Exact Crystalline Structure Recovery in X-ray Crystallography from Coded Diffraction Patterns

Exact Crystalline Structure Reco v ery in X-ra y Crystallograph y from Co ded Diffraction P atterns Sam uel Pinilla ∗ Dep artment of Ele ctr onic Engine ering, Universidad Industrial de Santander Jorge Bacca † Dep artment of Computer Scienc e, Universidad Industrial de Santander Cesar V argas F undaci´ on Universitaria Konr ad L or enz Juan P ov eda Dep artment of Chemistry, Universidad Industrial de Santander Henry Arguello Dep artment of Computer Scienc e., Universidad Industrial de Santander (Dated: Marc h 22, 2021) X-ra y crystallography (X C) is an exp erimen tal technique used to determine three-dimensional crystalline structures. The acquired data in X C, called diffraction patterns, is the F ourier magnitudes of the unkno wn crystalline structure. T o estimate the crystalline structure from its diffraction patterns, w e prop ose to mo dify the traditional system b y including an optical element called co ded ap erture which mo dulates the diffracted field to acquire co ded diffraction patterns (CDP). F or the prop osed co ded system, in con trast with the traditional, we derive exact reconstruction guarantees for the crystalline structure from CDP (up to a global shift phase). Additionally , exploiting the fact that the crystalline structure can b e sparsely represen ted in the F ourier domain, we develop an algorithm to estimate the crystal structure from CDP . W e show that this metho d requires 50% few er measurements to estimate the crystal structure in comparison with its comp etitive alternativ es. Sp ecifically , the prop osed method is able to reduce the exp osition time of the crystal, implying that under the prop osed setup, its structural in tegrity is less affected in comparison with the traditional. W e discuss further implementation of imaging devices that exploits this theoretical co ded system. I. INTR ODUCTION X-ra y Crystallograph y (XC) is kno wn as the leading tec hnique for molecular structure c haracterization in ma- terial analysis [1]. Sp ecifically , XC plays an essential role in fields as biology [2], drug design [3], and material sci- ences [4], among others. The traditional acquisition sys- tem in X C is illustrated in Fig.1, where the acquired data forms “the Ewald sphere” according to the p erturbation theory which is mathematically mo deled as the F ourier magnitude of the unkno wn crystalline structure called diffraction patterns [5]. Sp ecifically , in Fig.1 the result- ing diffraction patterns, when an X-ra y source irradiates a crystal, are recorded while b oth the sensor and the crystal gradually rotate [6]. The crystal structure can b e uniquely estimated from the phase of its diffraction patterns [5]. Although, the phase of the diffraction patterns cannot b e directly mea- sured, it can b e reco vered from their phaseless intensit y measuremen ts [7], problem that is known as phase re- triev al (PR) [7, 8]. Mathematically , the PR problem ∗ samuel.pinilla@correo.uis.edu.co; http://diffraction.uis.edu.co † jorge.bacca1@correo.uis.edu.co X-Ray Source Cry stal θ ' ɸ ' Detecto r Far Field Sensor Ew ald sphere FIG. 1. T raditional acquisition system of diffraction patterns in XC. An X-ra y source irradiates a crystal pro ducing diffrac- tion patterns that are recorded with a tw o-dimensional sensor. in XC consists on retrieving a discrete version of the crystalline structure x ∈ C n from the phase less mea- suremen ts y k = | f H k x | 2 where n is the discrete size of the crystal, k = 1 , · · · , m with m the total n um b er of measuremen ts, f k ∈ C n are the vectors that mo dels the three-dimensional discrete F ourier transform [9], and ( · ) H represen ts the conjugate transp ose op eration. The PR problem has b een challenging due to its infinite solutions [7, 10]. Sp ecifically , there are so-called trivial ambigu- ities that are alw ays presen t [7]. The following three 2 transformations (or any combination of them) conserve F ourier magnitude: 1) glob al shift phase: x [ a ] = x [ a ] e j φ 0 ; 2) c onjugate inversion : x [ a ] = x [ − a ]; 3) sp atial shift : x [ a ] = x [ a + a 0 ]. Moreov er, the num b er of measurements m m ust satisfy m ≥ 4 n − 1 to accurately retrieve the signal x (up to trivial am biguities) [10, 11]. i.e. a 4-fold time exp osition X-ray radiation is required, whic h leads to degradation of the crystalline structure [12 – 17]. Addi- tionally , if the crystal deteriorates under this irradiation, material characterization b ecomes more c hallenging [14]. Despite such challenges, several metho ds hav e b een dev elop to retriev e the crystalline structure. T o name a few, the error reduction metho d [18], dual space pro- grams SnB and ShelxD [19], iterated pro jections [20], and the charge flipping algorithm [21]. All these meth- o ds alternate betw een real and recipro cal space by the F ourier transform by imp osing constrain ts on the real- space charge density . All these algorithms require to o ver-expose the crystal to the X-ra y source in order to acquired the large enough amoun t of diffraction patterns needed to estimate the crystalline structure i.e. they w ork under the m ≥ 4 n − 1 regime. F urther, these ap- proac hes tend to return inaccurate estimates of the crys- talline structure when the measurements y k are corrupted b y noise [7]. Also, the conv ergence to the true crystalline structure of the ab o ve methods is not guaran teed. This work prop oses to mo dify the traditional system by including an optical element called co ded aperture which mo dulates the diffracted field to acquire coded diffraction patterns (CDP). The main adv an tage of the prop osed system is that only the global phase shift ambiguit y ap- p ears, that is, the conjugate in v ersion and spatial shift am biguities will not o ccur [22, 23]. This implies that retrieving the crystalline structure from CDP is muc h effortless compared with the traditional XC. F or the pro- p osed co ded system, w e deriv e exact reconstruction guar- an tees for the crystalline structure from CDP (up to a global shift phase). Additionally , giv en the fact that the crystalline structure is a p erio dic element, it can b e sparsely represen ted in the F ourier domain, i.e. the num- b er of non-zero coefficients of the F ourier transformed crystal is muc h smaller than n . Thus, exploiting the sparsit y prior of the crystal we develop an algorithm to estimate the crystalline structure from CDP that requires 50% fewer measuremen ts in comparison with its comp et- itiv e alternativ es. Sp ecifically , the prop osed method is able to reduce the exp osition time of the crystal, imply- ing that under the proposed setup, its structural integrit y is less affected in comparison with the traditional. Nu- merical sim ulations are conducted to ev aluate the p erfor- mance of the prop osed metho d using syn thetic data. I I. PR OBLEM F ORMULA TION In con trast to the traditional acquisition system illus- trated in Fig. 1, a co ded diffraction system, as sho wn in Fig. 2, is proposed. Notice that Fig. 2 includes an optical element kno wn as co ded ap erture, mo deled as D , at distance z p from the sensor. This optical elemen t mo dules the diffracted field before being measured by the sensor. Besides, c hanging the distance z p , this acquisi- tion system allows acquiring multiple snapshots, where p = 1 , · · · , P indexes the sensing distances. The signal ˜ x ∈ C n in Fig. 2 corresp onds with the F ourier transform of x , that is, ˜ x = Fx , where F represen ts the 3D discrete F ourier transform matrix. X-Ray Source Coded Aperture Sensor Crystal Far Field Near Field Ewald sph er e FIG. 2. Illustrativ e configuration to acquire co ded diffrac- tion patterns from a crystal. A co ded ap erture is lo cated at a distance z p from the sensor suc h that the acquired data corresp onds with diffraction patterns in the near field. Observ e that in Fig. 2, w e assume that the co ded ap erture D is lo cated at the far field from the crystal. Moreo ver, the distance z p b et w een the coded ap erture and the sensor is c hosen so that the co ded diffracted field, captured at the detector, b elongs to the near field. Addi- tionally , we assume that the co ded ap erture and the sen- sor join tly p erform a raster scanning across “the Ew ald sphere”, as illustrated in Fig. 3, while the crystal remains fixed. Thus, in con trast with the traditional arc hitecture, crystal rotation is a voided. Mathematically , a giv en high- ligh ted region in Fig. 3 is mo deled as ˜ x r = S r ˜ x where S r is a selection diagonal matrix where r = 1 , · · · , R with R as the n umber of regions. FIG. 3. Illustration of the prop osed scanning process to “the Ew ald sphere” of the system in Fig. 2. Each highlighted region in the dashed sphere corresp onds to a different rotation of the sensor and the co ded ap erture. In order to easily mo del the acquired data in Fig. 2, ˆ D ∈ C n × n is defined as a diagonal matrix whose entries are the elements of D . Th us, from the traditional diffrac- tion theory [23, 24], the measurements b eing captured at the detector for the r -th region and at a distance z p , are 3 giv en by g i p,r = |h a p,i , ˜ x r i| 2 , i = 1 · · · , n, (1) where a p,i are the sampling vectors defined as a p,i = ˆ DFT ( z p ) f u i , (2) where u i = ( i − 1) mo d n + 1, T ( z p ) is the spatial fre- quency transfer function which dep ends on the distance z p [23, 24], and ( · ) represents the conjugate op eration. No w, considering the definition of ˜ x r w e ha v e that (1) can b e equiv alen tly expressed as g i p,r = |h b r p,i , x i| 2 , i = 1 · · · , n, (3) where b r p,i = F H S r a p,i . Observe that under the prop osed acquisition system, the maxim um n umber of measure- men ts is given b y m ≤ nRP . Additionally , w e remark that the acquired measuremen ts g i p,r in Fig. 2 are differ- en t than y k in Fig. 1 due to the inclusion of the co ded ap erture. This pap er deals with the problem of recov ering x ∈ C n from the phaseless measuremen ts { g i p,r } . Additionally , w e assume that the en tries of a co ded aperture D are i.i.d copies of a discrete random v ariable d ob eying | d | ≤ 1. This assumption ov er the co ded ap erture is imp ortant b ecause under a random sensing pro cess uniqueness is guaran teed (up to a global phase shift) as will b e shown in Section I I I. This implies that retrieving the crystalline structure from CDP is m uc h easier compared with the traditional X C since the conjugate inv ersion and spatial shift am biguities will not o ccur [22, 23]. Moreov er, ex- ploiting the fact that the crystalline structure is perio dic, it can b e sparsely represented in the F ourier domain, i.e. k Fx k 0 = s  n where k·k 0 is the ` 0 pseudo-norm, w e form ulate the following optimization problem min x ∈ C n h ( x ) = 1 m n X i =1 R X r =1 P X p =1  q g i p,r − |h b r p,i , x i|  2 , s.t k Fx k 0 ≤ s (4) T o solve (4) w e prop ose a gradient descend metho d that will b e describ ed in Section IV. I II. EXACT RECO VER Y GUARANTEES Observ e that (3) can be equiv alen tly expressed as g i p,r = ( b r p,i ) H xx H b r p,i . (5) No w, consider the linear op erator B : S n × n → R m ( S n × n is the space of self-adjoint matrices) defined as B ( W ) = h ( b 1 1 , 1 ) H Wb 1 1 , 1 , · · · , ( b R P,n ) H Wb R P,n i T , (6) and stac king the measuremen ts { g i p,r } as g := [ g 1 1 , 1 , · · · , g n P,R ] T , then w e hav e that g = B ( xx H ) . (7) Th us, to prov e that the signal x can b e exactly reco vered from the measurements g i p,r in (5), from (7) we ha ve that B ( · ) m ust b e injectiv e [25, 26]. More precisely , this work follo ws the strategy in [27] that considers T x =  xw H + wx H | w ∈ C n  , (8) as the tangent space of the manifold of all rank-1 Her- mitian matrices at the point xx H . Thus, if the opera- tor B satisfies the follo wing condition, which is pro ved in Theorem 1, one can guarantee reco very with high probabilit y[25]. Condition 1. F or any δ ∈ (0 , 1) and some c onstant β > 0 the line ar op er ator B satisfies (1 − δ ) k W k 1 ≤ 1 β kB ( W ) k 1 ≤ (1 + δ ) k W k 1 , (9) for al l matric es W ∈ T x , wher e k W k 1 = P i σ i ( W ) with σ i ( W ) as the i -th singular value of W . Theorem 1. Fix any δ ∈ (0 , 1) and the coded ap erture ˆ D ∈ C n × n , with i.i.d copies of a random v ariable d such that | d | ≤ 1. Then, w e ha v e that P  1 m k B k 2 ∞ ≤ 1 + δ  ≤ 1 − 2 e − c 0 m 2 , (10) for some constan t c 0 > 0 provided that m ≥ C s where s is the sparsity of the crystalline structure with C > 0 and B is given by B =  b r 1 , 1 , · · · , b R P,n  H . Also, Condition 1 is satisfied with the same probabilit y taking β = m , where k B k ∞ denotes the spectral norm of B . The pro of is deferred to App endix A. Theorem 1 estab- lishes tw o asp ects: first, the only am biguity that appears in the prop osed system is the global phase shift. Second, the condition m ≥ 4 n − 1 imp osed by the traditional sys- tem in XC can b e defeated by the prop osed arc hitecture since no w the num ber of measuremen ts m dep ends on the sparsity ( s  n ) of the crystal, i.e. m ≥ C s for some constan t C > 0. The v alue of C > 0 will b e n umerically estimated in Section V. IV. CR YST ALLINE STR UCTURE RECONSTR UCTION METHODOLOGY In order to solve (4), this w ork adapted the Sparse Phase Retriev al via Smo othing F unction (SPRSF) metho d in tro duced in [28]. This algorithm claims to ha ve b etter p erformance in terms of the num ber of measure- men ts compared with recent approaches in the state-of- the-art. SPRSF uses a sp ecial mapping called smo othing function, whic h is useful to eliminate the non-smo othness of h ( · ). Sp ecifically , the optimization problem solved by SPRSF is form ulated as min x ∈ C n f ( x ) = 1 m n X i =1 R X r =1 P X p =1  q g i p,r − ϕ µ ( |h b r p,i , x i| )  2 , s.t k Fx k 0 ≤ s (11) 4 where ϕ µ ( w ) is defined as ϕ µ ( w ) = p w 2 + µ 2 . The pro- p osed reconstruction algorithm consists on t wo stages as follo ws: • Initialization step: this pro cedure consists on es- timating the crystalline structure x as the leading eigen vector of a carefully designed matrix. • R efining step: the outcome of the first step is refined up on a sequence of Wirtinger gradient iter- ations. These tw o stages are summarized in Algorithm 1. Algorithm 1 Crystalline reconstruction algorithm 1: Input: Data { ( b r p,i ; g i p,r ) } . The step size τ ∈ (0 , 1), con trol v ariables γ , γ 1 ∈ (0 , 1), µ (0) ∈ R ++ , n umber of iterations T and the sparsity s . 2: Initialization: ˆ J set of s largest indices of ( 1 m n X i =1 P X p =1 R X r =1 g i p,r | ( b r p,i ) q | 2 ) 1 ≤ q ≤ n . Let ˜ z (0) b e the leading eigenv ector of the matrix H := 1 m n X i =1 P X p =1 R X r =1 g i p,r ( b r p,i ) ˆ J ( b r p,i ) H ˆ J 1 { g i p,r ≤ α 2 y φ 2 } , where α y = 3 and φ 2 = 1 m n X i =1 P X p =1 R X r =1 g i p,r . 3: z (0) ← F H  r n m φ 2  ˜ z (0) 4: for t = 0 : T − 1 do 5: ˜ z ( t +1) = H s  F  z ( t ) − τ ∂ f ( z ( t ) , µ ( t ) )  6: z t +1 ← F H ˜ z ( t +1) 7: if k ∂ f  z ( t +1) , µ ( t )  k 2 ≥ γ µ ( t ) then 8: µ ( t +1) = µ ( t ) 9: else 10: µ ( t +1) = γ 1 µ ( t ) 11: end 12: Output: z ( T ) Algorithm 1 requires the sampling vectors and the ac- quired co ded diffraction patterns as mo deled in (3) (Line 1). The initialization step is presented in Lines 2-3. F ur- ther, a thresholding step is calculated in Line 5, where the op erators H s ( w ) set all the en tries in the v ector w ∈ C n to zero, except its s largest absolute v alues. Additionally if the condition in Line 7 is not satisfied the smo othing parameter µ is updated in Line 9 to obtain a new p oint. Remark that eac h v ector ∂ f ( z ( t ) , µ ( t ) ) in Algorithm 1 is calculated using the Wirtinger deriv ative [29] as ∂ f  z ( t ) , µ ( t )  = 2 m n X i =1 R X r =1 P X p =1  ( b r p,i ) H z ( t ) − q g i p,r ( b r p,i ) H z ( t ) ϕ µ ( t ) ( |h b r p,i , z ( t ) i| ) ! b r p,i . (12) The theoretical guarantees of conv ergence of this algo- rithm are demonstrated in [28]. V. SIMULA TIONS AND RESUL TS The p erformance of the prop osed algorithm is pre- sen ted. Three differen t tests are p erformed: first, the empirical success of the proposed metho d is analyzed, among 100 trial runs. The second examines the recon- struction p erformance of Algorithm 1 for reco vering the crystalline structure. The third test studies the stability b eha vior of the recov ery algorithm under additiv e noise. The default v alues of the parameters of Algorithm 1 w ere determined using a cross-v alidation strategy . They w ere fixed as τ = 0 . 3, γ = 0 . 8, γ 1 = 0 . 5, µ (0) = 60 and T = 800. The p erformance metric used is relativ e error := dist ( z , x ) k x k 2 , where dist ( z , x ) is defined as dist ( z , x ) = min θ ∈ [0 , 2 π ) k x e − j θ − z k 2 , with j = √ − 1. The simulated co ded aperture was a blo c k-un blo c k ensem ble as used in our previous w orks [23, 30]. All sim ulations w ere implemen ted in Matlab R2019a on an Intel Core i7 3.41Ghz CPU with 32 GB RAM. T o examine the reco very success rate against the level of sparsit y s and the n umber of measurements m under a noiseless scenario, we randomly simulate 1000 crystalline structures. The success rate is determined ov er 100 trial runs for each crystalline structure when a relativ e error of 10 − 5 is reached. The results are shown in Fig. 4, where the success rate is plotted in gra y scale when white and blac k represen t 100% and 0% probability of success, resp ectiv ely . Measur ements (m/ n) Sparsit y (s/ n) Proba bi l i t y of Success FIG. 4. Empirical success rate of Algorithm 1 when the spar- sit y s and the num ber of measurements m are v aried. Each v alue is determined ov er 100 trial runs when a relativ e error of 10 − 5 is reached. The white and black colors represent 100% and 0% probability of success, resp ectively . F rom Fig. 4, it can b e concluded that the prop osed al- gorithm is able to estimate the crystalline structure when m/n ≥ 2 for all the tested sparsit y lev els. This result im- plies that the proposed acquisition system, along with its 5 Cryst all i ne Unit Cryst all i ne Struct ure Origi nal R econstruc ted 1x10 -15 relati v e error = 20x1 0 -2 relati v e error = Noisel ess ca se Noisy case FIG. 5. Returned crystalline structure using Algorithm 1 for both noiseless and noisy scenarios when m/n = 2. F or the noisy case the SNR = 30dB. The sim ulated crystal is NaCl. reconstruction approach, is able to estimate the crystal structure using 50% less measurements than the state- of-the-art metho ds that require m/n ≥ 4 [10, 11]. Th us, considering the fact that s  n , then the constant C of Theorem 1 is b ounded from ab ov e as C ≤ 2. Observ e that in practice we can ac hiev e this reduction b y choosing the num ber of sensing distances as P = 2. Finally , the reconstruction accuracy of Algorithm 1 for b oth noiseless and noisy scenarios is analyzed. The tested crystalline structure is the so dium c hloride (NaCl) as il- lustrated in Fig. 5. Specifically the green spheres in Fig. 5 mo dels the Cl atoms and the y ellow ones the so dium atoms. F or the noisy case, we study the stability b e- ha vior of Algorithm 1 under additiv e white noise with a Signal-to-Noise-Ratio (SNR) fixed as S N R = 30 dB where SNR= 20 log ( k g k 2 / ( mσ )) with σ the v ariance of the noise. The n umber of measurements m used for this exp erimen ts satisfies m/n = 2. The attained reconstruc- tions are shown in Fig. 5 suggesting the effectiveness of the prop osed metho d to estimate the crystalline struc- ture from both noiseless and noisy measurements. VI. CONCLUSION AND DISCUSSION This work presented an algorithm to recov er the 3D structure of a crystal, under a system that records co ded diffraction patterns. Our approach tak es adv an tage of the fact that the crystalline structure can b e sparsely represen ted in the F ourier domain to significan tly reduce the n um b er of measurements. Sim ulations sho w that our approac h is able to reconstruct the crystalline struc- ture with up to 50% less amount of measurements com- pared with the traditional metho ds in the state-of-the- art. Also, the results suggest that the proposed approach allo ws reconstructing the crystalline structure ev en in noisy scenarios. T o implement the prop osed acquisition system in Fig. 2 a block-un block co ded ap erture is feasible [23, 30]. Sp ecifically , the blo cking elements of these co ded ap er- tures can b e fabricated using tungsten, since this material can stop an x-ra y beam, resulting in lo w fabrication costs [31 – 33]. App endix A: Pro of of Theorem 1 T o prov e Theorem 1 we divide it in to tw o parts. First, w e prov e the right inequality in 9, and then as a second part we prov e the left inequalit y . Thus, let W ∈ T x . As W has rank at most tw o, we can choose normalized v ectors u , v ∈ C n suc h that W = λ 1 uu H + λ 2 vv H . Then considering the definition of the linear map B in (6) we ha ve that kB ( W ) k 1 = n X i =1 R X r =1 P X p =1   λ 1 |h b r p,i , u i| 2 + λ 2 |h b r p,i , v i| 2   ≤ n X i =1 R X r =1 P X p =1 | λ 1 ||h b r p,i , u i| 2 + | λ 2 ||h b r p,i , v i| 2 = | λ 1 |k Bu k 2 2 + | λ 2 |k Bv k 2 2 ≤ k W k 1 k B k 2 ∞ , (A1) in whic h the first and second inequalities are obtained using the triangular inequalit y , and matrix B as defined in Theorem 1. F urther, considering definition of matrix B we ha ve that B H B = n X i =1 R X r =1 P X p =1 F H S r ˆ DFT ( z p ) f i f H i T ( z p ) F H ˆ DS r F = ˆ D ˆ D , (A2) since F H F = FF H = I , T ( z p ) is a diagonal orthogonal matrix, and P R r =1 S r S r = I . Thus, using the fact that the admissible random v ariable d is assumed | d | ≤ 1 we ha ve from (A2) that B is an isotropic subgaussian matrix [34]. Then, from Theorem 5.39 in [34] we ha ve that P  k B k ∞ ≥ √ m + C √ s + t  ≤ 2 e − c 0 t 2 , (A3) for constants c 0 , C > 0 and an y t > 0. Then, taking m ≥ C 2  − 2 s and t = √ m for any  ∈ (0 , 1 / 2), we hav e from (A3) that P  1 m k B k 2 ∞ ≤ 1 + δ  ≤ 1 − 2 e − c 0 m 2 , (A4) 6 for δ = 2  . Thus, we conclude that 1 m kB ( W ) k 1 ≤ (1 + δ ) k W k 1 , (A5) for any δ ∈ (0 , 1). On the other hand, from (A1) we can also conclude that kB ( W ) k 1 ≥ n X i =1 R X r =1 P X p =1 λ 1 |h b r p,i , u i| 2 + λ 2 |h b r p,i , v i| 2 = λ 1 k Bu k 2 2 + λ 2 k Bv k 2 2 = ( λ 1 + λ 2 ) = k W k 1 , (A6) in which the third equality comes from observ ation in (A2), using that W is assumed to b e p ositive semidefi- nite. Th us, we hav e that 1 m kB ( W ) k 1 ≥ 1 m (1 − δ ) k W k 1 , (A7) for any δ ∈ (0 , 1). Th us, combining (A5) and (A7) the result holds. [1] M. Sm ythand J. Martin, x ray crystallography , Journal of Clinical Pathology 53 , 8 (2000). [2] M. 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