Today Defocus Deconvolution / inverse filters MIT.7/.70 Optics //05 wk5-a-
MIT.7/.70 Optics //05 wk5-a- Defocus
MIT.7/.70 Optics //05 wk5-a-3 0 th Century Fox
Focus in classical imaging in-focus defocus MIT.7/.70 Optics //05 wk5-a-4
Focus in classical imaging in-focus defocus MIT.7/.70 Optics //05 wk5-a-5
Intensity distribution near the focus of an ideal lens (rotationally symmetric wrt axis) x = 0.6 λ ( NA) = λ ( NA) MIT.7/.70 Optics //05 wk5-a-6
x Back to the basics: 4F system f f x x f f object Fourier image MIT.7/.70 Optics //05 wk5-a-7
x Back to the basics: 4F system f f x x a f f x max (NA) (NA) object Fourier image MIT.7/.70 Optics //05 wk5-a-8 ( NA) = x f max ( NA) = x f max
x Back to the basics: 4F system f f x x f f object g( x) Fourier x G λ f image f g x f MIT.7/.70 Optics //05 wk5-a-9
x 4F system with defocused input f f x x f f g( x) object Fourier image MIT.7/.70 Optics //05 wk5-a-0
x 4F system with defocused input f f x x f f object Fourier image g x ( x) exp iπ λ MIT.7/.70 Optics //05 wk5-a- + y
x 4F system with defocused input f f x x f f object Fourier image g ( x) exp iπ G ( ) λ exp iπ λ λf MIT.7/.70 Optics //05 wk5-a- x I x x λf
x 4F system with defocused input f f x x f f object Fourier image g x ( ) x exp iπ ( ) ( ) x λ x G exp iπ λf λf MIT.7/.70 Optics //05 wk5-a-3 I
Effect of defocus on the Fourier mild defocus cos π λ f ( ) ( ) x x λ f 0 λ( ) MIT.7/.70 Optics //05 wk5-a-4
Effect of defocus on the Fourier cos π λ f strong defocus ( ) ( ) x x λ f 0 MIT.7/.70 Optics //05 wk5-a-5 λ( )
Effect of defocus on the Fourier cos π λ f mild defocus ( ) ( ) x x G λ f x λ f 0 MIT.7/.70 Optics //05 wk5-a-6 λ( ) ( ) ( ) G x x cos π f λ f not too different from x λ G λ f
Effect of defocus on the Fourier strong defocus ( ) ( ) x cos π λ f x G λ f 0 x λ f MIT.7/.70 Optics //05 wk5-a-7 x G λ f cos π λ f ( ) ( ) x very different from x G λ f
Depth of field cos π λ f ( ) ( ) x λ( ) x λ f 0 MIT.7/.70 Optics //05 wk5-a-8 system is defocus-insensitive as long as frequencies that pass through the system are confined within this region
x λf max ( ) λ ( ) λ ( x f ) max Depth of field cos π λ f ( ) ( ) x λ( ) x λ f 0 MIT.7/.70 Optics //05 wk5-a-9 system is defocus-insensitive as long as frequencies that pass through the system are confined within this region
x λf max ( ) λ ( ) λ ( NA) Depth of field Depth of field 0 cos π λ f ( ) ( ) x λ( ) x λ f MIT.7/.70 Optics //05 wk5-a-0 system is defocus-insensitive as long as ( ) is small enough that frequencies that pass through the system can be confined within this region
x Depth of field & Depth of focus f f x x f f (NA) (NA) object MIT.7/.70 Optics //05 wk5-a- ( ) ( NA) λ x max = f ( NA) Depth of field Fourier ( ) ( NA) λ x max = f ( NA) Depth of focus image
NA trade offs high NA narrow PSF in the lateral direction (PSF width ~/NA) sharp lateral features narrow PSF in longitudinal direction (PSF depth ~/NA ) poor depth of field low NA broad PSF in the lateral direction (PSF width ~/NA) blurred lateral features broad PSF in longitudinal direction (PSF depth ~/NA ) good depth of field MIT.7/.70 Optics //05 wk5-a-
Depth of focus: Geometrical Optics viewpoint x f f x x a f f x max (NA) (NA) x MIT.7/.70 Optics //05 wk5-a-3 From similar triangles: = x ( NA) Now require defocused spot diffraction spot: Therefore: x 0.6 ( NA) defocused image λ λ 0.6 ( NA)
Defocus and Deconvolution (Inverse filters) MIT.7/.70 Optics //05 wk5-a-4
x Imaging a ½D object f f x x f f ½D object Depth Of Focus Defocus worsens away from focal MIT.7/.70 Optics //05 wk5-a-5
x Imaging a ½D object f f x x f f portion of object defocused by Δ MIT.7/.70 Optics //05 wk5-a-6
x Imaging a ½D object f f x x f f is equivalent to same portion in-focus PLUS fictitious quadratic phase mask on the Fourier MIT.7/.70 Optics //05 wk5-a-7 exp i ( x + y ) π λf (applied locally )
Example (DoF)s 4 (DoF)s focal MIT.7/.70 Optics //05 wk5-a-8
Raw image (collected by camera noise-free) Distance between s Depths of Field left-most M : image blurred by diffraction only center and right-most M s : image blurred by diffraction and defocus MIT.7/.70 Optics //05 wk5-a-9
Raw image explanation: convolution M convolved with standard diffraction PSF M convolved with diffraction PSF and defocus M convolved with diffraction PSF and more defocus MIT.7/.70 Optics //05 wk5-a-30
Raw image explanation: Fourier domain I {" M" } H diffraction I {"M"} H diffraction I{ "M"} H H ( DoF) H ( 4DoF) MIT.7/.70 Optics //05 wk5-a-3 defocus defocus diffraction
Can diffraction and defocus be undone? Effect of optical system (expressed in the Fourier ): I {" M" } H system where Hsystem = H diffraction H defocus To undo the optical effect, multiply by the inverse transfer function {"M"} ( I H ) = I{ "M"}!!! system H system MIT.7/.70 Optics //05 wk5-a-3
Can diffraction and defocus be undone? Effect of optical system (expressed in the Fourier domain): I {" M" } Hsystem where H = H system diffraction H defocus To undo the optical effect, multiply by the inverse transfer function {"M"} ( I H ) = I{ "M"}!!! system H system Problems Transfer function goes to ero outsie the system pass-band Inverse transfer function will multiply the FT of the noise as well as the FT of the original signal transfer function MIT.7/.70 Optics //05 wk5-a-33 u max 0 u max
Solution: Tikhonov regulariation final I = image I original object H system µ + H * system H system raw image (formed by the optics) post-processing ( inverse filter ) x f f x x f f original object MIT.7/.70 Optics //05 wk5-a-34 I CCD camera image raw image original I H object system final image COMPUTATIONAL IMAGING
On Tikhonov regulariation final I = image I original object H system µ + H * system H system µ is the regularier or regulariation parameter choice of µ : depends on the noise and signal energy for Gaussian noise and image statistics, optimum µ is µ optimum = SNR Wiener filter power More generally, the optimal inverse filters are nonlinear and/or probabilistic (e.g. maximum likelihood inversion) For more details:.77 MIT.7/.70 Optics //05 wk5-a-35
Deconvolution: diffraction and defocus noise free MIT.7/.70 Optics //05 wk5-a-36 Deconvolution using Tikhonov regularied inverse filter Utilied a priori knowledge of depth of each digit (alternatively, needs depth-from defocus algorithm) Artifacts due primarily to numerical errors getting amplified by the inverse filter (despite regulariation)
SNR=0 Noisy raw image MIT.7/.70 Optics //05 wk5-a-37
Deconvolution in the presence of noise SNR=0 MIT.7/.70 Optics //05 wk5-a-38 Deconvolution using Wiener filter (i.e. Tikhonov with µ=/snr) Noise is destructive away from focus (4DOFs) Utilied a priori knowledge of depth of each digit Artifacts due primarily to noise getting amplified by the inverse filter