Wigner Distributions and How They Relate to the Light Field - PowerPoint PPT Presentation

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ Wigner distribution of wave function

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ Wigner distribution Wigner distribution of wave function of aperture window

Observable Light Field 2 � � � l ( T ) U ( x ) T ( x − s ) e − j 2 π u λ x dx � � obs ( s, u ) = � � � � blur trades off resolution in position with direction l ( T ) s, u − s, u � � � � obs ( s, u ) = W U ⊗ W T λ λ Wigner distribution Wigner distribution of wave function of aperture window

Observable Light Field at zero wavelength limit (regime of ray optics) l ( T ) s, u � � − s, u � � obs ( s, u ) = W U W T ⊗ λ λ Wigner distribution Wigner distribution of wave function of aperture window

Observable Light Field at zero wavelength limit (regime of ray optics) l ( T ) s, u � � obs ( s, u ) = W U ⊗ δ ( − s, u ) λ Wigner distribution of wave function

Observable Light Field at zero wavelength limit (regime of ray optics) l ( T ) s, u � � obs ( s, u ) = W U λ observable light field and Wigner equivalent!

Observable Light Field • observable light field is a blurred Wigner distribution with a modified coordinate system • blur trades off resolution in position with direction • Wigner distribution and observable light field equivalent at zero wavelength limit

Application - Refocusing u s light field

Application - Refocusing Isaksen et. al u 2000 s light field

Application - Refocusing Isaksen et. al u 2000 s light field image at z=0

Application - Refocusing Isaksen et. al u 2000 s light field image at z=z 0

Application - Refocusing Isaksen et. al u f u 2000 light s f s light Fourier field field spectrum

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=0

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=z 0

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum f ξ ξ Wigner ambiguity f x x Fourier distribution function

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=0 f ξ ξ Wigner ambiguity f x x Fourier distribution function

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=z 0 f ξ ξ Wigner ambiguity f x x Fourier distribution function

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=0 f ξ ξ Wigner ambiguity f x x Fourier distribution function

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum image at z=z 0 f ξ ξ Wigner ambiguity f x x Fourier distribution function

Application - Refocusing Isaksen Ng et. al u f u 2005 2000 light s f s light Fourier field field spectrum Papoulis f ξ ξ 1974 Wigner ambiguity f x x Fourier distribution function

Application - Wavefront Coding Dowski and Cathey 1995 same aberrant blur regardless of depth of focus

Application - Wavefront Coding Dowski and Cathey 1995 point in scene same aberrant blur regardless of depth of focus

Application - Wavefront Coding Dowski and Cathey 1995 point cubic in scene phase plate same aberrant blur regardless of depth of focus

Application - Wavefront Coding Dowski and Cathey 1995 point small change cubic in scene in blur shape phase plate same aberrant blur regardless of depth of focus

Application - Wavefront Coding slices corresponding to various depths ambiguity function

Application - Wavefront Coding

Application - Wavefront Coding u s point

Application - Wavefront Coding u u s s point before phase plate

Application - Wavefront Coding u u s s point after phase plate

Application - Wavefront Coding u u u s s s at image point after phase plate plane

Application - Wavefront Coding u u u s s s at image point after phase plate plane

Application - Wavefront Coding u • refocusing in ray space is shearing • shearing of a parabola results in translation s • blur shape invariant to refocusing