New Renaissance Institute’s founder is credited as the first to discover and report the fractional Fourier transform properties of lenses in the benchmark text The Fractional Fourier Transform with Applications in Optics and Signal Processing (p.386) some five years prior to the independent engulfing rise of the topic to widespread attention beginning in 1993. The recognized paper was accepted for presentation only as poster session at the 1988 SPIE Spatial Light Modulator Conference. Many hundreds of papers, including dozens that have since become celebrated, have been subsequently published beginning in 1993, and the work of two leading individuals in the 2003initiated Fractional Fourier Transform publication “wave” were awarded the 1998 Prize of the International Commission for Optics.
New Renaissance Institute has continued its work in fractional Fourier transforms in the contexts of formal mathematical operator theory, fractional Hilbertspace operators, the theory of Special Functions, computational imaging, electron microscopy, coherent optics, visiblelight imaging, and monolithic optical computing engines.
The items below pertain to NRI’s work in the area of lightfield imaging and computational image refocusing involving phasereconstruction for computational refocusing of legacy coherent images leveraging operator group and operator algebra properties of the fractional Fourier transform operator, for example for use in refocusing of legacy and realtime highenergy electron microscopy images but with many other potential optical applications. Mathematical aspects of this work include several aspects of the discrete fractional Fourier transform operator both in structure for proper approximate propagation modeling as well as numeric computation as well as complexvalued powers of the fractional Fourier transform operator.
NRI’s founder and NRI have performed other innovative work lightfield imaging and computational imaging, including its lensless lightfield imaging camera work
NRI is pleased to have hosted summer interns from U.C. Santa Cruz, U.C. Irvine, U.C. Berkeley, and U.C. San Diego who have provided laboratory work for in this technology area.
Issued Patents
Title  Patent Number  Application Number  Priority Dates  Text Only  Related Patents  

Correction of overfocus in digital images using centered discrete imaginarypower fractional Fourier transformations with highaccuracy orthonormal eigenvectors  9,031,347  14/207,186  11/15/2009  Text  Focus Correction 

Variable focusing of electron microscopy image data utilizing fractional powers of the Fourier transform operator  8,934,731  13/846,810  02/25/1999 02/25/2000 09/18/2003  Text  Focus Correction  
Variable focusing of electron microscopy image data utilizing origincentered discrete fractional fourier transform  8,897,590  13/846,760  02/25/1999 02/25/2000 09/18/2003  Text  Focus Correction  
HighAccuracy Centered Fractional Fourier Transform Matrix for Optical Imaging and Other Applications  8,712,185  13/959,604  11/15/2009  PDF  Text  Focus Correction 
Correction of Misfocus in Recorded Images Using Centered Discrete Fractional Fourier Transformations with HighAccuracy Orthonormal Eigenvectors  8,542,945  12/945,902  11/15/2009  Text  Focus Correction  
Generation of Image Data with Correction For Optical Misfocus Utilizing Fractional Powers of the Fourier Transform Operator  8,442,342  13/346,619  02/25/1999 02/25/2000 09/18/2003  Text  Focus Correction  
Correction of Unfocus and Misfocus via OriginCentered Discrete Fractional Fourier Transform  8,442,341  13/346,459  04/11/2008  Text  Focus Correction  
Discrete Fractional Fourier Numerical Environments for Computer Modeling of Image Propagation Through a Physical Medium in Restoration and other Applications  8,094,969  13/037,342  02/25/1999 02/25/2000 09/18/2003  Text  Focus Correction  
Generation of Image Data with Correction For Optical Misfocus Utilizing Fractional Powers of the Fourier Transform Operator  8,094,961  12/754,587  02/25/1999 02/25/2000  Text  Focus Correction  
Computing Arbitrary Fractional Powers of a Transform Operator from Selected Precomputed Fractional Powers of the Operator  7,697,777  11/929,360  02/25/1999 02/25/2000  Text  Focus Correction  
Focus Correction Using Fractional Fourier Operator Approximations  7,627,195  11/697,624  02/25/1999 02/25/2000  Text  Focus Correction  
Computing Arbitrary Fractional Powers Of A Transform Operator From Selected Precomputed Fractional Powers Of The Operator  7,203,377  10/937,192  02/25/1999 02/25/2000  Text  Focus Correction  
Relative Optical Path Phase Reconstruction In The Correction Of Misfocused Images Using Fractional Powers Of The Fourier Transform  7,054,504  10/665,439  02/25/1999 02/25/2000  Text  Focus Correction  
Iterative Approximation Environments for Modeling the Evolution of an Image Propagating Through a Physical Medium in Restoration and Other Applications Iterative Approximation Environments for Modeling the Evolution of an Image Propagating Through a Physical Medium in Restoration and Other Applications  7,039,252 RE42,187  10/980,744 12/101,878  02/25/1999 02/25/2000 09/18/2003 02/25/1999 02/25/2000 09/18/2003  PDF  Text Text  Focus Correction Focus Correction 
Correction of Image Misfocus via Fractional Fourier Transform  6,687,418  09/512,775  02/25/1999  Text  Focus Correction 
Pending Patents
Title  Publication Number  Application Number  Priority Dates  Text Only  Related Patents 
Pending Unpublished Applications
Title  Application Number  Priority Dates  Related Patents 