Prof. Dr. Anderson Rocha

Microsoft Research Faculty Fellow Affiliate Member, Brazilian Academy of Sciences

Reasoning for Complex Data (Recod) Lab. Institute of Computing, University of Campinas (Unicamp)

Campinas, SP, Brazil

anderson.rocha@ic.unicamp.br http://www.ic.unicamp.br/~rocha

MC851 - Projetos em Computação Visão Computacional

Aula #4

Pixels and Image Filters This lecture slides were made based on slides of several researchers such as James Hayes, Derek Hoiem, Alexei Efros, Steve Seitz, David Forsyth and many others. Many thanks to all of these authors.

Reading: Szeliski, Chapter 3, Secs. 3.2

Filtering  

•  Image  filters  in  spa0al  domain  – Filter  is  a  mathema0cal  opera0on  of  a  grid  of  numbers  

– Smoothing,  sharpening,  measuring  texture    

•  Templates  and  Image  Pyramids  – Filtering  is  a  way  to  match  a  template  to  the  image  

– Detec0on,  coarse-­‐to-­‐fine  registra0on  

Image  filtering  

•  Image  filtering:  compute  func0on  of  local  neighborhood  at  each  posi0on  

•  Really  important!  –  Enhance  images  

•  Denoise,  resize,  increase  contrast,  etc.  –  Extract  informa0on  from  images  

•  Texture,  edges,  dis0nc0ve  points,  etc.  –  Detect  paHerns  

•  Template  matching  

 

1 1 1

1 1 1

1 1 1

Slide credit: David Lowe (UBC)

],[g ⋅⋅

Example:  box  filter  

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

++=∑

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

++=∑

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10 20

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

++=∑

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10 20 30

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

++=∑

0 10 20 30 30

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

++=∑

0 10 20 30 30

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

Credit: S. Seitz

?

],[],[],[,

lnkmflkgnmhlk

++=∑

0 10 20 30 30

50

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1

1 1 1

1 1 1

],[g ⋅⋅

Credit: S. Seitz

?

],[],[],[,

lnkmflkgnmhlk

++=∑

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 90 0 90 90 90 0 0

0 0 0 90 90 90 90 90 0 0

0 0 0 0 0 0 0 0 0 0

0 0 90 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 10 20 30 30 30 20 10

0 20 40 60 60 60 40 20

0 30 60 90 90 90 60 30

0 30 50 80 80 90 60 30

0 30 50 80 80 90 60 30

0 20 30 50 50 60 40 20

10 20 30 30 30 30 20 10

10 10 10 0 0 0 0 0

[.,.]h[.,.]f

Image  filtering  1 1 1 1 1 1 1 1 1 ],[g ⋅⋅

Credit: S. Seitz

],[],[],[,

lnkmflkgnmhlk

++=∑

What does it do? •  Replaces each pixel with

an average of its neighborhood

•  Achieve smoothing effect (remove sharp features)

1 1 1

1 1 1

1 1 1

Slide credit: David Lowe (UBC)

],[g ⋅⋅

Box  Filter  

Smoothing  with  box  filter  

One  more  on  board…  

Prac0ce  with  linear  filters  

0 0 0

0 1 0

0 0 0

Original

?

Source: D. Lowe

Prac0ce  with  linear  filters  

0 0 0

0 1 0

0 0 0

Original Filtered (no change)

Source: D. Lowe

Prac0ce  with  linear  filters  

0 0 0

1 0 0

0 0 0

Original

?

Source: D. Lowe

Prac0ce  with  linear  filters  

0 0 0

1 0 0

0 0 0

Original Shifted left By 1 pixel

Source: D. Lowe

Prac0ce  with  linear  filters  

Original

1 1 1 1 1 1 1 1 1

0 0 0 0 2 0 0 0 0 - ?

(Note that filter sums to 1)

Source: D. Lowe

Prac0ce  with  linear  filters  

Original

1 1 1 1 1 1 1 1 1

0 0 0 0 2 0 0 0 0 -

Sharpening filter -  Accentuates differences with local average

Source: D. Lowe

Sharpening  

Source: D. Lowe

Other  filters  

-1 0 1

-2 0 2

-1 0 1

Vertical Edge (absolute value)

Sobel

Other  filters  

-1 -2 -1

0 0 0

1 2 1

Horizontal Edge (absolute value)

Sobel

•  Weight  contribu0ons  of  neighboring  pixels  by  nearness  

   

0.003 0.013 0.022 0.013 0.003 0.013 0.059 0.097 0.059 0.013 0.022 0.097 0.159 0.097 0.022 0.013 0.059 0.097 0.059 0.013 0.003 0.013 0.022 0.013 0.003

5 x 5, σ = 1

Slide credit: Christopher Rasmussen

Important  filter:  Gaussian  

Smoothing  with  Gaussian  filter  

Smoothing  with  box  filter  

Gaussian  filters  •  Remove  “high-­‐frequency”  components  from  the  image  (low-­‐pass  filter)  –  Images  become  smoother  

•  Convolu0on  with  self  is  another  Gaussian  –  So  can  smooth  with  small-­‐width  kernel,  repeat,  and  get  same  result  as  larger-­‐width  kernel  would  have  

–  Convolving  two  0mes  with  Gaussian  kernel  of  width  σ  is  same  as  convolving  once  with  kernel  of  width  

•  Separable  kernel  –  Factors  into  product  of  two  1D  Gaussians  

Source: K. Grauman

�p2

How  big  should  the  filter  be?  •  Values  at  edges  should  be  near  zero  •  Rule  of  thumb  for  Gaussian:  set  filter  half-­‐width  to  about  3  σ  

Prac0cal  maHers  

Prac0cal  maHers  •  What  about  near  the  edge?  

–  the  filter  window  falls  off  the  edge  of  the  image  – need  to  extrapolate  – methods:  

•  clip  filter  (black)  •  wrap  around  •  copy  edge  •  reflect  across  edge  

Source: S. Marschner

Prac0cal  maHers  

– methods  (MATLAB):  •  clip  filter  (black):    imfilter(f,  g,  0)  •  wrap  around:  imfilter(f,  g,  ‘circular’)  •  copy  edge:      imfilter(f,  g,  ‘replicate’)  •  reflect  across  edge:    imfilter(f,  g,  ‘symmetric’)  

Source: S. Marschner

Q?

Prac0cal  maHers  •  What  is  the  size  of  the  output?  •  MATLAB:  filter2(g,  f,  shape)  

–  shape  =  ‘full’:  output  size  is  sum  of  sizes  of  f  and  g  –  shape  =  ‘same’:  output  size  is  same  as  f  –  shape  =  ‘valid’:  output  size  is  difference  of  sizes  of  f  and  g    

f

g g

g g

f

g g

g g

f

g g

g g

full same valid

Source: S. Lazebnik

Class  Implementa0on  Gaussian Filter!

Laplacian Filter!

 A.  Oliva,  A.  Torralba,  P.G.  Schyns,    “Hybrid  Images,”  SIGGRAPH  2006  

Gaussian unit impulse Laplacian of Gaussian

Take-­‐home  messages  •  Image  is  a  matrix  of  numbers  

•  Linear  filtering  is  sum  of  dot  product  at  each  posi0on  –  Can  smooth,  sharpen,  translate  (among  many  other  uses)  

   

•  Be  aware  of  details  for  filter  size,  extrapola0on,  cropping  

   

1 1 1

1 1 1

1 1 1

0.92   0.93   0.94   0.97   0.62   0.37   0.85   0.97   0.93   0.92   0.99  

0.95   0.89   0.82   0.89   0.56   0.31   0.75   0.92   0.81   0.95   0.91  

0.89   0.72   0.51   0.55   0.51   0.42   0.57   0.41   0.49   0.91   0.92  

0.96   0.95   0.88   0.94   0.56   0.46   0.91   0.87   0.90   0.97   0.95  

0.71   0.81   0.81   0.87   0.57   0.37   0.80   0.88   0.89   0.79   0.85  

0.49   0.62   0.60   0.58   0.50   0.60   0.58   0.50   0.61   0.45   0.33  

0.86   0.84   0.74   0.58   0.51   0.39   0.73   0.92   0.91   0.49   0.74  

0.96   0.67   0.54   0.85   0.48   0.37   0.88   0.90   0.94   0.82   0.93  

0.69   0.49   0.56   0.66   0.43   0.42   0.77   0.73   0.71   0.90   0.99  

0.79   0.73   0.90   0.67   0.33   0.61   0.69   0.79   0.73   0.93   0.97  

0.91   0.94   0.89   0.49   0.41   0.78   0.78   0.77   0.89   0.99   0.93  

=