambiguit y separately . The first example shows an ambiguit y resulting from the arithmetically progressing blo c k phases whic h make p ositive and negativ e imprin ts on the ob ject and phase estimates, re- sp ectiv ely . Example 4.1. F or q = 3 , τ = m/ 2 , let f =   f 00 f 10 f 20 f 01 f 11 f 21 f 02 f 12 f 22   g =   f 00 e i2 π / 3 f 10 e i4 π / 3 f 20 e i2 π / 3 f 01 e i4 π / 3 f 11 f 21 e i4 π / 3 f 02 f 12 e i2 π / 3 f 22   b e the obje ct and its r e c onstruction, r esp e ctively, wher e f ij ∈ C n/ 3 × n/ 3 . L et µ kl =  µ kl 00 µ kl 10 µ kl 01 µ kl 11  , ν kl =  µ kl 00 e − i2 π / 3 µ kl 10 e − i2 π / 3 µ kl 01 e − i4 π / 3 µ kl 11  , k , l = 0 , 1 , 2 , b e the ( k , l ) -th shift of the pr ob e and estimate, r esp e ctively, wher e µ kl ij ∈ C n/ 3 × n/ 3 . L et f ij and g ij b e the p art of the obje ct and estimate il luminate d by µ ij and ν ij , r esp e ctively. It is verifie d e asily that ν ij  g ij = e i( i + j )2 π / 3 µ ij  f ij . The next example illustrates the p erio dic artifact called raster grid pathology . 8 Example 4.2. F or q = 3 , τ = m/ 2 and any ψ ∈ C n 3 × n 3 , let f =   f 00 f 10 f 20 f 01 f 11 f 21 f 02 f 12 f 22   g =   e − i ψ  f 00 e − i ψ  f 10 e − i ψ  f 20 e − i ψ  f 01 e − i ψ  f 11 e − i ψ  f 21 e − i ψ  f 02 e − i ψ  f 12 e − i ψ  f 22   b e the obje ct and its r e c onstruction, r esp e ctively, wher e f ij ∈ C n/ 3 × n/ 3 . L et µ kl =  µ kl 00 µ kl 10 µ kl 01 µ kl 11  , ν kl =  e i ψ  µ kl 00 e i ψ  µ kl 10 e i ψ  µ kl 01 e i ψ  µ kl 11  , k , l = 0 , 1 , 2 , b e the ( k , l ) -th shift of the pr ob e and estimate, r esp e ctively, wher e µ kl ij ∈ C n/ 3 × n/ 3 . L et f ij and g ij b e the p art of the obje ct and estimate il luminate d by µ ij and ν ij , r esp e ctively. It is verifie d e asily that ν ij  g ij = µ ij  f ij . The com bination of the ab ov e t w o types of am biguity gives rise to the general am biguities for blind ptyc hograph y with the raster scan as stated next. Theorem 4.3. Supp ose that supp( f ) = Z 2 n . Consider the r aster sc an T and supp ose that an obje ct estimate g and a pr ob e estimate ν 00 satisfy the r elation ν kl  g kl = e i θ kl µ kl  f kl , θ kl = θ 00 + r · ( k , l ) (25) as given by The or em 3.1. Then the fol lowing statements hold. (I). F or τ ≤ m/ 2 , if ν 00 00 = e i ψ  µ 00 00 , ψ ∈ C τ × τ , (26) then ν 00 kl = e − i r · ( k,l ) e i ψ  µ 00 kl , k , l = 0 , . . . , p − 1 (27) g kl = e i θ 00 e i r · ( k,l ) e − i ψ  f kl , k , l = 0 , . . . , q − 1 . (28) (I I). F or τ > m/ 2 , if  ν 00 00 ν 00 10 ν 00 01 ν 00 11  = e i ψ   µ 00 00 µ 00 10 µ 00 01 µ 00 11  (29) for some ψ =  ψ 00 ψ 10 ψ 01 ψ 11  ∈ C τ × τ , then  g kl 00 g kl 10 g kl 01 g kl 11  = e i θ 00 e i r · ( k,l ) e − i ψ   f kl 00 f kl 10 f kl 01 f kl 11  (30) 9 for al l k , l = 0 , . . . , q − 1 . Mor e over, ν 00 2 j = e − i r 1 e i ψ 0 j  µ 00 2 j , j = 0 , 1 (31) ν 00 j 2 = e − i r 2 e i ψ j 0  µ 00 j 2 , j = 0 , 1 (32) ν 00 22 = e − i( r 1 + r 2 ) e i ψ 00  µ 00 22 (33) and henc e g kl 2 j = e i θ 00 e i r · ( k +1 ,l ) e − i ψ 0 j  f kl 2 j , j = 0 , 1 (34) g kl j 2 = e i θ 00 e i r · ( k,l +1) e − i ψ j 0  f kl j 2 , j = 0 , 1 (35) g kl 22 = e i θ 00 e i r · ( k +1 ,l +1) e − i ψ 00  f kl 22 . (36) Remark 4.4. Sinc e ψ is any c omplex τ × τ matrix, (26) and (29) r epr esent the maximum de gr e es of ambiguity over the r esp e ctive initial sub-blo cks. This ambiguity is tr ansmitte d to other sub-blo cks, forming p erio dic artifacts c al le d the r aster grid p atholo gy. On top of the p erio dic artifacts, ther e is the non-p erio dic ambiguity inherite d fr om the affine blo ck phase pr ofile. The non-p erio dic arithmetic al ly pr o gr essing ambiguity is differ ent fr om the affine phase ambiguity (1) - (2) as they manifest on differ ent sc ales: the former on the blo ck sc ale while the latter on the pixel sc ale. Pr o of. (I). F or τ ≤ m/ 2, recall the decomp osition ν kl =      ν kl 00 ν kl 10 · · · ν kl p − 1 , 0 ν kl 01 µ kl 11 · · · ν kl p − 1 , 1 . . . . . . . . . . . . ν kl 0 ,p − 1 ν kl 1 ,p − 1 · · · ν kl p − 1 ,p − 1      , g =   g 00 . . . g q − 1 , 0 . . . . . . . . . g 0 ,q − 1 . . . g q − 1 ,q − 1   , with ν kl ij , g ij ∈ C m/p × m/p , in analogy to (6) and (10). g 00 = e i θ 00 e − i ψ  f 00 b y restricting (25) to M 00 00 . F or n ∈ M 10 00 , we ha v e ν 10 00  g 10 = e i θ 10 µ 10 00  f 10 , b y (25), and ν 10 00 ( n ) = ν 00 00 ( n − ( τ , 0)) = ( e i ψ  µ 00 00 )( n − ( τ , 0)) = ( e i ψ  µ 10 00 )( n ) b y (26). Hence g 10 = e i θ 10 e − i ψ  f 10 implying ν 00 10  g 10 = e i θ 10 e − i ψ ν 00 10  f 10 = e i θ 00 µ 00 10  f 10 b y (25) and consequently ν 00 10 = e i θ 00 e − i θ 10 e i ψ µ 00 10 . 10 Rep eating the same argumen t for the adjacent blo cks in b oth directions, we obtain ν 00 kl = e i θ 00 e − i θ kl e i ψ  µ 00 kl g kl = e i θ kl e − i ψ  f kl whic h are equiv alent to (27) and (28) in view of the blo ck phase profile in (24). (I I). First recall µ kl =   µ kl 00 µ kl 10 µ kl 20 µ kl 01 µ kl 11 µ kl 21 µ kl 02 µ kl 12 µ kl 22   , g = q − 1 _ k,l =0 g kl , g kl =   g kl 00 g kl 10 g kl 20 g kl 01 g kl 11 g kl 21 g kl 02 g kl 12 g kl 22   in analogy to (11) and (15). Since ν kl ( n ) = ν 00 ( n − τ ( k , l ))) , µ kl ( n ) = µ 00 ( n − τ ( k , l )) , (37) (30) follows from (29) and (25). By (29) and restricting (25) to M 10 0 j , j = 0 , 1 , we obtain g 00 2 j = g 10 0 j = e i θ 10 e − i ψ 0 j  f 10 0 j = e i θ 10 e − i ψ 0 j  f 00 2 j , j = 0 , 1 , whic h implies b y (25) ν 00 2 j = e i θ 00 e − i θ 10 e i ψ 0 j  µ 00 2 j , j = 0 , 1 , ν 00 j 2 = e i θ 00 e − i θ 01 e i ψ j 0  µ 00 j 2 , j = 0 , 1 , and consequently (31) and (32). By (37) and restricting (25) to M kl 2 j , M kl j 2 , j = 0 , 1 , we ha v e (34) and (35). F or (36) with ( k , l ) = (0 , 0), the blo ck M 10 02 = M 00 22 is masked by µ 10 02 , a translate of µ 00 02 . By restricting (25) to M 10 02 , g 00 22 = g 10 02 = e i( θ 10 + θ 01 − θ 00 ) e − i ψ 00  f 00 22 . (38) whic h is equiv alent to (36) with ( k , l ) = (0 , 0). Then (25) and (38) imply ν 00 22 = e i( θ 00 − θ 10 ) e i( θ 00 − θ 01 ) e i ψ 00  µ 00 22 (39) whic h is equiv alent to (33). F or (36) with general k , l , b y restricting (25) to M kl 22 and (39) we ha ve g kl 22 = e i θ kl e i( θ 10 − θ 00 ) e i( θ 01 − θ 00 ) e − i ψ 00  f kl 00 and hence (36).  When τ = 1, the non-p erio dic, arithmetically progressing ambiguit y and the affine phase am biguity b ecome the same. In addition, for τ = 1 the raster grid pathology b ecomes a constan t phase factor which can b e ignored [17]. Corollary 4.5. If τ = 1 (i.e. q = n, p = m ) and (25) holds, then the pr ob e and the obje ct c an b e uniquely and simultane ously determine d. 11 (a) Perturbed grid (40) (b) Perturbed grid (41) Pr o of. F or τ = 1, µ 00 consists of just one pixel and ψ is a n um b er. Hence µ 00 = ν 00 up to a constan t phase factor and (27)-(28) then imply that the affine phase am biguity is the only am biguity mo dulo the constant phase factor.  5. Slightl y per turbed raster scan In this section, w e demonstrate a simple wa y for remo ving all the raster scan am biguities except for the scaling factor and the affine phase ambiguit y . F or the rest of the pap er, we assume that f do es not v anish in Z 2 n . W e consider the p erturb ed raster scan (Fig. 5(a)) t kl = τ ( k , l ) + ( δ 1 k , δ 2 l ) , k, l = 0 , . . . , q − 1 (40) where δ 1 k , δ 2 l are small in tegers relative to τ and m − τ (see Theorem 5.5 for details). More general than (40) is the p erturb ed grid pattern (Fig. 5(b)): t kl = τ ( k , l ) + ( δ 1 kl , δ 2 kl ) , k, l = 0 , . . . , q − 1 , (41) whic h is harder to analyze and implemen t in exp erimental practice (w e will only presen t n umerical sim ulation for it). Without loss of generality we set δ 1 0 = δ 2 0 = 0 and hence t 00 = (0 , 0). Let us express the prob e and ob ject errors in terms of ν 00 ( n ) /µ 00 ( n ) := α ( n ) exp (i φ ( n )) , n ∈ M 00 (42) h ( n ) := ln g ( n ) − ln f ( n ) , n ∈ Z 2 n , (43) where we assume α ( n ) 6 = 0 for all n ∈ M 00 , and rewrite (4) as h ( n + t ) = i θ t − ln α ( n ) − i φ ( n ) mo d i2 π , (44) for n ∈ M 00 . 12 By (44) with t = (0 , 0), h ( n ) = i θ 00 − ln α ( n ) − i φ ( n ) , ∀ n ∈ M 00 (45) and hence for all t ∈ T and n ∈ M 00 h ( n + t ) − h ( n ) = i θ t − i θ 00 mo d i2 π . (46) W e wish to generalize such a relationship to the case where t in (60) is replaced b y e 1 = (1 , 0) and e 2 = (0 , 1). 5.1. A simple p erturbation. Let us first study the simple example of the tw o-shift per- turbation to the raster-scan with δ 1 2 = δ 2 2 = − 1 but all other δ j k = 0, i.e. t kl = τ ( k , l ) for ( k , l ) 6 = (2 , 0) , (0 , 2). Then h ( n + 2 t 10 − t 20 ) = h ( n + (1 , 0)) (47) h ( n + 2 t 01 − t 02 ) = h ( n + (0 , 1)) . (48) There are sev eral routes of reduction from (1 , 0) to (0 , 0) via the shifts in T . F or example, w e can pro ceed from (1 , 0) = 2 t 10 − t 20 to (0 , 0) along the path (2 t 10 − t 20 ) − → ( t 10 − t 20 ) − → t 10 − → (0 , 0) (49) b y rep eatedly applying (46) where the direction of the second step is to b e rev ersed since − t 20 6∈ T ( T is no longer a group even under the p erio dic b oundary condition). The direction is imp ortan t for k eeping track of the domain of v alidity of (46) along the path. Hence for all n ∈ ( M 00 + t 20 − t 10 ) ∩ M 00 (50) w e hav e h ( n + 2 t 10 − t 20 ) = h ( n + t 10 − t 20 ) + i θ 10 − i θ 00 = h ( n + t 10 ) + i θ 10 − i θ 20 = h ( n ) + 2i θ 10 − i θ 20 − i θ 00 and hence h ( n + (1 , 0)) = h ( n ) + i∆ 1 , ∆ 1 := 2 θ 10 − θ 20 − θ 00 (51) mo dulo i2 π . Let us consider another alternative route for reduction: (2 t 10 − t 20 ) − → 2 t 10 − → t 10 − → (0 , 0) (52) where the proper direction for the first step in applying (46) is rev ersed. Keeping trac k of the domain of v alidity along the path, w e hav e h ( n + 2 t 10 − t 20 ) = h ( n + 2 t 10 ) − i θ 20 + i θ 00 = h ( n + t 10 ) + i θ 10 − i θ 20 = h ( n ) + i∆ 1 for all n ∈ ( M 00 − 2 t 10 + t 20 ) ∩ ( M 00 − t 10 ) ∩ M 00 . (53) 13 In summary , (51) holds for all n in the union of (50) and (53), i.e. D 1 = ( J 0 , m − τ − 1 K ∪ J τ − 1 , m − 1 K ) × J 0 , m − 1 K . Clearly including other routes for reducing 2 t 10 − t 20 to (1 , 0) in D 1 can enlarge the domain of v alidity for (51). F or simplicit y of argumen t, we omit them here. By rep eatedly applying (46) w e hav e the following result. Prop osition 5.1. The r elation (51) holds true in the set [ t ∈T  t + D 1 ∩ M 00 ∩ ( M 00 − e 1 )  (54) which c ontains Z 2 n if τ ≤ ( m − 2) ∧ [( m + 1) / 2] . (55) Pr o of. F or n ∈ D 1 ∩ M 00 ∩ ( M 00 − e 1 ), we ha v e h ( n + t ) = h ( n + e 1 ) − i∆ 1 + i θ t − i θ 00 , (56) b y (51) and (46). Hence, by (46) and (56), h ( n + e 1 + t ) = h ( n + e 1 ) + i θ t − i θ 00 = h ( n + t ) + i∆ 1 . In other w ords, (51) has b een extended to t + D 1 ∩ M 00 ∩ ( M 00 − e 1 ). T aking the union o ver all shifts, we obtain (54). F or the second part of the prop osition, let us write the set (54) explicitly as q − 1 [ k,l =0 { τ ( k , l ) + [( J 0 , m − τ − 1 K ∪ J τ − 1 , m − 1 K ) ∩ J 0 , m − 2 K ] × J 0 , m − 1 K } . Note that [( J 0 , m − τ − 1 K ∪ J τ − 1 , m − 1 K ) ∩ J 0 , m − 2 K ] = J 0 , m − τ − 1 K ∪ J τ − 1 , m − 2 K = J 0 , m − 2 K under m − τ − 1 ≥ τ − 2 or, equiv alen tly , (55). T o complete the argument, observ e that the adjacen t rectangles among ( τ ( k , l ) + J 0 , m − 2 K ) × J 0 , m − 1 K , k , l = 0 , . . . , q − 1 , ha ve zero gap if τ ≤ m − 2.  By the same argumen t under (55), it follo ws from (48) that for all n ∈ Z 2 n h ( n + (0 , 1)) = h ( n ) + i∆ 2 mo d i2 π , ∆ 2 := 2i θ 01 − i θ 02 − i θ 00 . (57) In conclusion, h ( n ) = h (0) + i n · r mo d i2 π , ∀ n ∈ Z 2 n , (58) 14 where r = (∆ 1 , ∆ 2 ). 5.2. General p erturbation. Next we consider more general p erturbations { δ i k } to the raster scan and deriv e (58). Let us rewrite (46) in a different form: Subtracting the respectiv e (46) for t and t 0 , w e obtain the equiv alent form h ( n + t ) − h ( n + t 0 ) = i θ t − i θ t 0 mo d i2 π , (59) for any n ∈ M 00 and t , t 0 ∈ T , which can also b e written as h ( n + t − t 0 ) = h ( n ) + i( θ t − θ t 0 ) mod i2 π , (60) for n ∈ M t 0 b y shifting the argument of h . Consider the triplets of shifts ( t kl , t k +1 ,l , t k +2 ,l ) , ( t kl , t k,l +1 , t k,l +2 ) for which w e hav e 2( t k +1 ,l − t kl ) − ( t k +2 ,l − t kl ) = (2 δ 1 k +1 − δ 1 k − δ 1 k +2 , 0) := ( a 1 k , 0) , 2( t k,l +1 − t kl ) − ( t k,l +2 − t kl ) = (0 , 2 δ 2 l +1 − δ 2 l − δ 2 l +2 ) := (0 , a 2 l ) . Analogous to (52) and (49) the paths of reduction (2 t k +1 ,l − t kl − t k +2 ,l ) − → 2( t k +1 ,l − t kl ) − → ( t k +1 ,l − t kl ) − → (0 , 0) and (2 t k +1 ,l − t kl − t k +2 ,l ) − → ( t k +1 ,l − t k +2 ,l ) − → ( t k +1 ,l − t kl ) − → (0 , 0) lead to h ( n + ( a 1 k , 0)) = h ( n ) + 2i θ k +1 ,l − i θ k +2 ,l − i θ kl mo d i2 π (61) for all n ∈ D 1 kl where D 1 kl :=  M kl ∩ [ M kl − 2 t k +1 ,l + t k +2 ,l + t kl ] ∩ [ M kl − t k +1 ,l + t kl ]  [  M kl ∩ [ M kl + t k +2 ,l − t k +1 ,l ]  =  M kl ∩ [ M kl − ( a 1 k , 0)] ∩ [ M kl − ( τ + δ 1 k +1 − δ 1 k , 0)]  [  M kl ∩ [ M kl + ( τ + δ 1 k +2 − δ 1 k +1 , 0)]  Lik ewise, rep eatedly applying (60) along the paths, (2 t k,l +1 − t kl − t k,l +2 ) − → 2( t k,l +1 − t kl ) − → ( t k,l +1 − t kl ) − → (0 , 0) and (2 t k,l +1 − t kl − t k +2 ,l ) − → ( t k +1 ,l − t k +2 ,l ) − → ( t k +1 ,l − t kl ) − → (0 , 0) w e get h ( n + (0 , a 2 l )) = h ( n ) + 2i θ k,l +1 − i θ k,l +2 − i θ kl mo d i2 π (62) 15 for n ∈ D 2 kl where D 2 kl :=  M kl ∩ [ M kl − 2 t k,l +1 + t k,l +2 + t kl ] ∩ [ M kl − t k,l +1 + t kl ]  [  M kl ∩ [ M kl + t k,l +2 − t k,l +1 ]  =  M kl ∩ [ M kl − (0 , a 2 l )] ∩ [ M kl − (0 , τ + δ 2 l +1 − δ 2 l )]  [  M kl ∩ [ M kl + (0 , τ + δ 2 l +2 − δ 2 l +1 )]  . Lemma 5.2. L et k, l b e fixe d. The r elations (61) and (62) hold true in the sets [ t ∈T  t + D 1 kl ∩ M kl ∩ ( M kl − ( a 1 k , 0))  (63) and [ t ∈T  t + D 2 kl ∩ M kl ∩ ( M kl − (0 , a 2 l ))  , (64) r esp e ctively. Both sets c ontain Z 2 n if the fol lowing c onditions hold: max i =1 , 2 {| a i k | + δ i k +1 − δ i k } ≤ τ (65) 2 τ ≤ m − max i =1 , 2 { δ i k +2 − δ i k } (66) max k 0 max i =1 , 2 {| a i k | + δ i k 0 +1 − δ i k 0 } ≤ m − 1 − τ . (67) Remark 5.3. Pr op osition 5.1 c orr esp onds to ( k , l ) = (0 , 0) with (65) , (66) and (67) r e duc e d to 1 ≤ τ , 2 τ ≤ m + 1 , τ ≤ m − 2 , r esp e ctively. Remark 5.4. Ine qualities (65) and (67) ar e smal lness c onditions for the p erturb ations r el- ative to the aver age step size and the overlap b etwe en the adjac ent pr ob es. The most c onse- quential c ondition (66) suggests an aver age overlap r atio of at le ast 50% , i.e. under-shifte d r aster sc an. Pr o of. The argument follows the same pattern as that for Prop osition 5.1. F or n ∈ D 1 kl ∩ M 00 ∩ ( M 00 − ( a 1 k , 0)), we ha v e h ( n + t ) = h ( n + ( a 1 k , 0)) − i(2 θ k +1 ,l − θ k +2 ,l − θ kl ) + i θ t − i θ 00 b y (61) and (46). Hence, by (46) and (56), h ( n + ( a 1 k , 0) + t ) = h ( n + ( a 1 k , 0)) + i θ t − i θ 00 = h ( n + t ) + i(2 θ k +1 ,l − θ k +2 ,l − θ kl ) . T aking the union ov er all shifts, w e obtain the set in (63). The case for (64) is similar. 16 F or the second part of the prop osition, note that M kl ∩ ( M kl − ( a 1 k , 0)) = J 0 , m − 1 − | a 1 k | K × J 0 , m − 1 K if a 1 k ≥ 0 (68) or J | a 1 k | , m − 1 K × J 0 , m − 1 K , if a 1 kl < 0 . In the former case in (68) the set (63) contains [ t ∈T  t + t kl + ( J 0 , m − 1 − τ − δ 1 k +1 + δ 1 k K ∪ J τ + δ 1 k +2 − δ 1 k +1 , m − 1 K ) (69) × J 0 , m − 1 K ∩  J 0 , m − 1 − | a 1 k | K × J 0 , m − 1 K  under the condition | a 1 k | + δ 1 k +1 − δ 1 k ≤ τ ≤ m − 1 − | a 1 k | − δ 1 k +1 + δ 1 k . (70) The set in (69) b ecomes [ t ∈T  t + t kl +  J 0 , m − 1 K ∩ J 0 , m − 1 − | a 1 k | K  × J 0 , m − 1 K  (71) = [ t ∈T  t + t kl + J 0 , m − 1 − | a 1 k | K × J 0 , m − 1 K  under the condition m − 1 − τ − δ 1 k +1 + δ 1 k ≥ τ + δ 1 k +2 − δ 1 k +1 − 1 . (72) The set in (71) contains Z 2 n if for each l 0 the adjacent sets among τ ( k 0 + k , l 0 + l ) + ( δ 1 k 0 + δ 1 k , δ 2 l 0 + δ 2 l ) + J 0 , m − 1 − | a 1 k | K × J 0 , m − 1 K , for k 0 = 0 , . . . , q − 1 , hav e no gap b et w een them, whic h is the case if τ + δ 1 k 0 +1 − δ 1 k 0 ≤ m − 1 − | a 1 k | , ∀ k 0 . (73) Note that (73) subsumes the second inequality in (70). Lik ewise for the latter case in (68) the set in (63) contains [ t ∈T  t + t kl + ( J | a 1 k | , m − 1 − τ − δ 1 k +1 + δ 1 k K ∪ J τ + δ 1 k +2 − δ 1 k +1 , m − 1 K ) (74) × J 0 , m − 1 K ∩  J | a 1 k | , m − 1 K × J 0 , m − 1 K  under the condition | a 2 l | + δ 2 l +1 − δ 2 l ≤ τ ≤ m − 1 − | a 2 l | − δ 2 l +1 + δ 2 l . (75) The set in (74) in turn b ecomes [ t ∈T  t + t kl + J | a 1 k | , m − 1 K × J 0 , m − 1 K  under the condition m − 1 − τ − δ 1 k +1 + δ 1 k ≥ τ + δ 1 k +2 − δ 1 k +1 − 1 . The set in (76) contains Z 2 n if for each l 0 the adjacent sets among τ ( k 0 + k , l 0 + l ) + ( δ 1 k 0 + δ 1 k , δ 2 l 0 + δ 2 l ) + J | a 1 k | , m − 1 K × J 0 , m − 1 K , 17 for k 0 = 0 , . . . , q − 1 , hav e no gap b et w een them, which is the case under the same condition (73) which subsumes the second inequalit y in (75). The case with (64) can b e pro ved by the same argumen t as ab ov e.  Since M kl o verlaps with M k +1 ,l and M k,l +1 whic h in turn o verlap with M k +2 ,l and M k,l +2 , resp ectiv ely (and so on), the quan tities ∆ 1 k := 2 θ k +1 ,l − θ k +2 ,l − θ kl (76) ∆ 2 l := 2 θ k,l +1 − θ k,l +2 − θ kl (77) on the righthand side of (61) and (62) dep end only on one index and w e can write Supp ose further that there exist c 1 k , c 2 l ∈ Z such that q − 1 X k =0 c 1 k a 1 k = q − 1 X l =0 c 2 l a 2 l = 1 , (78) i.e. { a j i } are co-prime in tegers for eac h j = 1 , 2. Then by rep eatedly using (61)-(62) we arrive at h ( n + (1 , 0)) = h n + ( X k c 1 k a 1 k , 0) ! = h ( n ) + i r 1 mo d i2 π h ( n + (0 , 1)) = h n + (0 , X l c 2 l a 2 l ) ! = h ( n ) + i r 2 mo d i2 π where r 1 = q − 1 X k =0 c 1 k ∆ 1 k , r 2 = q − 1 X l =0 c 2 l ∆ 2 l . (79) Therefore, we obtain (58) with r = ( r 1 , r 2 ) giv en by (79). F ollowing through the rest of argumen t we can prov e the following result. Theorem 5.5. Supp ose f do es not vanish in Z 2 n . F or the p erturb e d r aster sc an (40) , let { δ i j k } b e the subset of p erturb ations satisfying τ ≥ max i =1 , 2 {| a i j k | + δ i j k +1 − δ i j k } (80) 2 τ ≤ m − max i =1 , 2 { δ i j k +2 − δ i j k } (81) m − τ ≥ 1 + max k 0 max i =1 , 2 {| a i j k | + δ i k 0 +1 − δ i k 0 } (82) wher e a i j = 2 δ i j +1 − δ i j − δ i j +2 . Supp ose gcd j k  | a i j k |  = 1 , i = 1 , 2 . (83) L et r = ( r 1 , r 2 ) ∈ R 2 b e given by (79) and { c j i } b e any solution to (78) such that { c i j k } ar e the only nonzer o entries. 18 (c) f ’s real part (d) f ’s imaginary part (e) Randomly phased prob e Figure 2. The real part (a) and the imaginary part (b) of the ob ject and (c) randomly phased prob e µ 00 . Then b oth the obje ct and pr ob e err ors have a c onstant sc aling factor and an affine phase pr ofile: g ( n ) /f ( n ) = α − 1 (0) exp(i n · r ) , (84) ν 0 ( n ) /µ 0 ( n ) = α (0) exp(i φ (0) − i n · r ) . (85) F urther the blo ck phases have an affine pr ofile: θ kl = θ 00 + t kl · r mo d 2 π , (86) for k , l = 0 , · · · , q − 1 . Remark 5.6. It c an b e verifie d thr ough a te dious c alculation that (86) (with (76) - (77) , (78) and (79) ) is an under determine d line ar system for { θ kl } , which is c onsistent with the fact that the affine phase ambiguity (1) - (2) is inher ent to any blind ptycho gr aphy. Pr o of. It remains to v erify (85) and (85) which follo w immediately from (44) and (84). The blo ck phase relation (86) follo ws up on substituting t = t kl and (84) into (44). T o summarize, we ha v e shown that the scaling factor in (85) and the affine phase ambiguit y , in (84) and (85), are the only ambiguities for the slightly p erturb ed raster scan (40).  6. Numerical experiments In this section we demonstrate geometric con v ergence for blind ptyc hograph y with the p er- turb ed raster scan (40). Let F ( ν, g ) ∈ C N b e the totalit y of the F ourier (magnitude and phase) data corresp onding to the prob e ν and the ob ject g suc h that |F ( µ, f ) | = b where b is the noiseless pt yc hographic data. Since F ( · , · ) is a bilinear function, A k h := F ( µ k , h ) , k ≥ 1 , defines a matrix A k for the k -th prob e estimate µ k and B k η := F ( η , f k +1 ) , k ≥ 1 , for the ( k + 1)-st image estimate f k +1 suc h that A k f j +1 = B j µ k , j ≥ 1 , k ≥ 1. Let P k = A k A † k b e the orthogonal pro jection on to 19 (a) RE with (40) (b) RE with (41) Figure 3. RE for v arious boundary conditions with the sampling sc heme (a) (40) and (b) (41). the range of A k and R k = 2 P k − I the corresp onding reflector. Lik ewise, let Q k = B k B † k b e the orthogonal pro jection onto the range of B k and S k the corresp onding reflector. Algorithm 1: Alternating minimization (AM) 1: Input: initial prob e guess µ 1 . 2: Up date the ob ject estimate f k +1 = arg min L ( A k g ) s.t. g ∈ C n × n . 3: Up date the prob e estimate µ k +1 = arg min L ( B k ν ) s.t. ν ∈ C m × m . 4: T erminate if k| B k µ k +1 | − b k 2 stagnates or is less than tolerance; otherwise, go bac k to step 2 with k → k + 1 . W e use the ob jective function L ( y ) = 1 2 k| y | − b k 2 2 and a randomly c hosen initial prob e guess satisfying <  µ 1 ( n )  µ 00 ( n )  > 0 , ∀ n , i.e. each pixel of the prob e guess is aligned with the corresponding pixel of the true prob e p ositiv ely . The inner lo ops for up dating the ob ject and prob e estimates are carried out by the Douglas-Rac hford splitting metho d as detailed in [3, 11]: At ep o c h k , for l = 1 , 2 , 3 , . . . u l +1 k = 1 2 u l k + 1 2 b  sgn  R k u l k  , u 1 k = u ∞ k − 1 v l +1 k = 1 2 v l k + 1 2 b  sgn  S k v l k  , v 1 k = v ∞ k − 1 with the ob ject estimate f k +1 = A † k u ∞ k and the prob e estimate µ k +1 = B † k v ∞ k where u ∞ k and v ∞ k are terminal v alues of the k -th ep o ch of the inner lo ops. In the simulation for Fig. 3 w e k eep the maxim um num b er of iterations in the inner lo op to 30. 20 T o discoun t the constant amplitude offset and the linear phase ambiguit y w e consider the follo wing relativ e error (RE) for the recov ered image f k and prob e µ k at the k th ep o ch: RE( k ) = min α ∈ C , k ∈ R 2 k f ( k ) − αe − ı 2 π k · r /n f k ( k ) k 2 k f k 2 (87) The image is 256-by-256 Cameraman+ i Barbara (CiB). W e use the randomly phased prob e µ 00 ( n ) = exp[i φ ( n )] where [ φ ( n )] are 60 × 60 i.i.d. uniform random v ariables o v er [0 , 2 π ). W e let δ 1 k (resp. δ 1 kl ) and δ 2 l (resp. δ 2 kl ) to b e i.i.d. uniform random v ariables o v er J − 4 , 4 K . In other words, the adjacen t prob es o v erlap by an av erage of 1 − τ /m = 50%. When the prob e steps outside of the boundary of the ob ject domain, the area M \ Z 2 n needs sp ecial treatment in the reconstruction pro cess. The p erio dic b oundary condition forces the slope r in the linear phase am biguity to b e in tegers. The dark-field and bright-field boundary conditions assume zero and nonzero (= 100 in the simulation) v alues, resp ectiv ely , in M \ Z 2 n . When the bright-field b oundary condition is present in the sim ulation data and enforced in reconstruction, the linear phase am biguity disapp ears from the ob ject estimate. On the con trary , enforcing the dark-field b oundary condition can not remo ve the linear phase ambiguit y . In b oth cases, how ever, enforcemen t of b oundary condition in reconstruction sp eeds up the conv ergence as shown in Figure 3. Figure 3 sho ws that the sampling scheme (41) generally outp erforms (40) with a faster con vergence rate, indicating that higher lev el of disorder in the grid pattern is b etter for blind ptyc hohgraphy . 7. Conclusions W e ha ve studied the artifacts in blind pt yc hographic reconstruction from the p ersp ective of uniqueness theory of in v erse problems and iden tified the p erio dic am biguities in the raster scan ptyc hography as the raster grid pathology rep orted in the optics literature. W e hav e giv en a complete c haracterization of blind ptyc hographic ambiguities for the raster scan including the p erio dic and non-p erio dic am biguities. The non-p erio dic am biguity ha ve an affine profile mirroring that of the blo ck phases. T o the b est of our kno wledge, suc h an am biguity has not b een rep orted in the literature. W e ha ve presented a slightly p erturb ed under-shifted raster scan and prov ed that such a sc heme can remov e all the ambiguities except for those inherent to any blind pt yc hography , namely the scaling factor and the affine phase ambiguit y . In comparison, the same goal is approac hed in [1] not by c hanging the raster scan but b y considering only a set of generic ob jects. F or the p erturb ed under-shifted raster scan (40) with small random δ i j , it is highly probable that the co-prime condition (83) holds for large q and hence only the scaling factor and the affine phase ambiguit y are presen t under (65)-(67) [19]. It w ould b e interesting to see if the analysis presented in Section 5 can b e extended to other scan patterns in practice 21 suc h as the concen tric circles [6, 36, 37], the F ermat spiral [13] and those designed for F ourier pt ychograph y [13]. In a noisy ptyc hographic exp eriment with the raster scan, as the step size shrinks, raster grid pathology b ecomes less apparent and ev en tually invisible b efore the step size reaches 1 [14] (cf. Corollary 4.5). The affine phase am biguity and the raster grid pathology can also b e suppressed b y additional prior information suc h as the F ourier intensities of the prob e [23]. A cknowledgment This w ork w as supp orted b y the National Science F oundation under Gran t DMS-1413373. I thank National Cen ter for Theoretical Sciences (NCTS), T aiw an, where the presen t w ork w as completed, for the hospitality during my visits in June and August 2018. I am grateful for Zhiqing Zhang for preparing Fig. 3. References [1] T. Bendory , D. Edidin and Y. C. Eldar, “Blind phaseless short-time F ourier transform reco very ,” [2] O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, F. Pfeiffer, “Influence of the ov erlap parameter on the conv ergence of the ptyc hographical iterativ e engine,” Ultr amicr osc opy 108 (5) (2008) 481-487. [3] P . Chen and A. F annjiang, “ Co ded-ap erture ptyc hography: uniqueness and reconstruction”, Inverse Pr oblems 34 (2018) 025003. [4] H.H. Con wa y & N.J.A. Sloane, Spher e Packings, L attic es and Gr oups , 3rd ed., Berlin, New Y ork: Springer-V erlag, 1999. [5] M. Dierolf, A. Menzel, P . Thibault, P . Schneider, C. M. Kewish, R. W epf, O. Bunk, and F. Pfeiffer, “ Pt yc hographic x-ra y computed tomograph y at the nanoscale,” Natur e 467 (2010), 436-439. [6] M. Dierolf, P . Thibault, A. Menzel, C. Kewish, K. Jefimovs, I. Schlic hting, K. Kong, O. Bunk, and F. Pfeiffer, “Ptyc hographic coherent diffractive imaging of w eakly scattering sp ecimens,” New J. Phys. 12 (2010), 035017. [7] C. F alldorf, M. Agour, C. V. Kopylo w and R. B. Bergmann, “Phase retriev al b y means of a spatial ligh t mo dulator in the F ourier domain of an imaging system,” Appl. Opt. 49 , 1826-1830 (2010). [8] R. Egami, R. Horisaki, L. Tian & J. T anida, “Relaxation of mask design for single-shot phase imaging with a co ded ap erture,” Appl. Opt. 55 (2016) 1830-1837. [9] A. F annjiang and P . Chen, “Blind ptyc hography: uniqueness & ambiguities,” [10] A. F annjiang and W. Liao, “Phase retriev al with random phase illumination,” J. Opt. So c. A 29 , 1847-1859 (2012). [11] A. F annjiang and Z. Zhang, “Blind pt yc hography by Douglas-Rachford splitting,” [12] M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. W epf, and O. Bunk, “Phase tomograph y from x-ra y coheren t diffractiv e imaging pro jections,” Opt. Exp. 19 (2011), 21345-21357. [13] K. Guo, S. Dong, P . Nanda and G. Zheng, “Optimization of sampling pattern and the design of F ourier pt yc hographic illuminator,” Opt. Exp. 23 (2015) 6171-6180. [14] X. Huang, H. Y an, M. Ge, H. ¨ Ozt¨ ork, E. Nazaretski, I. K. Robinson and Y.S. Chu, “Artifact mit- igation of pt yc hograph y in tegrated with on-the-fly scanning prob e microscopy ,” Appl. Phys. L ett. 111 (2017) 023103. [15] X. Huang, H. Y ang, R. Harder, Y. Hwu, I.K. Robinson & Y.S. Ch u,“Optimization of ov erlap uni- formness for pt yc hograph y ,” Opt. Expr ess 22 (2014), 12634-12644. [16] R. Horisaki, R. Egami & J. T anida, “Single-shot phase imaging with randomized light (SPIRaL)”. Opt. Expr ess 24 , 3765-3773 (2016). 22 [17] M. A. Iwen, A. Viswanathan, and Y. W ang, “F ast phase retriev al from local correlation measure- men ts,” SIAM J. Imaging Sci. 9(4) (2016), pp. 1655-1688. [18] Y. Jiang, Z. Chen, Y. Han, P . Deb, H. Gao, S. Xie, P . Purohit, M. W. T ate, J. Park, S. M. Gruner, V. Elser & D. A. Muller “Electron ptyc hograph y of 2D materials to deep sub-angstrom resolution,” Natur e 559 343-349 (2018). [19] A. M. Maiden, M. J. Humphry , F. Zhang and J. M. Ro denburg, “Superresolution imaging via pt y- c hograph y ,” J. Opt. So c. Am. A 28 (2011), 604-612. [20] A. Maiden, D. Johnson and P . Li, “F urther improv ements to the ptyc hographical iterative engine,” Optic a 4 (2017), 736-745. [21] A.M. Maiden, G.R. Morrison, B. Kaulic h, A. Gianoncelli & J.M. Ro denburg, “Soft X-ray sp ectromi- croscop y using pt yc hograph y with randomly phased illumination,” Nat. Commun. 4 (2013), 1669. [22] A.M. Maiden & J.M. Rodenburg, “An impro v ed pt ychographical phase retriev al algorithm for diffrac- tiv e imaging,” Ultr amicr osc opy 109 (2009), 1256-1262. [23] S. Marchesini, H. Krishnan, B. J. Daurer, D.A. Shapiro, T. Perciano, J. A. Sethian and F.R.N.C. Maia, “SHARP: a distributed GPU-based pt ychographic solver,” J. Appl. Cryst. 49 (2016), 1245-1252. [24] Y. S. G. Nashed, D. J. Vine, T. Peterk a, J. Deng, R. Ross and C. Jacobsen, “P arallel ptyc hographic reconstruction,” Opt. Expr ess 22 (2014) 32082-32097. [25] P .D. Nellist, B.C. McCallum & J.M. Ro denburg, “Resolution b eyond the information limit in trans- mission electron microscop y , ” Natur e 374 (1995) 630-632. [26] P .D. Nellist and J.M. Rodenburg, “ Electron ptyc hography . I. Exp erimental demonstration beyond the con v en tional resolution limits,” A cta Cryst. A 54 (1998), 49-60. [27] K.A. Nugent, “Coheren t metho ds in the X-ray sciences, ” A dv. Phys. 59 (2010) 1-99. [28] X. Ou, G. Zheng and C. Y ang, “Em b edded pupil function reco v ery for F ourier pt ychographic mi- croscop y ,” Opt. Exp. 22 (2014) 4960-4972. [29] X. P eng, G.J. Ruane, M.B. Quadrelli & G.A. Swar tzlander, “ Randomized ap ertures: high resolution imaging in far field,” Opt. Expr ess 25 (2017) 296187. [30] F. Pfeiffer, “X-ray pt ychograph y ,” Nat. Photon. 12 (2017) 9-17. [31] J.M. Rodenburg, “Ptyc hography and related diffractiv e imaging methods,” A dv. Imaging Ele ctr on Phys. 150 (2008) 87-184. [32] M.H. Seab erg, A. d’Aspremon t & J.J. T urner, “Coherent diffractive imaging using randomly co ded masks,” Appl. Phys. L ett. 107 (2015) 231103. [33] M. Sto ckmar, P . Clo etens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P . Thibault,“Near-field pt yc hograph y: phase retriev al for inline holography using a structured illumination,” Sci. R ep. 3 (2013), 1927. [34] D. Sylman, V. Mic´ o, J. Garca & Z. Zalevsky , “Random angular co ding for sup erresolved imaging,” Appl. Opt. 49 (2010), 4874-4882. [35] P . Thibault, M. Dierolf, O. Bunk, A. Menzel, F. Pfeiffer, “Prob e retriev al in pt ychographic coherent diffractiv e imaging,” Ultr amicr osc opy 109 (2009), 338-343. [36] P . Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, F. Pfeiffer, “High-resolution scanning X-ray diffraction microscop y”, Scienc e 321 (2008), 379-382. [37] P . Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measuremen ts,” Natur e 494 (2013), 68-71. [38] F. Zhang, B. Chen, G. R. Morrison, J. Vila-Comamala, M. Guizar-Sicairos & I. K. Robinson, “Phase retriev al b y coheren t mo dulation imaging,” Nat. Comm. 7 (2016):13367. [39] X. Zhang, J. Jiang, B. Xiangli, G.R. Arce, “Spread spectrum phase mo dulation for coherent X-ray diffraction imaging,” Optics Expr ess 23 (2015), 25034-25047. [40] G. Zheng, R. Horstmeyer and C.Y ang, “Wide-field, high-resolution F ourier ptyc hographic mi- croscop y ,” Natur e Photonics 7 (2013), 739-745. DEP AR TMENT OF MA THEMA TICS, UNIVERSITY OF CALIFORNIA, D A VIS, CALIFORNIA 95616, USA. EMAIL: FANNJIANG@MATH.UCDAVIS.EDU 23