Iterative Data Refinement for Soft X-ray Microscopy

Transcription

1 Iterative Data Refinement for Soft X-ray Microscopy Presented by Joanna Klukowska August 2, 2013

2 Outline Iterative Data Refinement Transmission X-ray Microscopy Numerical Tests Joanna Klukowska IDR for Soft X-ray Microscopy Current: 2, total: 24

3 Outline Iterative Data Refinement Transmission X-ray Microscopy Numerical Tests Joanna Klukowska IDR for Soft X-ray Microscopy Current: Outline, total: 24

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5 Iterative data refinement (IDR) is a general procedure for producing a sequence of estimates of the data that would be collected by a measuring device which is idealized to a certain extent, starting from the data that are collected by an actual device.

6 Iterative data refinement (IDR) is a general procedure for producing a sequence of estimates of the data that would be collected by a measuring device which is idealized to a certain extent, starting from the data that are collected by an actual device. The discrepancy between the actual data and data which are to a certain extent idealized, henceforth called ideal data, can sometimes be estimated from the actual data (based on knowledge of the measuring process), leading to a better approximation of the ideal data. This new approximation can then be used to estimate the new discrepancy, and the process can be repeated.

7 v start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k ideal measuring device operator I? actual measuring device operator D recovery operator R discrepancy operator U v Up k correction operator + µp dev + Up k

8 v start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k ideal measuring device operator I actual measuring device operator D recovery operator R discrepancy operator U = (I µd)r v Up k correction operator + µp dev + Up k

9 v start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k ideal measuring device operator I actual measuring device operator D recovery operator R discrepancy operator U = (I µd)r v Up k correction operator + µp dev + Up k

10 v start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k ideal measuring device operator I actual measuring device operator D recovery operator R discrepancy operator U = (I µd)r v Up k correction operator + µp dev + Up k

11 v start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k ideal measuring device operator I actual measuring device operator D recovery operator R discrepancy operator U = (I µd)r v Up k µ correction operator + µp dev + Up k

12 v choice of p 0 start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k ideal measuring device operator I actual measuring device operator D recovery operator R discrepancy operator U = (I µd)r v Up k µ correction operator + µp dev + Up k

13 IDR Algorithm Given: device operator D Select: ideal device operator I recovery operator R the first iterate p 0 Compute improved projection data: p k+1 = µp dev + Up k = µp dev + (I µd) Rp k Joanna Klukowska IDR for Soft X-ray Microscopy Current: 7, total: 24

14 IDR Algorithm Given: device operator D Select: ideal device operator I recovery operator R the first iterate p 0 Compute improved projection data: p k+1 = µp dev + Up k = µp dev + (I µd) Rp k Alternative formulation: Iterative Object Refinement v 0 = Rp 0 v k+1 = µrp dev + R (I µd) v k Joanna Klukowska IDR for Soft X-ray Microscopy Current: 7, total: 24

15 Applications of IDR Sorzano, C. O. S., Marabini, R., Herman, G. T., Censor, Y., & Carazo, J. M. (2004). Transfer function restoration in 3D electron microscopy via iterative data refinement. Physics in medicine and biology, 49(4), 509. Joanna Klukowska IDR for Soft X-ray Microscopy Current: 8, total: 24

16 CTF correction in 3DEM using IDR phantom: cubic grid, with 3.5 A between grid points, based on an atomic structure Halobacterium halobium bacteriorhodopsin (PDB id: 1BRD, Henderson et al (1990)) Joanna Klukowska IDR for Soft X-ray Microscopy Current: 9, total: 24

17 CTF correction in 3DEM using IDR projection data: 2000 noisy projections convolved with CTF Joanna Klukowska IDR for Soft X-ray Microscopy Current: 9, total: 24

18 CTF correction in 3DEM using IDR (a) (b) (c) (d) (e) reconstruction slices: (a) phantom, (b) reconstruction without CTF correction, (c) reconstruction with phase flipping, (d,e) reconstruction with IDR after one and six iterations. Joanna Klukowska IDR for Soft X-ray Microscopy Current: 9, total: 24

19 Outline Iterative Data Refinement Transmission X-ray Microscopy Numerical Tests Joanna Klukowska IDR for Soft X-ray Microscopy Current: Outline, total: 24

20 Image Formation Model Image formation model is a mathematical description of the physical process by which the projections are obtained. Joanna Klukowska IDR for Soft X-ray Microscopy Current: 10, total: 24

21 Image Formation Model Image formation model is a mathematical description of the physical process by which the projections are obtained. CT: ray transform models computed tomography: p θ (x 1, x 2 ) = R v θ (x 1, x 2, x 3 ) dx 3. Joanna Klukowska IDR for Soft X-ray Microscopy Current: 10, total: 24

22 Image Formation Model Image formation model is a mathematical description of the physical process by which the projections are obtained. CT: ray transform models computed tomography: p θ (x 1, x 2 ) = R v θ (x 1, x 2, x 3 ) dx 3. TEM: (distance-dependent) ray transform models transmission electron microscopy: p TEM θ (x 1, x 2 ) = R v θ (x 1, x 2, x 3 ) h (x 1, x 2, x 3 ) dx 3. x 1,x 2 }{{} blurring Joanna Klukowska IDR for Soft X-ray Microscopy Current: 10, total: 24

23 Image Formation Model Image formation model is a mathematical description of the physical process by which the projections are obtained. CT: ray transform models computed tomography: p θ (x 1, x 2 ) = R v θ (x 1, x 2, x 3 ) dx 3. TEM: (distance-dependent) ray transform models transmission electron microscopy: p TEM θ (x 1, x 2 ) = R v θ (x 1, x 2, x 3 ) h (x 1, x 2, x 3 ) dx 3. x 1,x 2 }{{} blurring TXM: distance-dependent attenuated ray transform models transmission x-ray microscopy: p TXM θ (x 1, x 2 ) = R c (x 1, x 2 ) v θ (x 1, x 2, x 3 ) e ( x 3 }{{}}{{} h (x 1, x 2, x 3 ) dx 3. x 1,x 2 calibration attenuation }{{} v θ(x 1,x 2,t)dt) blurring Joanna Klukowska IDR for Soft X-ray Microscopy Current: 10, total: 24

24 Outline Iterative Data Refinement Transmission X-ray Microscopy Numerical Tests Joanna Klukowska IDR for Soft X-ray Microscopy Current: Outline, total: 24

25 Phantom size: 5.51 µm 5.51 µm digitization: array, pixel size: 0.01 µm 0.01 µm features: a large disk with LAC of 0.2 µm 1 smaller disks of different sizes (0.3 µm, 0.2 µm, 0.15 µm, 0.08 µm, 0.06 µm, 0.04 µm) with LAC of 0.4 µm 1 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 11, total: 24

26 IDR Algorithm - Our Implementation Given: device operator D - simulated soft x-ray microscope Select: ideal device operator I - ray transform (line integrals) recovery operator R - ART, 10 iterations, relaxation parameter 0.5 the first iterate p 0 - soft x-ray microscopy data corrected for attenuation Compute the iterates using: v 0 = Rp 0 v k+1 = µrp dev + R (I µd) v k Joanna Klukowska IDR for Soft X-ray Microscopy Current: 12, total: 24

27 IDR Reconstruction v v 0 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

28 IDR Reconstruction v v 1 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

29 IDR Reconstruction v v 2 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

30 IDR Reconstruction v v 3 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

31 IDR Reconstruction v v 4 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

32 IDR Reconstruction v v 5 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

33 IDR Reconstruction v v 6 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

34 IDR Reconstruction v v 7 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

35 IDR Reconstruction v v 8 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

36 IDR Reconstruction v v 9 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

37 IDR Reconstruction v v 10 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

38 IDR Reconstruction v v 11 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

39 IDR Reconstruction v v 12 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

40 IDR Reconstruction v v 13 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

41 IDR Reconstruction v v 14 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 13, total: 24

42 v 0 v 4 v 7 v 12

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47 Large Phantom size µm µm digitization: array, pixel size: 0.01 µm 0.01 µm features: large disk with LAC of 0.2 µm 1 smaller disks of different sizes (0.3 µm, 0.2 µm, 0.15 µm, 0.08 µm, 0.06 µm, 0.04 µm) with LAC of 0.4 µm 1 two large ellipsis with LAC of 0.3 µm 1 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 17, total: 24

48 IDR Reconstruction (Large Phantom) v v 0 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 18, total: 24

49 IDR Reconstruction (Large Phantom) v v 1 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 18, total: 24

50 IDR Reconstruction (Large Phantom) v v 2 Joanna Klukowska IDR for Soft X-ray Microscopy Current: 18, total: 24

51 v start with k = 0 p k ideal measuring device operator I actual measuring device operator D recovery operator R increment k p p dev v k recovery operator R ideal measuring device operator I discrepancy operator U = (I µd)r actual measuring device operator D v Up k correction operator + µp dev + Up k v k+1 = µrp dev + R (I µd) v k

52 Summary 1 We have a new forward problem modeling image formation in soft x-ray microscopy. 2 The problem does not have, as of now, a theoretical inversion method. 3 IDR approach provides some improvement for small size samples,... but does not seems to produce desirable quality reconstructions for large samples. Joanna Klukowska IDR for Soft X-ray Microscopy Current: 20, total: 24

53 Things To Do 1 Is there an inverse (in mathematical sense) for the forward problem of soft x-ray microscopy? If it exist and we find it, it will provide basis for development of reconstruction algorithms that are specifically designed for soft x-ray microscopy (not computed tomography or electron microscopy as it is done currently). Joanna Klukowska IDR for Soft X-ray Microscopy Current: 21, total: 24

54 Things To Do (cont.) 2 Are there any other choices for the sarting point, the ideal and device operators that can produce better results? Joanna Klukowska IDR for Soft X-ray Microscopy Current: 22, total: 24

55 Things To Do (cont.) 3 Data collected from multiple defocus points. 1 if we can use MANY defocus points, we can produce good reconstruction > but in practice we cannot get data from enough defocus points; 2 if we have data from only a few defocus points the reconstruction is not so good > BUT we think that it is good enough to provide a better starting point for the IDR. Joanna Klukowska IDR for Soft X-ray Microscopy Current: 23, total: 24

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57 Thank You!