imgp roc 04 local op

Image Processing:Local Operators

Local operatorsConvolutionsSmooting and BluringEdge detectionSharpeningNon Linear filters

© 2004 Rolf Ingold, University of Fribourg

Image processing operations

Classification of image processing operations Point operation Local operationGlobal operation

A operator is local if each pixel in the result image depends only on this pixel and its neighborhood in the source image

(x,y)V(x,y)


Shift-invariant operator

An local operator φ is said shift-invariant if it has the following property

A shift invariant local operator can be characterized by a function fsuch as

)]},[({],[ ],[],[ yxVvuvuIfyxJ ∈=

]),[(],[]),[(],[ 0000 yyxxIyyxxJyxIyxJ −−φ=−−⇒φ=


Linear operators

An operator φ is said linear if it has the following property

As special cases

( ) )(.)(... JbIaJbIa φ+φ=+φ

( ))()(

)()(IaaI

JIJIφ=φ

φ+φ=+φ


Convolution in continuous space

The convolution of two bidimensional signals is defined as followsin a 1D continuous space

example

in a 2D continuous space

∫ ∫∞+

∞−

∞+

∞−−−=⊗ dudvvyuxgvufyxgyxf ),(),(),(),(

x

f(x)

0

1

x

g(x)

0

1

x

f(x) *g(x)

0

1

∫∞+

∞−−=⊗ duuxgufxgxf )()()()(


Convolution in discrete space

The convolution of two discrete signals is defined as followsin the 1D case

in the 2D case

The convolution can be understood as a sum of shift and multiplyoperations

( ) ∑ ∑∞+

−∞=

∞+

−∞=−−=⊗

u vvyuxgvufyxgf ],[],[],[

( ) ∑ ∞+

−∞=−=⊗

uuxgufxgf ][][][


Convolution kernel

In digital image processing, convolution based operators are usually defined on windows

centered at origin [0,0]bounded to [-m,+m]×[-n,+n]

The convolution of an image I by a kernel h is expressed as

the output pixel value is the weighted sum of the pixels in the neighborhood of correponding input pixelthe matrix containing these weights is called convolution kernel or convolution mask

( ) ∑ ∑+

−=

+

−=−−=⊗

m

mu

n

nvvyuxIvukyxIk ],[],[],[

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

++++−

+−

−+−−−

=

],[],0[],[

]0,[]0,0[]0,[

],[],0[],[

nmknknmk

mkkmk

nmknknmk

k

KK

MOMNM

KK

MNMOM

KK


Convolution at the border

The convolution of an image I with dimension w×h is well defined in a domain (w-m)×(h-n)What values should be assigned to I[x,y] for x<0, x≥w, y<0, or y≥h ? There are several solutions

extend the images with a constant extend the image periodicallyextend the image by mirroring it at its boundaries extend the values at the boundaries indefinitely


Convolution normalization

Result values should stay in the same intensity range as the input image

they can be truncated to upper and lower boundsthey can be normalized by a factor and an offset

In the case of only positive coefficients the factor should be

∑vu

vuk,

],[1


Convolution properties

The convolution is linear

The convolution is commutative and associative (provided that, at the border kernel and image are extended with a 0 constant)

The convolution is shift independent

( ) )*()*(..* JkbIkaJbIak +=+

)*(*)*(**)*(** 12212121 IkkIkkIkkIkk ===

],)[*(],)[*(],[],[ 0000 yyxxIkyxJkyyxxIyxJ −−=⇒−−=


Complexity of the convolution

Using the definition, computing the convolution ofan image I with dimension w×h with a convolution kernel of size (2m+1)×(2n+1)

has a complexity of O(m×n×w×h)


Separable kernel

A convolution kernel is said separable if it can be written as

where hx is a matrix of size 1×(2n+1) and hy a matrix of size (2m+1)×1Thus the convolution can be computed using

In this case the complexity of the convolution is reduced to O((m+n)×w×h)

yx kkk ⋅=

( ) ( )∑ ∑+

−=

+

−=−−=⊗⊗=⊗

m

mu

n

nv yxyx vyuxIvkukIkkyxIk ],[][][)(],[


Applications of local operators

Local operators are used to perform image filtering such as smooting and bluringnoise removal or noise reducing edge detectionsharpening


Mean Filter

The mean filter (or uniform filter) is a based on a simple convolution that computes for each pixel the mean value of its neighbors

kernels vary in shape and sizeMean filters are used for smoothing and noise reducing

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111111111

91

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1111111111111111111111111

251

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

0111011111111111111101110

211


Pyramidal and Cone filters

Pyramidal and cone filters generalize mean filters by using weightswheights decrease with the distance to the center

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

111141111

121

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

0121012321234321232101210

401

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1232124642369632464212321

811


Gaussian smoothing

The 2D Gaussian has the following equation

For smoothing, Gaussian filters have become very popularseveral kernel may be used, for different values of σ

Gaussian kernels are separable

22

22

221),( σ

+−

πσ=

yx

eyxG

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

01210158512813821585101210

811

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

121242121

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

131383131

241

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

1353131423143523382353142314313531

2341

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⊗

121242121

121

121 ( )

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

≅

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⊗

1353131423143523382353142314313531

834.0739.3164.6739.3834.0

834.0739.3164.6739.3834.0


Gradient kernels

Gradient kernels (first order derivative) are directional andused to detect horizontal or vertical edges

The gradient can be computed as

its magnitude is

The simplest gradient kernels would bethey lead to phase shifts cannot computed horizontaly and verticaly at the same point !

As alternative the Roberts kernel may be used

( )101−=xh⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

101

yh

⎟⎟⎠

⎞⎜⎜⎝

⎛⊗⊗

=∇],)[(],)[(

],[yxIhyxIh

yxIy

x

22 ]),)[((]),)[((],[ yxIhyxIhyxI yx ⊗+⊗=∇

( )11−=xh ⎟⎠

⎞⎜⎝

⎛−=

11

yh

⎟⎠

⎞⎜⎝

⎛−

=0110

2h⎟⎠

⎞⎜⎝

⎛−=

1001

1h


Prewitt and Sobel filters

Prewitt and Sobel have proposed two filters which combine derivation and smoothing in orthogonal directions

Prewitt kernels

Sobel kernels

An offset of 0.5 is often added to display the results

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

=−⊗⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101101101

31101

111

31

xh ( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−⊗=

111000111

31

101

11131

yh

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

=−⊗⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

101202101

41101

121

41

xh ( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −−−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−⊗=

121000121

41

101

12131

yh


Illustration of gradient filters

Originale image, filtered horizontaly and verticaly by simple gradient and Sobel kernels, then combined together to get the magnitude


Laplacian kernel

Laplacian kernels (second order derivative) consist of

The most popular Laplacian filter is

The following alternative kernels are also used for Laplacian filters

Laplacian filters are very sensitive to noiseTherefore they are often combined with a Gaussian smoothing filter

fhfhyf

xff yx ⊗+⊗=

∂∂

+∂∂

=∇ 22222 ( )1212 −=xh

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=12

1

2yh

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=+

010141010

)( 22 yx hh

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

111181111

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−−

121242121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

101040101


Laplacian kernel

The effect of a Laplacian filter is explained by the 2nd derivative

Illustration of the Laplacian filter with different additive offsets: 0, 1, 0.5


Laplacian of Gaussian

The Laplacian of the 2D Gaussian has the following equation

A possible kernels is

22

22

2

22

4 211),( σ

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛σ+

−πσ

−=∇yx

eyxyxG

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

01110130311024011303101110


Illustration of edge detection by Laplacian

Original images, the result of a Lapacian kernel, then thresholded


Sharpening

Sharpening the edges of an image consists inisolating the edges and amplifying them then adding them back to the original image

Result is obtained by adding the identity to the inverse of the Laplacian

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−

010151

010

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−−−−−−

111191111

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−

121252

121

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−

0202112

020

31

Original images, inverse of Lapacian, addition, and result of right kernel above


Statistical filters

Consider the ordered set of pixel values in the neighborhood of [x,y]

where z(1) § z(2) § ... § z(N×N). Given a weight vector W, the result value can be computed as a weighted sum

As special cases, we considerminimum filter : w1 = 1, wi = 0 "i ∫ 1maximum filter : wN×N = 1, wi = 0 "i ∫ N × Nmédian filter : w(N×N+1)/2 = 1, wi = 0 "i ∫ (N × N + 1) / 2

Z[x, y] = {I[u,v]}[u,v]∈V[x ,y] = {z(1) [x,y], z(2)[x, y],..., z(N× N )[x, y]}

StatW (I)[x, y] = wi ⋅z(i )[x,y]

i =1

N×N

∑


Median Filter

The median filter consists in replacing each pixel by then the median value of its neighbors

the neighborhood is determined by a binary mask

Original noisy image, filtered by a mean filter and a median filter


Conservative Smoothing

Conservative smoothing simply ensures that each pixel value is bounded within the range of intensities defined by its neighbors

the neighborhood is determined by a binary mask


Morphological operators

Given an image I and a mask Merosion is defined as

dilation is defined as ]}0,0[,|],[],[{],)[( VvuvuMvyuxIMaxyxIDilM ∈−−−=

]}0,0[,|],[],[{],)[( VvuvuMvyuxIMinyxIEroM ∈−−−=

Original noisy, the results of erosion and dilation

imgp roc 04 local op

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