2. Introduction
What is Image denoising?
The removing of noise from the
image is called Image denoising.
The algorithms are used for
Image denoising are called Image
denoising algorithms.
3. What is Image?
A n image is generally encoded as a matrix of
grayscale or color values. Each pair
(i, u(i)), where u(i) is the value at i, is called a
pixel.
In the case of grayscale images, i is a point on a
two-dimensional (2D) grid and u(i) is a real
value. In the case of classical color images, u(i)
is a triplet of values for the red, green, and blue
components.
4. What is noise?
Each one of the pixel values u(i) is the result of
a light intensity measurement, usually made by
a charge coupled device (CCD) matrix coupled
with a light focusing system.
Each captor of the CCD is roughly a square in
which the number of incoming photons is being
counted for a fixed period corresponding to the
obturation time.
5. What is Noise?
When the light source is constant, the number
of photons received by each pixel fluctuates
around its average in accordance with the
central limit theorem.
In other words, one can expect fluctuations of
order √n for n incoming photons. In
addition, each captor, if not adequately
cooled, receives heat photons. This is usually
called “noise.”
6. Noise model
All denoising algorithm are based on Noise
Model.
Noise Model
v(i) = u(i) + n(i) ;iϵI
v(i): observed value,
u(i): true value,
n(i): noise value
7. Method noise
( ℎ,v) = v – ℎ(v)
•V: noise image
•Dh: denoise method
•Dh(v) is more smooth than v (Smooth part )
•n(Dh,v): the noise guessed by the method
(Non-smooth part (contains both noise and texture))
8. Types of Denoising Algorithms
All the denoising algorithms are achieved by
averaging. The most common types are:-
Spatial domain filter
•Gaussian filtering
•Anisotropic filtering (AF)
•Neighboring filtering
•Total Variation minimization
Non-Local-Means (NL-means) algorithm
9. Gaussian Filtering
The image isotropic linear filtering boils down to
the convolution of the image by a linear symmetric
gaussian kernel.
The image method noise of the convolution with a
gaussian kernel Gh is
u − Gh ∗ u = −h²Δu + o(h²),
for h small enough.
10. Gaussian Filtering
Gaussian convolution is optimal in flat
parts of the image.
Drawback of Gaussian Filtering
Edges and textures are blurred.
11. Anisotropic filtering (AF)
Attempt to avoid the blurring effect of the Gaussian.
Convolve the image at only in the direction
orthogonal to ( ).
u(x) − AFhu(x) = −½h²|Du|curv(u)(x) + o(h²),
where the relation holds when Du(x) = 0.
12. Anisotropic filtering (AF)
The Straight edges are well restored.
Drawbacks of AF
Flat and texture regions are degraded
13. Total Variation minimization
In total variation minimization, the original image u
is supposed to have a simple geometric
description, namely, a set of connected sets, the
objects, along with their smooth contours, or
edges. The image is smooth inside the objects but
with jumps across the boundaries.
u(x) − TVF[λ](u)(x) = − ½λcurv(TVF[λ](u))(x).
where TV (u) denotes the total variation of u and λ
is a given Lagrange multiplier.
14. Total Variation
minimizationStraight edges are maintained because of their small
curvature.
Drawback of Total Variation minimization
Textures can be over smoothed if λ is too small.
15. Neighborhood filtering
The previous filters are based on a notion of
spatial neighborhood or proximity. Neighborhood
filters instead take into account grayscale values
to define neighboring pixels. In the simplest and
more extreme case, the denoised value at pixel i
is an average of values at pixels which have a
grayscale value close to u(i). The grayscale
neighborhood is therefore
B(i, h) = {j ∈ I | u(i) −h < u(j) < u(i) + h}
16. Neighborhood filtering
This is a fully nonlocal algorithm, since pixels
belonging to the whole image are used for the
estimation at pixel i.
Drawback of Neighborhood filtering
Comparing only grey level values in as single pixel
is NOT so robust when these values are noisy.
18. NL-Means Algorithm
The NL-means algorithm tries to take advantage
of the high degree of redundancy of any natural
image. By this, we simply mean that every small
window in a natural image has many similar
windows in the same image. This fact is patent for
windows close by, at one pixel distance, and in
that case we go back to a local regularity
assumption.
19. NL-Means
Algorithm NL-means not only compares the grey level in a
single point but also the geometrical configuration
in a whole neighborhood.
More robust than neighborhood filter.
20. NL-Means Algorithm
P has the same grey level value of q3
But, the neighborhoods are much
different.
Therefore the weight w(p, q3) is
nearly 0
