Modal Reconstruction Methods Pros and Cons. Jim Schwiegerling, Ph.D. Department of Ophthalmology and Optical Sciences The University of Arizona

Transcription

1 odal Reconstructon ethods Pros and Cons Jm Schwegerlng Ph.D. Department of Ophthalmolog and Optcal Scences The Unverst of Arzona

2 Introducton Elevaton Data Cornea Topograph (usuall Proflometr Interferometr Slope Data Shack-Hartmann Tschernng Retnal Ratracng Spatall-resolved refractometr

3 Applcatons Wavefront sensng. Ablaton pattern desgn and analss. Dnamc analss of aberratons and accommodaton. Trackng bomechancal changes and healng effects wth tme. Feature detecton (e.g. keratoconus Optcal desgn - PALs and Custom Contacts

4 Curve Fttng

5 Low-order Polnomal Ft x R

6 Hgh-order Polnomal Ft

7 Splnes

8 Surface Fttng Extenson of curve fttng to two dmensons. Fttng functons are now a lnear combnaton of fundamental surfaces. Splnes can also be extended to two dmensons. -.3 x +. x Sphercal Refractve Error Clndrcal Refractve Error +. x Coma

9 Orthogonal Polnomals Complete Sets of Orthogonal polnomals are sets of surfaces whch have some nce mathematcal propertes for surface fttng. Examples are ernke polnomals and Fourer seres. Talor polnomals (.e. x x x. are not orthogonal. ÚÚ A V V dxd Ï Constant Ì Ó Otherwse

10 W ÚÚ A a ÚÚ A Â a WV V Â Orthogonalt a WV dxd WV dxd V V Âa ÚÚ Æ A Â k V V dxd W k k V k A s a crcle of unt radus for ernke polnomals A s a square for Fourer seres k

11 Speed The long part of calculatng ernke polnomals s calculatng factoral functons. m n ( r q ÔÏ Ì ÔÓ - m n m n R R m n m n ( rcos mq ( rsn mq ;for m ;for m < R m n ( r (n- m / Â s s! (- s (n s! [.5(n + m - s][!.5(n - m - s] - r! n-s

12 Speed Chong et al.* developed a recurrence relatonshp that avods the need for calculatng the factorals. The results gve a blazng fast algorthm for calculatng ernke expanson coeffcents usng orthogonalt. *Pattern Recognton 36;73-74 (3.

13 Least Squares Ft ˆ Ë Ê ˆ Ë Ê ˆ Ë Ê f f f a a a a nm m n m n m n L L L L ( F A F A F A T T T T - Conceptuall eas to understand although ths can be relatvel slow for hgh order fts.

14 Gram-Schmdt Orthogonalzaton Examnes set of dscrete data and creates a seres of functons whch are orthogonal over the data set. Orthogonalt s used to calculate expanson coeffcents. These surfaces can then be converted to a standard set of surfaces such as ernke polnomals. Advantages umercall stable especall for low samplng denst. Dsadvantages Can be slow for hgh-order fts Orthogonal functons depend upon data set so a new set needs to be calculated for ever ft.

15 Elevaton Ft Comparson 3 Orders or 56 total polnomals.5 Seconds Chong Algorthm Seconds Gram-Schmdt

16 Keratoconus Detecton Keratoconcs ormals

17 Keratoconus Detecton Keratoconus Feature ormal Feature

18 Kˆ ˆ v A < a Keratoconus Detecton unt vector of average hgher order coeffcents of Keratoconus Pate unt vector of average hgher order coeffcents of ormal Patents 3-3a 3 -a 3a 33a > The dot product of the feature vectors and the vector under test gves a measure of how much the test vector looks lke the feature vector. Kˆ ˆ

19 Keratoconus Detecton /4 (87% Cones Correctl Classfed ormal-lke..5. Keratoconcs ormals Decson Lne 4/4 (% ormals Correctl Classfed Cone-Lke

20 sclassfed Cones

21 Progressve Addton Lenses Progressve Lens manufacturers mnmze the detals of ther desgns for propretar reasons. Tpcall PAL patents gven a 7 x 7 grd of elevaton data. US Patent #68863 Sphercal Power Astgmatsm

22 Orthogonalt Slope data s a lttle trcker snce the dervatves of ernke polnomals are not orthogonal. Gavreldes* developed a set of functons that are orthogonal to the gradent of the ernke polnomals so expanson coeffcents can be calculated as follows: v W v v W G ÚÚ A a ÚÚ Â v v W G A a v v W G v V Â dxd a v v V G dxd Âa ÚÚ Æ A Â k v v V G v W k k dxd v G k *Optcs Letters 7;56-58 (98. k

23 Least Squares Ft È dv dv dv dv dv Î dv / dx / dx / d / dx / d / d dv dv dv dv dv dv / dx / dx / dx / d / d / d L L L L L L dv dv dv J dv dv dv J J J J J / dx / dx Èa / dx a / d / d Îa J / d È dw dw dw dw dw Î dw / dx / dx / dx / d / d / d A T A F Agan conceptuall eas to understand although A T F ( T - T F ths can be relatvel slow for hgh order fts.

24 Dscusson A pror knowledge should be used to determne the degree of ft needed for a gven applcaton. Orthogonalt allows fast but nos fts and can be used n cases where the gross features of the surface are to be analzed. Least squares and Gram-Schmdt technques provde more accurate fts but are slower.