The “4F system” (telescope) : a general model for imagingImaging a 2½D objectImaging a 2½D objectImaging a 2½D objectExample: multi-surface objectRaw image (collected by camera – noise-free )Deconvolution using depth as prior (noise-free)Effect of regularizer – low noiseEffect of regularizer – moderate noiseEffect of regularizer – strong noiseQuadratic error vs μ – moderate noiseQuadratic error vs μ – strong noiseCubic Phase Mask (the George B explanation)Raw image (collected by camera – noise-free )Deconvolution (noise-free) – no prior necessaryEffect of regularizer on cubic phase – low noiseEffect of regularizer on cubic phase – moderate noiseEffect of regularizer – strong noiseQuadratic error vs μ – moderate noiseQuadratic error vs μ – strong noise2.71704/20/05 – wk11-b-1©©2020ththCentury FoxCentury Fox2.71704/20/05 – wk11-b-22.71704/20/05 – wk11-b-3The “4F system” (telescope) : a general model for imaging 2f2fx′′x′x1f1fazimageimageplaneobjectobjectplaneFourierFourierplaneplaneplane plane2.71704/20/05 – wk11-b-4Imaging a 2½D object2f2fx′′x′x1f1fDepthDepthof of FocusFocusDefocusDefocusz()2NA5.0λ≈a22½½DDobjectobject2.71704/20/05 – wk11-b-5Imaging a 2½D object2f2fx′′x′x1f1fz∆zportion of objectportion of objectdefocused by defocused by ΔΔzz2.71704/20/05 – wk11-b-6Imaging a 2½D object2f2fx′′x′x1f1fz……is equivalent to same portion is equivalent to same portion inin--focusfocusPLUS PLUS …………fictitious quadratic fictitious quadratic phase mask phase mask on the Fourier plane()⎭⎬⎫⎩⎨⎧∆′′+′′−21222expfzyxiλπ(applied (applied locally locally --shift variantshift variant))on the Fourier plane2.71704/20/05 – wk11-b-7Example: multi-surface object2 (DoF)s4 (DoF)sfocal plane2.71704/20/05 – wk11-b-8Raw image (collected by camera – noise-free )z=0 mmz=3.2 mmz=6.4 mmλ=0.5µm, f1=f2=20cm, a=2.5mm→ NA=0.0125, DoF=1.6mmSpatially incoherent illuminationDistance between planes = 2 Depths of FieldImage blurred by diffraction (lateral aperture) as well2.71704/20/05 – wk11-b-9Deconvolution using depth as prior (noise-free)z=0 mmz=3.2 mmz=6.4 mmDeconvolution using Tikhonov regularized inverse filter, µ=10-5Utilized a priori knowledge of depth of each digit (alternatively,needs depth-from defocus algorithm)Note:regularization necessary because of numerical noise!2.71704/20/05 – wk11-b-10Effect of regularizer – low noise2.71704/20/05 – wk11-b-11Effect of regularizer – moderate noise2.71704/20/05 – wk11-b-12Effect of regularizer – strong noise2.71704/20/05 – wk11-b-13Quadratic error vs µ – moderate noiseSNR=10SNR=102.71704/20/05 – wk11-b-14Quadratic error vs µ – strong noiseSNR=3SNR=32.71704/20/05 – wk11-b-15Cubic Phase Mask (the George B explanation)2f2fx′′x′x1f1fzextended depthextended depthof uniform blurof uniform blurdigital digital deblurringThe CDM solution (The CDM solution (DowskiDowski& & CatheyCathey, 1995), 1995)add cubic phase distortion in the Fourier planeadd cubic phase distortion in the Fourier planedeblurring(){}33exp yxi′′+′′−α2.71704/20/05 – wk11-b-16Raw image (collected by camera – noise-free )Image using the cubic phase apertureNote blur is almost independent of depth2.71704/20/05 – wk11-b-17Deconvolution (noise-free) – no prior necessaryDeconvolution using Tikhonov regularized inverse filterUniform quality, comparable to in-focus image using clear apertureNo a priori depth information was used (i.e. depth filter is shift invariant)2.71704/20/05 – wk11-b-18Effect of regularizer on cubic phase – low noise2.71704/20/05 – wk11-b-19Effect of regularizer on cubic phase – moderate noise2.71704/20/05 – wk11-b-20Effect of regularizer – strong noise2.71704/20/05 – wk11-b-21Quadratic error vs µ – moderate noiseSNR=10SNR=10& cubic& cubic2.71704/20/05 – wk11-b-22Quadratic error vs µ – strong noiseSNR=3SNR=3& cubic&