2.71704/20/05 – wk11-b-1

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2.71704/20/05 – wk11-b-2

2.71704/20/05 – wk11-b-3

The “4F system” (telescope) : a general model for imaging

2f 2fx ′′ x′x

1f 1f

az

imageimageplane

objectobjectplane

FourierFourierplane planeplane plane

2.71704/20/05 – wk11-b-4

Imaging a 2½D object

2f 2fx ′′ x′x

1f 1f

DepthDepthof of

FocusFocus

DefocusDefocus

z

( )2NA5.0 λ

≈

a

22½½DDobjectobject

2.71704/20/05 – wk11-b-5

Imaging a 2½D object

2f 2fx ′′ x′x

1f 1f

z∆z

portion of objectportion of objectdefocused by defocused by ΔΔzz

2.71704/20/05 – wk11-b-6

Imaging a 2½D object

2f 2fx ′′ x′x

1f 1f

z

…… is equivalent to same portion is equivalent to same portion inin--focusfocus PLUS PLUS ……

…… fictitious quadratic fictitious quadratic phase mask phase mask

on the Fourier plane

( )⎭⎬⎫

⎩⎨⎧ ∆′′+′′− 2

1

22

2expf

zyxiλ

π (applied (applied locally locally -- shift variantshift variant))on the Fourier plane

2.71704/20/05 – wk11-b-7

Example: multi-surface object

2 (DoF)s 4 (DoF)s

focal plane

2.71704/20/05 – wk11-b-8

Raw image (collected by camera – noise-free )

z=0 mm z=3.2 mm z=6.4 mm

λ=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 well

2.71704/20/05 – wk11-b-9

Deconvolution using depth as prior (noise-free)

z=0 mm z=3.2 mm z=6.4 mm

Deconvolution using Tikhonov regularized inverse filter, µ=10-5

Utilized 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-10

Effect of regularizer – low noise

2.71704/20/05 – wk11-b-11

Effect of regularizer – moderate noise

2.71704/20/05 – wk11-b-12

Effect of regularizer – strong noise

2.71704/20/05 – wk11-b-13

Quadratic error vs µ – moderate noise

SNR=10SNR=10

2.71704/20/05 – wk11-b-14

Quadratic error vs µ – strong noise

SNR=3SNR=3

2.71704/20/05 – wk11-b-15

Cubic Phase Mask (the George B explanation)

2f 2fx ′′ x′x

1f 1f

z

extended depthextended depthof uniform blurof uniform blur

digital digital deblurring

The CDM solution (The CDM solution (DowskiDowski & & CatheyCathey, 1995), 1995)add cubic phase distortion in the Fourier planeadd cubic phase distortion in the Fourier plane

deblurring( ){ }33exp yxi ′′+′′− α

2.71704/20/05 – wk11-b-16

Raw image (collected by camera – noise-free )

Image using the cubic phase apertureNote blur is almost independent of depth

2.71704/20/05 – wk11-b-17

Deconvolution (noise-free) – no prior necessary

Deconvolution 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-18

Effect of regularizer on cubic phase – low noise

2.71704/20/05 – wk11-b-19

Effect of regularizer on cubic phase – moderate noise

2.71704/20/05 – wk11-b-20

Effect of regularizer – strong noise

2.71704/20/05 – wk11-b-21

Quadratic error vs µ – moderate noise

SNR=10SNR=10& cubic& cubic

2.71704/20/05 – wk11-b-22

Quadratic error vs µ – strong noise

SNR=3SNR=3& cubic& cubic