Filtering – images restorationusers.polytech.unice.fr/~lingrand/Ens/up/Lesson6-noise.pdf ·...

Image processing, 2006, lingrand@polytech.unice.fr 1

Filtering – images restoration

University of the Philippines - Diliman2006

Diane Lingrandlingrand@polytech.unice.fr

http://www.polytech.unice.fr/~lingrand

Noise : causes and modeling

Observe silence !

http://memory.loc.gov/ammem/awlhtml/awlhome.html

Noise sources• Acquisition context

– Under- or over-lighting– Sensors disturbance

• Sensor– Distorsions (geometrical, intensity)

• Sampling– Aliasing effect (out of validity bound of Nyquist-Shannon)– Objects for which the size is the same as the pixel size: salt and

pepper noise• Scene content

– Clouds in satellite images, …• Quantification

– 256 grey levels: not really a limiting factor for human visual system -- or less

• Compression• Transmission

– Data loss, data corruption

Noise modeling

• Noise depending or not of the image data • Additive or multiplicative noise• Noise: high frequencies (usually higher than the

image frequencies) => low-pass filtering• Kinds of noise:

– Impulse noise: white Gaussian, exponential...– Salt and pepper noise

Salt and pepper noise• nth order : turn n pixels to white and n pixels

to black randomly in an image.– Often expressed as a percentage p (0p1)

• Physical counter-part: dust on a camera film or a scanner, small objects, data loss

• in Java: import java.util.Random– method: int nextInt(int n) gives an integer in [0 n[, with

uniform distribution

– 2 methods :• randomly choose w*h/p pixels to be changed to BLACK

and w*h/p others to be changed to WHITE

• for each pixel: chose randomly an int in [0 1/p[, then if 0, change pixel to WHITE, if 1 change it to BLACK

Example of salt and pepper noise

Original image Noisy salt and papper image with p(white) =p(black)=5%

Impulse noise

• Probability density function form:

• For �α = 1 : exponential noise

• For �α = 2 : Gaussian noise

Gaussian noise

• Additive Gaussian noise with null mean and variance σ 2

• Similar to the acquisition noise• How to implement a Gaussian variable ?

– in Java :

import java.util.Random;The public method double Random.nextGaussian() is a Gaussian random value with mean 0 and standard deviation 1 : to be multiplied by – in C/C++: use rand() from C system library

Gaussian noise in C++

• Initializationstruct timeval tv;

struct timezone tz;

gettimeofday(&tv, &tz);

srand(tv.tv_usec);

• Normal random variable distribution between 0 and 1 rand()/(RAND_MAX+1.0f);

• Gaussian variable b = rand()/(RAND_MAX+1.0f);

if(b < epsilon) b = epsilon;

c = rand()/(RAND_MAX+1.0f);

a = -2.0 * Math::log(b);

if(a < 0.0) a = 0.0;

else a *= Math::cos(2.0 * Math::pi * c);

Gaussian noise example

Original image Noisy image with a Gaussian noise (std dev 4)

Worse :

Standard deviation = 8 Standard deviation = 16

Filtering - Restoration

Spatial filtering

• Filter applied to the neighborhood of a pixel in an image

• Linear or non-linear filtering– linear : convolution, Fourier Transform– non linear : no convolution, no Fourier Transform

• Processing:– smoothing (blurring), contours strengthening

(sharpening) …

Smoothing filters

• Mean filters

• Gaussian filters

• Median filters

• Conservative smoothing

• others

Convolution

• Convolution operator with a kernel

Mean filtering

• Also known as averaging or Box filtering

• Principle : a pixel value is replaced by the mean value computed from its neighbors (and itself)

• Convolution with a kernel of size (2n+1 x 2n+1).– ex:

1/91/91/9

1/91/91/9 Be careful ifyou make divisionswith integers !

Example

• This is a low-pass filtering

• Impairment of contours (blurring)

Smoothing a salt and pepper noise :

Kernel size = 3 Kernel size = 5

Smoothing a Gaussian noise

Kernel size = 3 Kernel size =5

Gaussien filter (Gaussian smoothing)

• 2D Gaussian :– null mean

– variance σ2

• Principle: convolution with a Gaussian• In practice : Gaussian discretization on a

kernel with size (2p+1,2p+1)

Gaussian kernels examples

• 3x3[ 1 2 1 ] [ 2 4 2 ] [ 1 2 1 ]

• 7x7[1 1 2 2 2 1 1][1 2 2 4 2 2 1][2 2 4 8 4 2 2][2 4 8 16 8 4 2][2 2 4 8 4 2 2][1 2 2 4 2 2 1][1 1 2 2 2 1 1]

• 5x5[ 1 1 2 1 1 ] [ 1 2 4 2 1] [ 2 4 8 4 2 ][ 1 2 4 2 1] [ 1 1 2 1 1]

How to compute a Gaussian kernel ? (1)

• Kernel ? ((2p+1) x (2p+1))

– Integer coefficient (faster computations) followed by a normalization step (division by the sum of coefficient)

– Even more efficient: coefficient as powers of 2

– σ depends on the size of the kernel (p)

How to compute a Gaussian kernel ? (2)

• The last coefficient value is 1

• Its real value is obtained by dividing it with the sum of all coefficients ......

1=2q.e-p2/σ22q.e-p2/(2σ2)

2q.e-p2/(2σ2)

Another example of Gaussian kernel

Gaussian filtering after a salt and pepper noise on an

Kernel size = 3 Kernel size = 7

Gaussian filtering following a Gaussian noise of null mean

(σ = 16)

Conservative filtering

• Principle : only the pixels whose value falls in the interval defined by the neighboring pixels are preserved. If the pixel values falls out of this interval, it is replaced by the closest bound:– Upper bound if the value is higher to that

– Lower bound if the value is lower to that

Conservative filtering after a salt and pepper noise

Original image After conservative filtering

Conservative filtering after gaussian noise

Original image After conservative filtering

Conservative filtering

255140

0 6763

255255

0 6763

0 ∈ [0 ; 255]

255140

62 6763

255255

62 6763

0 ∉ [62 ; 255]

Median filters

• Principle : a pixel is replaced by his neighbors' values

• Median value = value in the middle of the sorted list of neighbors' values⇒Need to sort values (ex: quick sort ?)

• Enable the suppression of outliers• Good performance when processing images

impaired by salt and pepper noise

Median filtering

255255

0 6763

088 93 255255 110

00 255 25511063 67 88 93

255255

0 6763

sorting

median value

Some median filter kernels

• carrés (7x7)[1 1 1 1 1 1 1][1 1 1 1 1 1 1][1 1 1 1 1 1 1][1 1 1 1 1 1 1][1 1 1 1 1 1 1][1 1 1 1 1 1 1][1 1 1 1 1 1 1]

• diamand (7x7)[0 0 0 1 0 0 0][0 0 1 1 1 0 0][0 1 1 1 1 1 0][1 1 1 1 1 1 1][0 1 1 1 1 1 0][0 0 1 1 1 0 0][0 0 0 1 0 0 0]

• en croix (7x7)[0 0 0 1 0 0 0][0 0 0 1 0 0 0][0 0 0 1 0 0 0][1 1 1 1 1 1 1][0 0 0 1 0 0 0][0 0 0 1 0 0 0][0 0 0 1 0 0 0]

• en croix (3x3)[0 1 0]

[1 1 1]

[0 1 0]

• carrés (3x3)[1 1 1][1 1 1][1 1 1]

Median filter after pepper and salt noise

kernel 3x3, applied once kernel 3x3, applied twice

Median filtering after a gaussian noise

Original image After median filtering

Contrast enhancement (Sharpening)

• High-pass filtering (edge crispening)• To counter-balance the effects of

smoothing• Some convolution kernels:

[0 -1 0] [1 -2 1] [-1 -1 -1][-1 10 -1] ou [-2 5 -2] ou [-1 9 -1][0 -1 0] [1 -2 1] [-1 -1 -1]

• Strengthen the noise as well• By lowering the center value, the effect

is amplified (try 5 instead of 10)

Example 1:

Salt and pepper noise + 2 times median filtering (3x3)

Example 2:

White noise σ = 16 followed by Gaussian filtering (7x7)

Preliminary conclusions

• Salt and pepper noise:– Median filtering performs well

• White Gaussian noise :– Gaussian filtering performs rather well

• Draw-backs in both cases :– Contours smoothing

• To go further :– Smoothing only in areas of weak gradients

Exercise

• Add noise to images– Gaussian noise– Salt and pepper noise

• Implement restoration filters– Mean filters (2n+1x2n+1), Gaussian filters– Conservative filtering– Median filtering– Contrast enhancement filters

• Test noise and restoration procedures on images

Smoothing with contours preservation

Variational approach

Back to Gaussian filtering

• Convolution with a Gaussian :

• with

Evolution of an image• We add a temporal parameter t as the

image is seen as an object evolving through time:

u(x) u(x,t)

• Image evolution :– Initialization : u(x,t0) = u0(x)

– State at a given time t : u(x,t) = modified image

– evolution : speed ?

Example : heat equation (1)

• Describes heat propagation:

• Linear and parabolic equation

• Interesting property : there exists an closed-form solution which is a Gaussian form :

Heat equation (2)

• Exercise : – Check out that the solution is valid

• Consequences :– Correlation between Gaussian smoothing with

variance �σ2=2t and evolution of the image according to the heat PDE (Partial Differential Equation)

– A classical Gaussian filter can be defined as the result of a PDE

The Partial Differential Equations

• Old methods, many mathematical studies, …

• How to apply this technique in the image processing area?

• In the general case, there is no known closed-form solution– Numerical implementation of the evolution

PDE applied to images restoration

• The drawback of filters seen previously is the lack of contour conservation

• Heat equation :

– We expect : • To smooth part of the images with weak gradent• In strong gradient areas: to smooth the image in a

direction parallel to the contour– Idea : weight by a function depending on the

gradient [Perona-Malik]

c(|∇u|2)

Restoration with contours conservation

• Intuitive idea :– Weak |∇u| : c near to 1 : heat equation

– High |∇u| : c near to 0 : null speed, no smoothing

Contour conservation (continued)

• More precisely : divide in two terms (tangential and normal)

• Taking :

• One get :

Weighting function

• Constraints :

• Example :c(s) = 1 / 1 + s

decreasing

tends at infinity towards

A practical example

Histogramequaliza-

Smoothing with edge conservation

Bibliography

• Mathematical problems in image processing: Partial Differential Equations and the Calculus of Variations, par G. Aubert et P. Kornprobst, edition Springer, Applied Mathematical Sciences, vol. 147, 2002 http://www-sop.inria.fr/odyssee/team/Pierre.Kornprobst/book-pde/index.html

Filtering – images restorationusers.polytech.unice.fr/~lingrand/Ens/up/Lesson6-noise.pdf ·...

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