Demonstration of inverse scattering in optical coherence tomography

Transcription

1 Demontration of invere cattering in optical coherence tomography Tyler S. Ralton a,b, Dan Mark a,b, P. Scott Carney a,b, and Stephen A. Boppart a,b,c,* a Beckman ntitute for Advanced Science and Technology b Department of Electrical and Computer Engineering c Department of Bioengineering, nternal Medicine Univerity of llinoi at Urbana-Champaign ABSTRACT An object tructure can be better reolved in optical coherence tomography by uing invere cattering theory, which take into account the finite beam width and focuing. Specifically, we how experiment where catterer are reolved outide of the confocal region uch that reolution i uniform to the focued region. Numerical imulation demontrate the effectivene of thi technique. When the algorithm i applied to experimentally-acquired OCT data, the tranvere reolution outide of the confocal parameter i improved, extending the apparent confocal parameter range. The experimental reult validate improvement for capabilitie of OCT to perform high-reolution croectional imaging. Keyword: Optical coherence tomography, invere cattering, Gauian beam, tranvere reolution, focuing. NTRODUCTON Optical microcopy ha long relied on the deign of phyical optical element to produce image of ample. However, with the advent of canning modalitie uch a confocal microcopy, near-field canning optical microcopy and optical coherence tomography (OCT, image quality can determined a much by algorithm development a the quality of optical element. Data ynthei and image formation algorithm have been crucial in other non-optical imaging modalitie uch a ynthetic aperture radar (SAR where improved algorithm have dramatically increaed the performance of uch ytem. For example, the modeling of phyical parameter ha led to enhanced mode of tripmap and potlight SAR imaging. n thee problem field quantitie have been better defined thu providing a method for invere of the data into an image. Adaptive optic and axicon lene are a election of hardware for OCT which can help generate image with high tranvere reolution over relatively large canning depth in a pecimen. Thee optical technique like dynamic focuing or focu tracking are ueful for generating higher reolution cro-ectional imaging 3, where the tight focu i canned in depth into the pecimen 4. Dynamic focuing technique in a ytem deign may require pecific hardware modification that can be difficult to control in real time. Some author have deigned algorithm that improve the axial reolution by compenating for the nonlinear diperion between data in the temporal domain and the patial domain 5. Of thee method, ome are ued to correct for the limited bandwidth of the laer pectrum, while other correct for the diperion induced by the optical ytem or the pecimen. Modeling of the cattering procee ha been limited to a one-dimenional quai-monochromatic model 6. Thee model do not take into account the relationhip between data acquired at multiple tranvere poition of the beam nor the finite tranvere extent propertie of the medium. Some author have tried to correct for artifact produced by ample poitioning, refraction, and the canning procedure 7. n addition, other have detailed a theoretical model of OCT including a len and heterodyning 8. Our goal i to digitally reduce the ditortion outide of the confocal region by olving the invere problem baed on the phyic of the cattering proce. We formulate a mathematical model to connect the experimentally acquired OCT ignal with the three-dimenional object tructure, taking into account the finite beam width and focuing. Therefore, by olving the invere cattering problem, we can better determine the full tructure of an imaged object from a data acquiition. * For correpondence with Dr. Stephen A. Boppart boppart@uiuc.edu phone: ( Coherence Domain Optical Method and Optical Coherence Tomography in Biomedicine X edited by Valery V. Tuchin, Joeph A. att, Jame G. Fujimoto, Proc. of SPE Vol T, ( /6/$5 doi:.7/ Proc. of SPE Vol T-

2 OCT MODEL n order to implify the model for OCT data acquiition everal aumption are generally made about an OCT ytem. Thee aumption do not take into account the hape of wavefront produced by len optic, the pectrum of the ource, or even unbalanced diperion in the media. Thu, many OCT image exhibit poor tranvere reolution outide of the confocal region, which manifet a curved and blurred feature obtained in thee area. The apect of the invere problem thi work addree i the reolution of catterer outide of the confocal region in experimental OCT data. 3 An equation for the acquired OCT ignal S( r, k = A( k d r d rg (,, k g(, k η( g(, k r r r r r r r i V preented in term of our model, ee Fig.. Thi equation repreent the linear forward problem having the form of a type Fredholm integral. The adjoint and the normal operator can be diagonalied by a coordinate tranformation in the Fourier pace uch that a implified leat quare olution i formed. By olving the invere problem, we are able to produce image with more harply defined feature. More importantly, we are able to ditinguih cloely adjacent catterer, even thoe that produce interference in the raw OCT image. Thi i a crucial advantage of invere cattering over imple deconvolution of a real valued point pread function. The field at the output face of the fiber i given by U ( r = A( k φ( r. The field propagate to the plane i = = l where a thin len i located. To the left of the = l plane, the field i given by the expreion U ( r = A( k d r P(, φ( i r r r, (. where P (, rr i the free-pace diffraction kernel. The len i aumed to be a thin phae creen with tranfer function given by L( r, ee Fig.. Thu, jut to the right of the len ( = l the field i given by U ( r = A( k L( r d r P(, φ( i r r r. The field i focued into a Gauian beam with beam wait in the = plane. The field i given by the expreion U ( r = A( k g( r. Thu the normalied beam profile i alo given by the i expreion g ( r d r L ( r d r P ( r, r φ( r P ( r, r. (. = = = l The coupling of light back into the fiber i now derived in term of the normalied wait plane field. Conider a field that propagate back toward the fiber and i given in ome plane Σ by U ( r on the right ide of the len. The field may be propagated to the len where it acquire the factor L( r. The field incident on the fiber face U ( r will = then be given by the expreion, U ( r d r P( r, r L( r d r P( r, r U ( r. (.3 = = = Σ l The ignal coupled into the ingle mode fiber i then given by the inner product of thi field (Eq.(.3 and the fiber mode φ(r, S = d rφ ( r U ( r. (.4 = Equation (.3 and (.4 may be combined to yield S d r ( r d r P( r, r L( r d r P( r, r U ( r. (.5 = φ = = Σ l The field emerging from the fiber i aumed to be a diverging Gauian beam with beam wait at =. Thu, φ ( r = φ( r. By comparing Eq. (. and (.5 and noting P( r, r = P( r, r, it may be een that S = d rg( r U ( r. (.6 Σ Thi i the expreion for the ignal ued above. Proc. of SPE Vol T-

3 S( r, k r r r U (, rr, k η ( r G( r, r, k g( r r, k Fig.. The diagram of cattering from a Gauian beam, where r i the tranvere poition of the beam, G(r,r,k i the radiated Green Function, g(r -r,k i the tranlated incident field with a Gauian beam profile, and η(r i the uceptibility of the object. The vector r decribe a point in the object volume, and the vector r decribe point on the boundary of the volume. The ignal may be written compactly a S% ( Q, k = i πa( k d H( Q,, k % η( Q;, (.7 ( ( ( ( where (,, ik q (, ik Qq H Q k = dq e g% q ke g% ( Qq, k. k ( q Proc. of SPE Vol T-3

4 L( r Optical Fiber φ( r = = = l L( r (a Optical Fiber U ( r = = l (b Fig. llutration of the coupling of light (a out of, and (b into the fiber. The kernel H take the form ( Q α + i α ( i( k k k H( Q,, k = ( f f ( Q,, k = + i e e % %. (.8 k k Equation (.7 may be een to be a type Fredholm integral equation and the kernel of the operator K i apparent, S% ( ( Q, k = K % η( Q ; = i πa( k dd Q δ ( Q Q H ( Q,, k % η( Q ;. (.9 Proc. of SPE Vol T-4

5 i Subtituting ( ; d e β % η Q = β % η( Q ; β, we obtain a repreentation for K π ( S % ( Q, k = ia( k dβd Q' δ ( QQ H % ( Q, β, k % η ( Q ; β, (. where, u ( β = H k e e u k 4k α k ( β k i β % Q k ( Q, β, = β +. The Heaviide unit tep function i denoted by β >. β The kernel of the tranform in Eq. (.9, H(,, k = ( Q α + i α ( i( k k k + i e e Q, (. k k mut undergo a Fourier tranform to arrive at Eq. (.. Remember that our Fourier tranform and invere Fourier tranform are defined by % iβ β, (. H( Q,, k = d e H( Q,, k and iβ H( Q, k, = dβe H % ( Q, β, k π, (.3 repectively. Thu by ubtitution of Eq. (. into Eq. (., we attain ( Q α + i ( iβ α i( k k k H% ( Q, β, k = de + i e e. (.4 k k α ( Next, we can pull out the contant and rewrite + i k k to fit a well-known Fourier tranform form, Q α ( + i iβ i( k k k H% ( Q, β, k = de k e e, (.5 α k + i( By introducing a phae hift term, i e β, inide and outide of the integral, we can complete the Fourier tranform variable. Q α ( + i iβ iβ iβ i( k k k H% ( Q, β, k = e d e e k e e, α k + i( (.6 Grouping all the (- term to the right produce Q α iβ ( k iβ i k k α k + i( Q ( ( ( i H% ( Q, β, k = e d ke e e e Pulling all the integration, d, contant out of the integral give Combining all the (- term reduce H to Q α iβ ( k iβ i k k α k + i( Q ( ( ( i H% ( Q, β, k = e ke d e e e. (.7. (.8 Proc. of SPE Vol T-5

6 Q α Q i( k β + iβ 4 k k α k + i H% ( Q, β, k = e ke d e (. (.9 Since i contant, H i equivalently tated Q α Q i( k β + iβ 4 k k α k + i ( H% ( Q, β, k = e ke d e. (. ( The form of the Fourier tranform of a unit tep function time a decaying exponential, in thi cae for Re{a} >, i known and i arrived at in the following way: β ( a+ i aβ i β aβ i β ( a i β e ( = = + = = ( + + dβ e u β e dβ e e e dβ a i a i. (. Therefore, one can ee that aβ i β dβ e u( β e = π π a+ i, (. and thu aβ i β dβ πe u( β e = π a+ i. (.3 Thi fit the form of our Fourier tranform, hence Q α Q i( k β + iβ 4 k k α k + i ( H% ( Q, β, k = e ke d e (.4 ( can be evaluated to be Q α α Q β k + i k 4k Q β k H% ( Q, β, k = e ke πe u β + k. 4k (.5 The like term can be grouped, and then reduced to Q α Q α α ( β k i Q β 4k 4k k H% ( Q, β, k = e ke e πe u β + k, (.6 4k α k ( β k iβ Q k H% ( Q, β, k = e e u β + k. (.7 4k Remember that thi olution i poible when conidering a real, poitive, bandlimited k. NUMERCAL SMULATONS The goal of our numerical imulation i to demontrate cattering and invere cattering for a ynthetic object. The forward OCT model imulate fringe from ub-reolution-ied point catterer. Figure 3 diplay imulated and recontructed image of an imaging area of 4 µm by 4 µm, where the bandwidth i 34 nm, the focal length of the len i mm, the pot ie i 4 µm, the confocal parameter i 3 µm, and the NA i.. Thee parameter correpond to a NA and bandwidth larger than uually encountered in OCT ytem in order to accentuate the ditortion effect. Proc. of SPE Vol T-6

7 a ts e a,. -.- a _ _ S C P a.. a a L. a a - a S.... a- -.. Fig. 3. Simulated OCT image, SNR = 35 db (top, and Tikhonov regularied olution (bottom. EXPERMENTAL DATA A collection of 8 to 6 µm diameter catterer were upended in ilicone and imaged with a pectral-domain OCT (SD- OCT ytem. Figure 4 diplay original and adjoint image of an imaging area of 5 (tranvere by µm (axial, where the bandwidth i nm, the focal length of the len i mm, the pot ie i 8 µm, and the confocal parameter i 5 µm. The tranvere image reolution of point catterer outide of the confocal region for the original experimental image data i not contant, but for the adjoint olution, the tranvere reolution i relatively contant along the entire image with only amplitude variation. Proc. of SPE Vol T-7

8 (a (b Fig. 4 Experimental SD-OCT image, (a original (b adjoint olution. CONCLUSONS A full model of OCT i preented for which the forward operator i formulated, thu giving rie to adjoint, normal, and invere olution. The reult how that linear etimation of the uceptibility of an object from OCT data i poible when uing an accurate model for the probe beam. Uing regulariation and the minimum norm olution, we can recontruct underlying object tructure with patially invariant reolution in both imulated and experimental OCT image data et, both of which contain region of defocu. Two- and three-dimenional OCT data can be ued for reolving object and recovering high-reolution detail outide of the confocal region with minimal lo of reolution. The model preented account for the effect of the ource pectrum and the finite beam width, a well a diperion, diffraction, and defocu effect. REFERENCES. Cooper, P. S., Won, A. F. & Gakell, A. P. High reolution ynthetic aperture radar uing a multiple ub-band technique. EE Radar, (997.. Ding, Z., Ren, H., Zhao, Y., Nelon, J. S. & Chen, Z. High-reolution optical coherence tomography over a large depth range with an axicon len. Optic Letter 7, (. 3. Schmitt, J. M. Optical Coherence Tomography (OCT: A Review. EEE Journal of Selected Topic in Quantum Electronic 5, 5-5 ( Cobb, M. J., Liu, X. & Li, X. Continuou focu tracking for real-time optical coherence tomography. Optic Letter 3, (5. 5. Kulkarni, M. D., Thoma, C. W. & att, J. A. mage enhancement in optical coherence tomography uing deconvolution. Electronic Letter 33, ( Bruno, O. & Chaubell, J. One-dimenional invere cattering problem for optical coherence tomography. ntitute of Phyic Publihing, nvere problem, (5. Proc. of SPE Vol T-8

9 7. Podoleanu, A., Charalambou,., Pleea, L., Dogariu, A. & Roen, R. Correction of ditortion in optical coherence tomography imaging of the eye. ntitute of Phyic Publihing, Phyic in Medicine and Biology, (4. 8. Feng, Y. & Wang, R. K. Theoretical model of optical coherence tomography for ytem optimiation and characteriation. Journal of the Optical Society of America, (3. Proc. of SPE Vol T-9