smoothing techniques in image processing prof. phd. vasile gui polytechnic university of timisoara
TRANSCRIPT
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Smoothing Techniques in Image Processing
Prof. PhD. Vasile Gui
Polytechnic University of Timisoara
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Content
Introduction Brief review of linear operators Linear image smoothing techniquesNonlinear image smoothing techniques
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Introduction
Why do we need image smoothing?
What is “image” and what is “noise”?
– Frequency spectrum– Statistical properties
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Brief Review of Linear Operators[Pratt 1991]
Generalized 2D linear operator
Separable linear operator:
Space invariant operator:
Convolution sum
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Brief Review of Linear Operators
Geometrical interpretation of 2D convolution
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Matrix formulation
Unitary transform
Separable transform
Transform is invertible
Brief Review of Linear Operators
TRCFAAT
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Brief Review of Linear Operators
Basis vectors are orthogonal
Inner product and energy are conserved
Unitary transform: a rotation in N-dimensional vector space
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Brief Review of Linear Operators
2D vector space interpretationof a unitary transform
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Brief Review of Linear Operators
DFT unitary matrix
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Brief Review of Linear Operators
1D DFT
2D DFT
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Brief Review of Linear Operators
Circular convolution theorem
Periodicity of f,h and g with N are assumed
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
SeparableCan be computed recursively, resulting in
roughly 4 operations per pixel
L pixels
+
m,n
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
Optimality properties Signal and additive white noise
Noise variance is reduced N times
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
Unknown constant signal plus noiseMinimize MSE of the estimation g:
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
i.i.d. Gaussian signal with unknown mean.
Given the observed samples, maximize
Optimal solution: arithmetic mean
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
Frequency response:
Non-monotonically decreasing with frequency
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
An example
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
Image smoothed with 33, 55, 99
and 11 11 box filters
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Linear Image smoothing techniques Box filters. Arithmetic mean LL operator
Original Lena imageLena image filtered with
5x5 box filter
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Linear Image smoothing techniques Binomial 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
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Linear Image smoothing techniques Binomial filters. 1D versions
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As size increases, the shape of the filter is closer to a Gaussian one
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Linear Image smoothing techniques Binomial filters. 2D versions
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Linear Image smoothing techniques Binomial filters. Frequency response
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with frequency
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Linear Image smoothing techniques Binomial filters. Example
Original Lena image Lena image filtered
with binomial 5x5 kernel
Lena image filtered
with box filter 5x5
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Linear Image smoothing techniques Binomial and box filters. Edge blurring comparison
Linear filters have to compromise smoothing with edge blurring
Step edge
Result of size L box filter
L
Size L binomial
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Nonlinear image smoothingThe median filter [Pratt 1991]
Block diagram
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Nonlinear image smoothingThe median filter
Numerical example
6 7
3 7 8
2 3
4 6 7
2, 3, 3, 4, 6, 7, 7, 7, 8
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Nonlinearity
Nonlinear image smoothingThe median filter
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
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Nonlinear image smoothingThe median filter
Noisy image 5x5 median filtered 5x5 box filter
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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!
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Nonlinear image smoothingThe median filter
Caution: points, thin lines and corners are erased by the median filter
Test images
Results of 33 pixel median filter
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Nonlinear image smoothingThe median filter
Cross shaped window can correct some of the problems
Results of the 9 pixel cross shaped window median filter
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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
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Nonlinear image smoothingThe median filter
VLSI median block diagram
a
b
Min(a,b)
Max(a,b)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
x1
x2
x3
x4
x5
x6
x7
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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
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Nonlinear image smoothingThe median filter
Example of color median filtering
5x5 pixels window Up: original image Down: filtered image
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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
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Nonlinear image smoothingThe weighted median filter
Numerical example for the weighted median filter
6 7
3 7 8
2 3
4 6 7
2, 2, 3, 3, 3, 3, 4, 6, 6, 7, 7, 7, 7, 7, 8
3
1 1
1 1
2 2
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Nonlinear image smoothingThe multi-stage median filter
Better detail preservation
mi = median( Ri ), i =1,2,3,4.
Result = median( m1,m2,m3,m4,m5 )
R1
R2
R3
R4
m5
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Nonlinear image smoothingRank-order filters (L filters)
Block diagram
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Nonlinear image smoothingRank-order filters (L filters)
Some examples
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aN=1
Max
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am=1
Median
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Mid range. . .
Particular cases of grey level
morphological filters
Percentile
filters (rank selection):
0%,
50%
100%
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Nonlinear image smoothingRank-order filters (L filters)
Optimality
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k
N
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Minkovski distance
Case r = 1: best estimator is median.Case r = 2, best estimator is arithmetic mean.Case r , best estimator is mid-range.
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Nonlinear image smoothingRank-order filters (L filters)
Alpha-trimmed mean filter
otherwise
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),12/(1
= (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
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Nonlinear image smoothingRank-order filters (L filters)
Median of absolute differences trimmed mean Better smoothing than the median filter and good edge preservation
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M2 is a robust estimator of the variance
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Nonlinear image smoothingRank-order filters (L filters)
k nearest neighbour (kNN) median filterMedian of the k grey values nearest by rank
to the central pixel Aim: same as above
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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. the mean of the most homogeneous region
Matsujama & Nagao
Nonlinear image smoothingSelected area filtering [Nagao 1980]
reg 1 reg 2
reg 3
reg 4
reg 5 reg 6
reg 7reg 8
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Nonlinear image smoothingConditional mean
Pixels in a neighbourhood are averaged only if they differ from the central pixel by less than a given threshold:
otherwise
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L is a space scale parameter and th is a range scale parameter
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Nonlinear image smoothingConditional mean
Example with L=3, th=32
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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.
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s() and r() are space and range similarity functions (Gaussian functions of the Euclidian distance between their arguments).
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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:
));K(d()(
}||||
exp{)(2
shs
h
c
c
xxx
xxx
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Nonlinear image smoothingBilateral filter
Example of Bilateral filtering Low contrast texture has been removed Yet edges are well preserved
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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:
dinii R xx ...1}{
n
id r
Knr
f1
1)(ˆ ixxx
)(
2 1)(
)(ˆ)(ˆ
2 xxi
x
xxxx
x
ri Sr n
Mf
f
d
r
S is a sphere of radius r, centered on x and nx is the number of samples inside the sphere
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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.
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Nonlinear image smoothingMean shift filtering
Example1.
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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
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Nonlinear image smoothingMean shift filtering
Example 2
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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.
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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.