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Content

- Introduction
- Brief review of linear operators
- Linear image smoothing techniques
- Nonlinear image smoothing techniques

Introduction

Why do we need image smoothing?

What is “image” and what is “noise”?

- Frequency spectrum
- Statistical properties

Brief Review of Linear Operators[Pratt 1991]

- Generalized 2D linear operator
- Separable linear operator:
- Space invariant operator:

Convolution sum

h20

h21

L-1

h12

h11

h10

2

L-1

h02

h01

h00

h22

h20

h21

H

2

h12

h11

h10

h02

h01

h00

L-1

h22

G

h20

h21

2

L-1

h12

h11

h10

2

h02

h01

h00

N-L+1

F

N

N+L-1

h22

h20

h21

h12

h11

h10

h22

h20

h21

h02

h12

h01

h00

h11

h10

h02

h01

h00

Brief Review of Linear Operators- Geometrical interpretation of 2D convolution

Brief Review of Linear Operators

- Matrix formulation
- Unitary transform
- Separable transform
- Transform is invertible

Brief Review of Linear Operators

- Basis vectors are orthogonal
- Inner product and energy are conserved
- Unitary transform: a rotation in N-dimensional vector space

Brief Review of Linear Operators

- DFT unitary matrix

Brief Review of Linear Operators

- 1D DFT
- 2D DFT

Brief Review of Linear Operators

- Circular convolution theorem
- Periodicity of f,h and g with N are assumed

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

+

L pixels

m,n+1

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator- Separable
- Can be computed recursively, resulting in roughly 4 operations per pixel

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

Optimality properties

- Signal and additive white noise
- Noise variance is reduced N times

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

- Unknown constant signal plus noise
- Minimize MSE of the estimation g:

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

- i.i.d. Gaussian signal with unknown mean.
- Given the observed samples, maximize
- Optimal solution: arithmetic mean

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

- Frequency response:

Non-monotonically decreasing with frequency

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

- An example

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

- Image smoothed with 33, 55, 99

and 11 11 box filters

Linear Image smoothing techniquesBox filters. Arithmetic mean LL operator

Lena image filtered with

5x5 box filter

Original Lena image

Linear Image smoothing techniquesBinomial filters [Jahne 1995]

- Computes a weighted average of pixels in the window
- Less blurring, less noise cleaning for the same size
- The family of binomial filters can be defined recursively
- The coefficients can be found from (1+x)n

Linear Image smoothing techniquesBinomial filters. 1D versions

As size increases, the shape of the filter is closer to a Gaussian one

Linear Image smoothing techniquesBinomial filters. Frequency response

Monotonically decreasing

with frequency

Linear Image smoothing techniquesBinomial filters. Example

Original Lena image

Lena image filtered

with binomial 5x5 kernel

Lena image filtered

with box filter 5x5

Result of size L

box filter

Size L binomial

L

Linear Image smoothing techniquesBinomial and box filters. Edge blurring comparison- Linear filters have to compromise smoothing with edge blurring

f1

f(2)

f2

.

.

.

f(m)

.

.

.

Order samples

select

f(N)

fN

median

Nonlinear image smoothingThe median filter [Pratt 1991]- Block diagram

N is odd

Nonlinear image smoothingThe median filter

- Nonlinearity

median{f1 + f2} median{ f1} + median{ f2}.

However:

median{ c f} = c median{ f },

median{ c + f} = c + median{ f}.

- The filter selects a sample from the window, does not average
- Edges are better preserved than with liner filters
- Best suited for “salt and pepper” noise

Nonlinear image smoothingThe median filter

- Optimality
- Grey level plateau plus noise. Minimize sum of absolute differences:
- Result:
- If 51% of samples are correct and 49% outliers, the median still finds the right level!

Results of 33 pixel median filter

Nonlinear image smoothingThe median filter- Caution: points, thin lines and corners are erased by the median filter

Results of the 9 pixel cross shaped window median filter

Nonlinear image smoothingThe median filter- Cross shaped window can correct some of the problems

Nonlinear image smoothingThe median filter

- Implementing the median filter
- Sorting needs O(N2) comparisons
- Bubble sort
- Quick sort
- Huang algorithm (based on histogram)
- VLSI median

x(1)

x2

x(2)

x3

x(3)

a

Min(a,b)

x4

x(4)

b

Max(a,b)

x5

x(5)

x6

x(6)

x7

x(7)

Nonlinear image smoothingThe median filter- VLSI median block diagram

Nonlinear image smoothingThe median filter

- Color median filter
- There is no natural ordering in 3D (RGB) color space
- Separate filtering on R,G and B components does not guarantee that the median selects a true sample from the input window
- Vector median filter, defined as the sample minimizing the sum of absolute deviations from all the samples
- Computing the vector median is very time consuming, although several fast algorithms exist

Nonlinear image smoothingThe median filter

- Example of color median filtering
- 5x5 pixels window
- Up: original image
- Down: filtered image

Nonlinear image smoothingThe weighted median filter

- The basic idea is to give higher weight to some samples, according to their position with respect to the center of the window
- Each sample is given a weight according to its spatial position in the window.
- Weights are defined by a weighting mask
- Weighted samples are ordered as usually
- The weighted median is the sample in the ordered array such that neither all smaller samples nor all higher samples can cumulate more than 50% of weights.
- If weights are integers, they specify how many times a sample is replicated in the ordered array

2

1

2

2

3

1

2

1

3

7

8

2

3

7

6

4

6

7

2, 2, 3, 3, 3, 3, 4, 6, 6, 7, 7, 7, 7, 7, 8

Nonlinear image smoothingThe weighted median filter- Numerical example for the weighted median filter

R2

R4

m5

R1

mi = median( Ri ), i =1,2,3,4.

Result = median( m1,m2,m3,m4,m5 )

Nonlinear image smoothingThe multi-stage median filter- Better detail preservation

f(1)

f1

a1

f(2)

f2

a2

y

.

.

.

.

.

.

ordering

sum

f(N)

fN

aN

y = a1f(1) + a2f(2)+...+ aNf(N)

Nonlinear image smoothingRank-order filters (L filters)- Block diagram

. . .

Min

aN=1

. . .

Max

am=1

. . .

. . .

Median

a1=.5

aN=.5

. . .

Mid range

Nonlinear image smoothingRank-order filters (L filters)- Some examples

Percentile

filters (rank selection):

0%,

50%

100%

Particular cases of grey level morphological filters

Nonlinear image smoothingRank-order filters (L filters)

- Optimality

Minkovski distance

Case r = 1: best estimator ismedian.

Caser = 2, best estimator is arithmetic mean.

Caser , best estimator is mid-range.

Nonlinear image smoothingRank-order filters (L filters)

- Alpha-trimmed mean filter

- = (2Q + 1) / N, defines de degree of averaging
- = 1 corresponds to arithmetic mean

= 1 / N corresponds to the median filter

Properties: in between mean and median

Nonlinear image smoothingRank-order filters (L filters)

Median of absolute differences trimmed mean

- Better smoothing than the median filter and good edge preservation

M2 is a robust estimator of the variance

Nonlinear image smoothingRank-order filters (L filters)

- k nearest neighbour (kNN) median filter
- Median of the k grey values nearest by rank to the central pixel
- Aim: same as above

reg 1

reg 4

reg 3

reg 6

reg 5

reg 8

reg 7

Nonlinear image smoothingSelected area filtering [Nagao 1980]- Kuwahara type filtering
- Form regions Ri
- Compute mean, mi and variance, Si for each region.
- Result = mi : mi < mj for all j different from i, i,e. themean of the most homogeneous region
- Matsujama & Nagao

Nonlinear image smoothingConditional mean

- Pixels in a neighbourhood are averaged only if they differ from the central pixel by less than a given threshold:

L is a spacescale parameter and th is a range scale parameter

Nonlinear image smoothingConditional mean

- Example with L=3, th=32

Nonlinear image smoothingBilateral filter [Tomasi 1998]

- Space and range are treated in a similar way
- Space and range similarity is required for the averaged pixels
- Tomasi and Manduchi [1998] introduced soft weights to penalize the space and range dissimilarity.

s() and r() are space and range similarity functions (Gaussian functions of the Euclidian distance between their arguments).

Nonlinear image smoothingBilateral filter

- The filter can be seen as weighted averaging in the joint space-range space (3D for monochromatic images and 5D – x,y,R,G,B - for colour images)
- The vector components are supposed to be properly normalized (divide by variance for example)
- The weights are given by:

Nonlinear image smoothingBilateral filter

- Example of Bilateral filtering
- Low contrast texture has been removed
- Yet edges are well preserved

Nonlinear image smoothingMean shift filtering [Comaniciu 1999, 2002]

- Mean shift filtering replaces each pixel’s value with the most probable local value, found by a nonparametric probability density estimation method.
- The multivariate kernel density estimate obtained in the point x with the kernel K(x) and window radius r is:
- For the Epanechnikov kernel, the estimated normalized density gradient is proportional to the mean shift:

S is a sphere of radius r, centered on x and nx is the number of samples inside the sphere

Nonlinear image smoothingMean shift filtering

- The mean shift procedure is a gradient ascent method to find local modes (maxima) of the probability density and is guaranteed to converge.
- Step1: computation of the mean shift vector Mr(x).
- Step2: translation of the window Sr(x) by Mr(x).
- Iterations start for each pixel (5D point) and tipically converge in 2-3 steps.

Nonlinear image smoothingMean shift filtering

Detail of a 24x40

window from the

cameraman image

a) Original data

b) Mean shift paths for some points

c) Filtered data

d) Segmented data

Nonlinear image smoothingMean shift filtering

Comparison to bilateral filtering

- Both methods based on simultaneous processing of both the spatial and range domains
- While the bilateral filtering uses a static window, the mean shift window is dynamic, moving in the direction of the maximum increase of the density gradient.

REFERENCES

- D. Comaniciu, P. Meer: Mean shift analysis and applications. 7th International Conference on Computer Vision, Kerkyra, Greece, Sept. 1999, 1197-1203.
- D. Comaniciu, P. Meer: Mean shift: a robust approach toward feature space analysis. IEEE Trans. on PAMI Vol. 24, No. 5, May 2002, 1-18.
- M. Elad: On the origin of bilateral filter and ways to improve it. IEEE trans. on Image Processing Vol. 11, No. 10, October 2002, 1141-1151.
- B. Jahne: Digital image processing. Springer Verlag, Berlin 1995.
- M. Nagao, T. Matsuiama: A structural analysis of complex aerial photographs. Plenum Press, New York, 1980.
- W.K. Pratt: Digital image processing. John Wiley and sons, New York 1991
- C. Tomasi, R. Manduchi: Bilateral filtering for gray and color images. Proc. Sixth Int’l. Conf. Computer Vision , Bombay, 839-846.

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