Linear Image Processing and Filtering

n Math of 2-d linear systems n Separability n Shift-invariant linear systems, 2-d convolution n Integral image for fast box-ﬁltering n 2-d frequency response

n Blurring and sharpening

n Zoneplate 1 0.8 n Image sampling/aliasing 0.6 0.4

0.2 n Wiener ﬁltering 0

2 3 2 0 1 n 0 Nonlinear noise reduction/sharpening ω -1 y -2 -2 ω -3 x

n Image processing system S(.) is linear, iff superposition principle holds: S α ⋅ f #x, y% + β ⋅ g #x, y% = α ⋅ S f #x, y% + β ⋅ S g #x, y% for all α, β ∈! ( $ & $ &) ( $ &) ( $ &) n Any linear image processing system can be written as   g = Hf Note: matrix H need not be square. by sorting pixels into a column vector  T f = f ⎣⎡0,0⎦⎤ f ⎣⎡1,0⎦⎤  f ⎣⎡N −1,0⎦⎤ f ⎣⎡0,1⎦⎤  f ⎣⎡N −1,1⎦⎤   f ⎣⎡0, L −1⎦⎤  f ⎣⎡N −1, L −1⎦⎤ ( )

n Another way to represent any linear image processing scheme

N−1 L−1 g[α,β ] = ∑∑ f [x, y]⋅h[x,α, y,β ] x=0 y=0 impulse response n Input: unit impulse at pixel [a,b]

&$ 1 x = a ∧ y = b f [x, y] = δ [x − a, y − b] = % 0 else n Output: impulse response '&

N−1 L−1 g[α,β ] = ∑∑δ [x − a, y − b]⋅h[x,α, y,β ] = h[a,α,b,β ] x=0 y=0

δ [x − a, y − b] f [a,b] g[α,β] = h[a,α,b,β] f [a,b] a x + α

b x

y β a + x α

b x y β +

N −1 L−1 f [x, y]⋅h[x,α, y,β] g[α,β] = ∑∑ f [x, y]⋅h[x,α, y,β] x=0 y=0 x α

∑ x

y β x α

∑ x

y β

Digital Image Processing: Bernd Girod, © 2013 Stanford University -- Linear Image Processing and Filtering 5 Relationship of H and h[x,α,y,β ] y → x → x → x → •• • ↓ α ↓ α ↓α x → x → x → •• • ↓ α ↓ α ↓ α • • • β ↓ • • • • • • x → x → x → •• • ↓ α ↓ α ↓ α

n Impulse response is separable in [x,α] and [y,β], i.e., can be written as N−1 L−1 g[α,β ] = ∑∑ f [x, y]⋅hx [x,α ]hy [y,β ] x=0 y=0 n Processing can be carried out l row by row, then column by column L−1 N−1 g[α,β ] = ∑hy [y,β ]∑ f [x, y]⋅hx [x,α ] y=0 x=0

l column by column, then row by row

N−1 L−1 g[α,β ] = ∑hx [x,α ]∑ f [x, y]⋅hy [y,β ] x=0 y=0

n If the digital input and output images are written as a matrices f and g, we can conveniently write T g = Hy ⋅ f ⋅Hx

⎡ h 0,0 h 0,1  h ⎡0, L −1⎤ ⎤ ⎡ h 0,0 h 0,1  h ⎡0, N −1⎤ ⎤ ⎢ y [ ] y [ ] y ⎣ g ⎦ ⎥ ⎢ x [ ] x [ ] x ⎣ g ⎦ ⎥ ⎢ ⎥ ⎢ ⎥ h 1,0 h 1,1  h ⎡1, N −1⎤ hy [1,0] hy [1,1]  hy ⎡1, Lg −1⎤ ⎢ x [ ] x [ ] x ⎣ g ⎦ ⎥ H = ⎢ ⎣ ⎦ ⎥ Hx = y ⎢ ⎥ ⎢    ⎥    ⎢ ⎥ ⎢ ⎥ h L 1,0 h L 1,1 h ⎡L 1, L 1⎤ ⎢ hx [N −1,0] hx [N −1,1]  hx ⎡N −1, Ng −1⎤ ⎥ ⎢ y [ − ] y [ − ]  y ⎣ − g − ⎦ ⎥ ⎣ ⎣ ⎦ ⎦ ⎣ ⎦

n Output image g has size Lg x Ng

n If the operator does not change image size, Hx and Hy are square matrices

Original Bike 256x256 256x256 Haar transform

Hx=Hy=Hr256

n Haar transform matrix for sizes N=2,4,8 ⎛ 1 1 2 0 2 0 0 0 ⎞ 1 ⎛ 1 1 ⎞ ⎜ ⎟ Hr2 = ⎜ ⎟ ⎜ 1 1 2 0 −2 0 0 0 ⎟ 2 ⎝ 1 −1 ⎠ ⎜ ⎟ ⎜ 1 1 − 2 0 0 2 0 0 ⎟ ⎛ ⎞ 1 ⎜ 1 1 − 2 0 0 −2 0 0 ⎟ 1 1 2 0 Hr8 = ⎜ ⎟ ⎜ ⎟ 8 ⎜ 1 −1 0 2 0 0 2 0 ⎟ 1 ⎜ 1 1 − 2 0 ⎟ ⎜ ⎟ Hr 1 −1 0 2 0 0 −2 0 4 = ⎜ ⎟ ⎜ ⎟ 4 1 −1 0 2 ⎜ 1 −1 0 − 2 0 0 0 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 0 2 0 0 0 2 ⎟ ⎝⎜ 1 −1 0 − 2 ⎠⎟ ⎝ − − − ⎠

n Can be computed by taking sums and differences

n Fast algorithms by recursively applying Hr2

n Image subsampling 2:1 horizontally and vertically n Small input image of size 8x8, output image size 4x4

⎡ 1 0 0 0 ⎤ ⎢ ⎥ ⎢ 0 0 0 0 ⎥ ⎢ 0 1 0 0 ⎥ ⎢ 0 0 0 0 ⎥ Hx = Hy = ⎢ ⎥ ⎢ 0 0 1 0 ⎥ ⎢ 0 0 0 0 ⎥ ⎢ 0 0 0 1 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 ⎦

n A somewhat better technique for 2:1 image size reduction

⎡ 0.5 0 0 0 ⎤ ⎢ ⎥ ⎢ 0.5 0 0 0 ⎥ ⎢ 0 0.5 0 0 ⎥ ⎢ 0 0.5 0 0 ⎥ Hx = Hy = ⎢ ⎥ ⎢ 0 0 0.5 0 ⎥ ⎢ 0 0 0.5 0 ⎥ ⎢ 0 0 0 0.5 ⎥ ⎢ ⎥ ⎣ 0 0 0 0.5 ⎦

Hx = Hy = Hx = Hy = ! 1 0 0 0 $ ! 0.5 0 0 0 $ # 0 0 0 0 & # & # & # 0.5 0 0 0 & # 0 1 0 0 & # 0 0.5 0 0 & # 0 0 0 0 & # 0 0.5 0 0 & # & # & # 0 0 1 0 & # 0 0 0.5 0 & # 0 0 0 0 & # 0 0 0.5 0 & # 0 0 0 1 & # 0 0 0 0.5 & # & # & " 0 0 0 0 % " 0 0 0 0.5 %

n Each pixel is replaced by the average of two horizontally (vertically) neighboring pixels ⎡ 0.5 0  0 ⎤ ⎢ ⎥ ⎢ 0.5 0.5 ⎥ ⎢ 0 0.5 0.5 ⎥ ⎢ 0.5 0.5   ⎥ Hx = Hy = ⎢ ⎥ ⎢ 0.5 0.5 ⎥ ⎢   0.5 0.5 ⎥ ⎢ 0.5 0.5 0 ⎥ ⎢ ⎥ 0  0 0.5 0.5 ⎣ ⎦

n Shift-invariant operation (except for image boundary)

n For a separable, shift-invariant, linear system

hx [x,α]= hsiv/x [α − x] and hy [y,β]= hsiv/y [β − y]

n Matrices Hx and Hy are square, and Toeplitz matrices, e.g.,

⎡ h 0 h 1  h N −1 ⎤ ⎢ siv/x [ ] siv/x [ ] siv/x [ ] ⎥ ⎢ hsiv/x [−1] hsiv/x [0]  hsiv/x [N − 2] ⎥ Hx = ⎢ ⎥ ⎢    ⎥ ⎢ hsiv/x [1− N ] hsiv/x [2 − N ]  hsiv/x [0] ⎥ ⎣ ⎦ n Operation is a 2-d separable convolution („ﬁltering“)

N −1 L−1 g[α,β]= ∑∑ f [x, y]⋅hsiv/x [α − x]hsiv/y [β − y] x=0 y=0

n Convolution kernel of linear shift-invariant system („ﬁlter“) can also be non-separable

N −1 L−1 g[α,β]= ∑∑ f [x, y]⋅hsiv [α − x,β − y] x=0 y=0

n Viewed as a matrix operation . . .   g = Hf . . . H is a block Toeplitz matrix

Digital Image Processing: Bernd Girod, © 2013 Stanford University -- Linear Image Processing and Filtering 16 Structure of H for non-separable convolution y → x → x → x → •• • ↓ α ↓ α ↓ α x → x → x → •• • ↓ α ↓ α ↓ α • • • β ↓ • • • • • • x → x → x → •• • ↓ α ↓ α ↓ α

f [a,b]δ [x − a, y − b] g[x, y] = hsiv [x − a, y − b] f [a,b] a + x x

b x 1 x 5 2 3 4

y + y a + x x

b x 1 x 5 + 2 3 4 y y +

Digital Image Processing: Bernd Girod, © 2013 Stanford University -- Linear Image Processing and Filtering 18 Convolution: linear combination of neighboring pixel values N −1 L−1 g , f x, y h x, y f [x, y]⋅hsiv [α − x,β − y] [α β]= ∑∑ [ ]⋅ siv [α − β − ] x=0 y=0 α x x

4 3 2 5 x 1 ∑ xx β + y y x x

4 3 2 5 x 1 ∑ + x β y y α

Original Bike blurred by convolution Bike Impulse response „box ﬁlter“

⎛ 1 1 1 1 1 ⎞ ⎜ 1 1 1 1 1 ⎟ 1 ⎜ ⎟ ⎜ 1 1 [1] 1 1 ⎟ 25 ⎜ 1 1 1 1 1 ⎟ ⎜ ⎟ ⎝ 1 1 1 1 1 ⎠

Original Bike blurred horizontally Bike Filter impulse response

1 1 1 [1] 1 1 5 ( )

Original Bike blurred vertically Bike Filter impulse response

⎛ 1 ⎞ ⎜ 1 ⎟ 1 ⎜ ⎟ ⎜ [1] ⎟ 5 ⎜ 1 ⎟ ⎜ ⎟ ⎝ 1 ⎠

j[x, y]= j[x, y −1]+ f [x, y] i[x, y]= i[x −1, y]+ j[x, y]

x y f [x, y] i[x, y]= ∑∑ f [u,v] u=0 v=0

B D

C A

Sum = A + B – C – D

N times

…

Variance σ2 σ 12 / N

n Approximate convolution with Gaussian kernel by convolution with box kernel, repeated N times n Convolution with box kernel can be computed efﬁciently using integral image

Direct implementation: 6561 multiplications and 6560 additions per pixel

x-y separable ﬁltering: 162 multiplications and 160 additions per pixel

Original image Gaussian ﬁltered 500x500 σ = 20, 81x81 kernel 20 additions or subtractions per pixel

Box filtered Box filtered Box filtered Box filtered after N = 1 after N = 2 after N = 3 after N = 4

n Given a 1-d sequence s k , k ∈Z = ,−1,0,1,2,3, [ ] {… …} n Fourier transform ∞ jω − jωk S (e ) = ∑ s⎣⎡k ⎦⎤e ω ∈ℜ k=−∞ n Fourier transform is periodic with 2π n Inverse Fourier transform

π 1 jω jωk s⎣⎡k ⎦⎤ = S (e ) e dω 2π ∫−π

n Given a 2-d array of image samples s[m,n] , m,n ∈Z2 n Fourier transform ∞ ∞ j jω − jω m− jω n S e ω x ,e y = s⎡m,n⎤e x y ω ,ω ∈ℜ2 ( ) ∑ ∑ ⎣ ⎦ x y m=−∞ n=−∞ n Fourier transform is 2π-periodic both in ω x and ω y n Inverse Fourier transform

1 π π j jω jω m+ jω n s m,n S e ω x ,e y e x y d d ⎣⎡ ⎦⎤ = 2 ω x ω y 4π ∫−π ∫−π ( )

n DFT matrix is orthonormal

n conceptually write as DFT-matrix vector multiplication O(N^2) but in practice use FFT O(NlogN)

n FT of symmetric function is always real; FT of real function is always symmetric

n convolution theorem is super useful:

x ∗ g = F −1 {F{x}⋅ F{g}}

• What is this?! Filtering – Low-pass Filter!

• low-pass ﬁlter: convolution in primal domain! b = x ∗c x c b *

small! kernel! = Filtering – Low-pass Filter! F b = F x ⋅ F c • low-pass ﬁlter: multiplication in frequency domain! { } { } { }

big!

= Filtering – Low-pass Filter! F b = F x ⋅ F c • low-pass ﬁlter: hard cutoff! { } { } { }

= Filtering – Low-pass Filter!

• Bessel function of the ﬁrst kind or “jinc”!

F −1 { }

imagemagick.org Filtering – Low-pass Filter!

• hard frequency ﬁlters often introduce ringing!

ß Filtering – High-pass Filter!

• sharpening (possibly with ringing, but don’t see any here)!

ß Filtering – Unsharp Masking!

• sharpening (without ringing): unsharp masking, e.g. in Photoshop!

b = x *(δ − clowpass _ gauss ) = x − x *clowpass _ gauss

or!

b = x *(δ + chighpass ) = x + x *chighpass Filtering – Band-pass Filter!

ß Filtering – Oriented Band-pass Filter!

• edges with speciﬁc orientation (e.g., hat) are gone!!

ß Optical Filtering with Fourier Optics!

• can do all of this optically (with coherent light)! ! • Fourier optics – not part of this course!

http://en.wikipedia.org/wiki/Fourier_optics! Image Downsampling (& Upsampling) !

• “anti-aliasing” à before re-sampling, apply appropriate ﬁlter!!

• how much ﬁltering? Nyquist-Shannon sampling theorem: !

fs ≥ 2 fmax Other Forms of Aliasing!

• wagon wheel effect ! (temporal aliasing):! =jHS9JGkEOmA ! watch?v / youtube.com Other Forms of Aliasing!

• wagon wheel effect (temporal aliasing)!

• sampling frequency was lower than ! 2 fmax

wikipedia! Other Forms of Aliasing! • photography – optical AA ﬁlter removed (“hot rodding” camera)!

John Shafer! mosaicengineering.com! Other Forms of Aliasing! • photography – optical AA ﬁlter removed (“hot rodding” camera)!

without AA ﬁlter! with AA ﬁlter (standard)!

maxmax.com! Other Forms of Aliasing! • photography – optical AA ﬁlter removed (“hot rodding” camera)!

without AA ﬁlter! with AA ﬁlter (standard)!

maxmax.com! 5x5 box filter revisited

Original Bike blurred by convolution Bike Impulse response „box ﬁlter“

⎛ 1 1 1 1 1 ⎞ ⎜ 1 1 1 1 1 ⎟ 1 ⎜ ⎟ ⎜ 1 1 [1] 1 1 ⎟ 25 ⎜ 1 1 1 1 1 ⎟ ⎜ ⎟ ⎝ 1 1 1 1 1 ⎠

∞ ∞ j j m j n jω x ω y − ω x − ω y H (e ,e ) = ∑ ∑ h⎣⎡m,n⎦⎤e m−∞ n=−∞ 2 2 2 2 1 − jω m− jω n 1 j m − jω n = ∑ ∑ e x y = ∑ e− ω x ∑ e y 25 m=−2 n=−2 25 m=−2 n=−2 1 1 1 2cos 2cos(2 ) 1 2cos 2cos(2 ) = ( + ω x + ω x ) + ω y + ω y 0.8 25 ( )

0.6

0.4

0.2 Separable ﬁlter: 1-d frequency responses 0 are multiplied 2 3 2 0 1 0 ω -1 y -2 -2 ω -3 x

j jω 1 H e ω x ,e y 1 2cos 2cos(2 ) = ( + ω x + ω x ) ( ) 5

0.8

0.6

0.4

0.2

2 Bike blurred horizontally 3 2 0 1 Filter impulse response 0 ω y -2 -1 -2 ω x 1 -3 1 1 [1] 1 1 5 ( )

j jω 1 H e ω x ,e y 1 2cos 2cos(2 ) = ( + ω y + ω y ) ( ) 5

0.8

0.6

0.4

0.2

2 Bike blurred vertically 3 2 0 1 Filter impulse response 0 ω y -2 -1 -2 ω x ⎛ 1 ⎞ -3 ⎜ 1 ⎟ 1⎜ ⎟ ⎜ ⎣⎡1⎦⎤ ⎟ 5⎜ ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ ⎝ ⎠ Digital Image Processing: Bernd Girod, © 2013 Stanford University -- Linear Image Processing and Filtering 50 Sharpening filter

Original Bike sharpened Bike Filter impulse response

⎛ 0 −1 0 ⎞ 1 ⎜ −1 [8] −1 ⎟ 4 ⎜ ⎟ ⎝⎜ 0 −1 0 ⎠⎟

∞ ∞ j j m j n jω x ω y − ω x − ω y 3 H (e ,e ) = ∑ ∑ h⎣⎡m,n⎦⎤e m−∞ n=−∞ 2.5 1 j j − jω jω = 8 − e− ω x − e ω x − e y − e y 2 4 ( )

1.5 1 1 = 2 − cosω x − cosω y 2 2 1 4 2 4 0 2 0 ω y -2 -2 -4 -4 ω x

j jω H e ω x ,e y = 5− 2cosω − 2cosω ( ) x y

0 Bike sharpened 4 2 4 Filter impulse response 0 2 0 ω -2 y -2 ω ⎛⎞010− -4 -4 x ⎜⎟15 1 ⎜⎟−−[ ] ⎜⎟ ⎝⎠010−

Equation to generate pattern: 2 2 s(x, y) = sˆcos(ax x + ay y ) + s0 Local frequency at (x, y) ∂ a x2 + a y2 = 2a x ∂x ( x y ) x ∂ a x2 + a y2 = 2a y ∂y ( x y ) y x

interpolation s⎡x, y⎤ ⎯⎯⎯⎯→s(x, y) ⎣ ⎦←⎯sam⎯pling⎯⎯

n How is the Fourier transform of s[m,n] related to the Fourier transform of the continuous signal ∞ ∞ s x, y s m,n x m, y n ( ) = ∑ ∑ "! $# δ 2 ( − − ) m=−∞ n=−∞

„continuous“ continuous delta function discrete

n Continuous-space 2-d Fourier transform

∞ ∞  j x j y S ω ,ω = s⎡m,n⎤ δ x − m, y − n e− ω x − ω y dx dy ( x y ) ∫ ∫ ∑ ∑ ⎣ ⎦ 2 ( ) x y m=−∞ n=−∞ ∞ ∞ j m j n j − ω x − ω y jω x ω y = ∑ ∑ s⎣⎡m,n⎦⎤e = S (e ,e ) m=−∞ n=−∞

Original Zoneplate (512x512) Lowpass ﬁltered with 5x5 box ﬁlter

2d impulse response Frequency response ⎛ 1 1 1 ⎞ 1 ⎜ 16 8 16 ⎟ ⎜ ⎟ ⎜ 1 1 1 ⎟ ⎜ ⎟ 0.5 8 4 8 |H| ⎜ ⎟ ⎜ 1 1 1 ⎟ ⎝⎜ 16 8 16 ⎠⎟ 0 2 0 2 0 -2 -2 ω y ωx

Nearest neighbor interpolation Bilinear interpolation

• point source on focal plane maps to point!

focal plane! Lens as Optical Low-pass Filter!

• away from focal plane: out of focus blur! blurred point !

focal plane! Lens as Optical Low-pass Filter!

• shift-invariant convolution!

focal plane! Lens as Optical Low-pass Filter! diffraction-limited PSF of circular

aperture (aka “Airy” pattern):! c ! b = c∗ x

x b sharp image! measured, blurred image! point spread function (PSF): Deconvolution!

• given measurements b and convolution kernel c, what is x?! c x b * ? = Deconvolution with Inverse Filtering!

−1 −1 ⎧F{b}⎫ • naive solution: apply inverse kernel ! x! = c ∗b = F ⎨ ⎬ F c ⎩ { } ⎭ x x! Deconvolution with Inverse Filtering & Noise!

−1 −1 ⎧F{b}⎫ • naive solution: apply inverse kernel! x! = c ∗b = F ⎨ ⎬ F c ⎩ { } ⎭ x! • Gaussian noise,! σ = 0.05

! Image deconvolution

n Given an image f [x,y] that is a blurred version of original image s[x,y], recover the original image. jω jω n Assume linear shift-invariant blur, transfer function B e x ,e y ( )

j jω j jω B e ω x ,e y H e ω x ,e y s[x, y] ( ) f [x, y] ( ) g[x, y] blur recovery ﬁlter

n Naive solution: inverse ﬁlter

j jω 1 H e ω x ,e y = jω ( ) jω x y B(e ,e ) jω jω n Problem: B e x , e y might be zero, noise ampliﬁcation ( )

jω jω j jω Original Blurred w/ B e x ,e y Deblurred w/ B−1 e ω x ,e y ( ) ( )

+ white noise rms = 2.5

• results: terrible! !

• why? this is an ill-posed problem (division by (close to) zero in frequency domain) à noise is drastically ampliﬁed!!

• need to include prior(s) on images to make up for lost data! • for example: noise statistics (signal to noise ratio) ! Deconvolution with Wiener Filtering!

• apply inverse kernel and don’t divide by 0 !

⎧ 2 ⎫ −1 ⎪ F{c} F{b}⎪ x! = F ⋅ ⎨ 2 1 ⎬ ⎪ F{c} + F{c} ⎪ ⎩ SNR ⎭

amplitude-dependent ! mean signal ≈ 0.5 SNR = damping factor!! noise std = σ Deconvolution with Wiener Filtering!

naive! Wiener! x x! Deconvolution with Wiener Filtering!

σ = 0.01 σ = 0.05 σ = 0.1 Deconvolution with Wiener Filtering!

• results: not too bad, but noisy !

• this is a heuristic à dampen noise ampliﬁcation!

• take EE 367 “Computational Imaging and Display” for more advanced deconvolution! Nonlinear noise reduction/sharpening

n Noise reduction: smooth the image, lowpass filtering n Deblurring: sharpen edges, highpass filtering n How can both be achieved simultaneously? n Key insight: large amplitude of highpass filtered image indicates presence of edge

α f "x, y$ ( h # %) HPF

jωx jωy

Ze, e f x, y ( ) - ⎡ ⎤ h ⎣ ⎦ f ⎡x, y⎤ h ⎣ ⎦ f [x, y] + g[x, y] ∑ ∑

n Can be extended to multiple HPFs

HPF Soft coring α fi "x, y$ jω jω ( # %)

Z e x ,e y (.) 1 f ⎡x, y⎤ ( ) - 1 ⎣ ⎦ f [x, y] + g[x, y] ∑ ∑ f ⎡x, y⎤ - i ⎣ ⎦ HPF Soft coring j jω ω x y f ⎡x, y⎤ Z e ,e 2 ⎣ ⎦ (.) 2 ( )

blurred, noisy image noise-reduced and sharpened

log magnitude of log magnitude of log magnitude of log magnitude of image filtered with image filtered with image filtered with image filtered with ⎛ 0 0 0 ⎞ ⎛0 − 0.5 0⎞ ⎛− 0.5 0 0 ⎞ ⎛ 0 0 − 0.5⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜− 0.5 [1] − 0.5⎟ ⎜0 [1] 0⎟ ⎜ 0 [1] 0 ⎟ ⎜ 0 [1] 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 0 ⎠ ⎝0 − 0.5 0⎠ ⎝ 0 0 − 0.5⎠ ⎝− 0.5 0 0 ⎠

γ f ⎡x,y⎤ ⎛ − i ⎣ ⎦ ⎞ τ α f ⎡x, y⎤ = m⋅ f ⎡x, y⎤⋅⎜1− e ⎟ ( i ⎣ ⎦) i ⎣ ⎦ ⎜ ⎟ ⎝ ⎠ Example: m = 3 slope of linear part γ = 2 transition between suppression and linear τ = 10 width of suppressed area α f ⎡x, y⎤ ( i ⎣ ⎦)

Noise reduction by Sharpening by highpass Combined noise reduction and lowpass ﬁlter (linear) ﬁlter (linear) sharpening (nonlinear)