![Page 1: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/1.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-1
Today
• Defocus
• Deconvolution / inverse filters
![Page 2: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/2.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-2
Defocus
![Page 3: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/3.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-3
©© 2020thth Century FoxCentury Fox
![Page 4: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/4.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-4
Focus in classical imaging
inin--focusfocus
defocusdefocus
![Page 5: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/5.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-5
Focus in classical imaging
inin--focusfocus
defocusdefocus
![Page 6: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/6.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-6
Intensity distribution near the focus of an ideal lens
( )NA61.0 λ×=∆x
( )2NA2λ
=∆z
(rotationally symmetric wrt z axis)
z
![Page 7: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/7.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-7
Back to the basics: 4F system
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
![Page 8: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/8.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-8
Back to the basics: 4F system
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
maxxa ≡
(NA)1
( )1
max1 NA
fx
=
(NA)2
( )2
max2 NA
fx
=
![Page 9: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/9.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-9
Back to the basics: 4F system
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
( )xg ⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ ⎟⎟
⎠
⎞⎜⎜⎝
⎛′− x
ffg2
1
![Page 10: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/10.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-10
4F system with defocused input
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
( )xg
![Page 11: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/11.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-11
4F system with defocused input
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∆+
∗zyxixg
λπ
22
exp
![Page 12: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/12.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-12
4F system with defocused input
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∆
∗z
xixgλ
π2
exp ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′∆⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′′2
11
expf
xzif
xGλ
λπλ
ℑ
![Page 13: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/13.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-13
4F system with defocused input
2f 2fx ′′ x′x
1f 1f
imageimageplaneplane
objectobjectplaneplane
FourierFourierplaneplane
z
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛∆
∗z
xixgλ
π2
exp ( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′′2
1
2
1
expf
xzif
xGλ
πλ
ℑ
![Page 14: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/14.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-14
Effect of defocus on the Fourier plane
mild defocus
1fxλ′′
( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
( )z∆λ21
1
0
![Page 15: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/15.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-15
Effect of defocus on the Fourier plane
strong defocus ( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
( )z∆λ21
1
0
1fxλ′′
![Page 16: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/16.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-16
Effect of defocus on the Fourier plane
mild defocus ( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
( )z∆λ21
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ
( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ
× not too different from ⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ
1
0
1fxλ′′
![Page 17: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/17.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-17
Effect of defocus on the Fourier plane
1
0
⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ
× very different from
strong defocus ( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ ⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′′
1fxGλ
1fxλ′′
![Page 18: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/18.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-18
Depth of field
( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
( )z∆λ21
1
0
1fxλ′′
system is defocus-insensitiveas long as frequencies
that pass through the systemare confined within this region
![Page 19: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/19.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-19
Depth of field
( )z∆λ21
1
0
system is defocus-insensitiveas long as frequencies
that pass through the systemare confined within this region
( )
( )( )21max
1
max
2
21
fxz
zfx
′′≤∆⇒
⇒∆
≤′′
λλλ
1fxλ′′
( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
![Page 20: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/20.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-20
Depth of field
( )z∆λ21
1
0
system is defocus-insensitiveas long as (∆z) is small enough that
frequencies that pass through the systemcan be confined within this region
( )
( )( ) 1
2
1
max
NA2
21
λλλ
≤∆⇒
⇒∆
≤′′
z
zfx
1fxλ′′
( ) ( )⎥⎦
⎤⎢⎣
⎡ ′′∆ 2
1
2
cosf
xzλ
π
Depth of field
![Page 21: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/21.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-21
Depth of field & Depth of focus
2f 2fx ′′ x′x
1f 1f
objectobjectplaneplane
FourierFourierplaneplane
z
( )( )
( )1
max1
12
NA
NA2
fx
z
′′=
≤∆λ ( )
( )
( )2
max2
22
NA
NA2
fx
z
′′=
≤∆λ
imageimageplaneplane
Depth of fieldDepth of field Depth of focusDepth of focus
(NA)1
(NA)2
![Page 22: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/22.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-22
NA trade – offs• high NA
– narrow PSF in the lateral direction (PSF width ~1/NA)• sharp lateral features
– narrow PSF in longitudinal direction (PSF depth ~1/NA2)• poor depth of field
• low NA– broad PSF in the lateral direction (PSF width ~1/NA)
• blurred lateral features– broad PSF in longitudinal direction (PSF depth ~1/NA2)
• good depth of field
![Page 23: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/23.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-23
Depth of focus: Geometrical Optics viewpoint
2f 2fx ′′ x′x
1f 1f
z(NA)1
(NA)2
defocuseddefocusedimageimageplaneplane
∆x
∆z
maxxa ≡
( )2NAxz ∆
=∆From similar triangles:
Now require defocused spot ≈ diffraction spot: ( )2NA61.0 λ
≈∆x
Therefore: ( )22NA61.0 λ
≈∆z
![Page 24: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/24.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-24
Defocus and Deconvolution(Inverse filters)
![Page 25: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/25.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-25
2f 2fx ′′ x′x
1f 1f
DepthDepthOf Of
FocusFocus
DefocusDefocus
worsensworsensawayawayfromfromfocalfocalplaneplane
22½½DDobjectobject
z
Imaging a 2½D object
![Page 26: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/26.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-26
2f 2fx ′′ x′x
1f 1f
portion of objectportion of objectdefocused by defocused by ΔΔzz
z∆z
Imaging a 2½D object
![Page 27: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/27.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-27
2f 2fx ′′ x′x
1f 1f
…… 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 planeon the Fourier plane
( )⎭⎬⎫
⎩⎨⎧ ∆′′+′′− 2
1
22
2expf
zyxiλ
π (applied(appliedlocally locally ))
z
Imaging a 2½D object
![Page 28: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/28.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-28
Example
focal plane
2 (DoF)s 4 (DoF)s
![Page 29: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/29.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-29
Distance between planes ≈ 2 Depths of Fieldleft-most “M” : image blurred by diffraction onlycenter and right-most “M”s : image blurred by diffraction and defocus
Raw image (collected by camera – noise-free)
![Page 30: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/30.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-30
Raw image explanation: convolution
“M” convolvedwith standard
diffraction PSF
“M” convolvedwith diffraction PSF
and defocus
“M” convolvedwith diffraction PSFand more defocus
![Page 31: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/31.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-31
Raw image explanation: Fourier domain
{ } ndiffractioM"" H×ℑ { }( )2DoF
M""
defocus
ndiffractio
HH
××ℑ { }
( )4DoFM""
defocus
ndiffractio
HH
××ℑ
![Page 32: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/32.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-32
Can diffraction and defocus be “undone” ?• Effect of optical system (expressed in the Fourier plane):
• To undo the optical effect, multiply by the “inverse transfer function”
{ } defocusndiffractiosystemsystem e wherM"" HHHH ×=×ℑ
{ }( ) { } !!! M"" 1M"" system
system ℑ=××ℑH
H
![Page 33: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/33.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-33
Can diffraction and defocus be “undone” ?• Effect of optical system (expressed in the Fourier domain):
• To undo the optical effect, multiply by the “inverse transfer function”
• Problems– Transfer function goes to zero outsize the system pass-band– Inverse transfer function will multiply the FT of the noise as well
as the FT of the original signal
2umax–2umax
1
0
transfer function
{ } systemM"" H×ℑ
{ }( ) { } !!! M"" 1M"" system
system ℑ=××ℑH
H
defocusndiffractiosystem HHH ×=where
![Page 34: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/34.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-34
Solution: Tikhonov regularization
2
system
*system
systemobjectoriginal
imagefinal
H
HH
+×⎟⎟⎠
⎞⎜⎜⎝
⎛×
⎭⎬⎫
⎩⎨⎧
ℑ=⎭⎬⎫
⎩⎨⎧
ℑµ
“raw image”(formed by the optics)
post-processing(“inverse filter”)
2f 2fx ′′ x ′x
1f 1f
image planeimage plane““raw imageraw image””
originaloriginalobjectobject
z
⎭⎬⎫
⎩⎨⎧
×⎭⎬⎫
⎩⎨⎧
ℑℑ−system
1
objectoriginal H
CCDcamera
finalfinalimageimage
COMPUTATIONALCOMPUTATIONALIMAGINGIMAGING
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MIT 2.71/2.710 Optics12/12/05 wk15-a-35
On Tikhonov regularization
2
system
*system
systemobjectoriginal
imagefinal
H
HH
+×⎟⎟⎠
⎞⎜⎜⎝
⎛×
⎭⎬⎫
⎩⎨⎧
ℑ=⎭⎬⎫
⎩⎨⎧
ℑµ
• µ is the “regularizer” or “regularization parameter”• choice of µ : depends on the noise and signal energy• for Gaussian noise andand image statistics, optimum µ is
“Wiener filter”
• More generally, the optimal inverse filters are nonlinear and/orprobabilistic (e.g. maximum likelihood inversion)
• For more details: 2.717
poweroptimum SNR
1=µ
![Page 36: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/36.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-36
Deconvolution using Tikhonov regularized inverse filterUtilized a priori knowledge of depth of each digit (alternatively,needs depth-from defocus algorithm)
Artifacts due primarily to numerical errors getting amplifiedby the inverse filter (despite regularization)
Deconvolution: diffraction and defocusnoise free
![Page 37: Defocus • Deconvolution / inverse filtersweb.mit.edu/2.710/Fall06/2.710-wk15-a-sl.pdf · 2005-11-27 · Deconvolution using Tikhonov regularized inverse filter Utilized a priori](https://reader033.vdocuments.mx/reader033/viewer/2022050102/5f40db09a0e69a6c0302763a/html5/thumbnails/37.jpg)
MIT 2.71/2.710 Optics12/12/05 wk15-a-37
Noisy raw imageSNR=10
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MIT 2.71/2.710 Optics12/12/05 wk15-a-38
Deconvolution using Wiener filter (i.e. Tikhonov with µ=1/SNR)Noise is destructive away from focus (4DOFs)Utilized a priori knowledge of depth of each digit
Artifacts due primarily to noise getting amplified by the inverse filter
Deconvolution in the presence of noiseSNR=10