Image Restoration

CONTENT

• Overview• Noise Models

– Gaussian– Salt-and-pepper– Uniform– Rayleigh

• Noise Removal using Spatial Filters– Order filters– Mean filters

• Geometric Transforms

Overview

• Used to improve the appearance of an image by application of a restoration process that uses a mathematical model for image degradation

• Types of degradation :-– Blurring caused by motion or atmospheric

disturbance– Geometric distortion caused by imperfect lenses– Superimposed interference patterns caused by

mechanical systems– Noise from electronic sources

• We see that sample degraded images and knowledge of the image acquisition process are inputs to development of a degradation model

• After the model has been developed, the next step is the formulation of the inverse process

• This inverse degradation process is then applied to the degraded image, d(r,c), which results in the output image, Î(r,c),

• The output image Î(r,c), is the restored image which represents an estimate of original image, I(r,c)

d̂

• Once the estimated image has been created, any knowledge gained by observation & analysis of this image is used as additional input for further development of degradation model

• This process continues until satisfactory results are achieved

• With this perspective, can define image restoration as the process of finding an approximation to the degradation process & finding the appropriate inverse process to estimate the original image

System Model

• Degradation process model consists of 2 parts, the degradation function & the noise function

• General mode in spatial domain :d(r,c) = h(r,c) * I(r,c) + n(r,c)

where d(r,c) = degraded imageh(r,c) = degradation functionI(r,c) = original imagen(r,c) = additive noise function

System Model

• Frequency domain :D(u,v) = H(u,v) * I(u,v) + N(u,v)

where D(u,v) = Fourier transform of the degraded image

H(u,v) = Fourier transform of the degradation function

I(u,v) = Fourier transform of the original image

N(u,v) = Fourier transform of the additive noise function

Noise Models

• Any undesired information that contaminates an image• noise models is a random variable with a probability

density function (PDF) that describes its shape and distribution

• The actual distribution of noise in a specific image is the histogram of the noise

• Noise can be modeled with Gaussian (“normal”), uniform, salt-and-pepper (“impulse”), or Rayleigh distribution

• Gaussian model – occur from electronic noise in image acquisition system– Most problematic with poor lighting conditions or vary

high temperatures– Also valid for film grain noise

• Salt-and-pepper noise (also called impulse noise, shot noise or spike noise) typically caused by malfunctioning pixel element in camera sensors, faulty memory locations, or timing errors in digitization process

• Uniform noise is useful - it can be used to generate any other type of noise distribution, and is often used to degrade images for the evaluation of image restoration algo since provides the most unbiased or neutral noise model

Gaussian distribution 22 2/)(

22

1

mg

g eH

iancevar

deviation standard

(average)mean m

levelgray g

2

• A bell-shapped• 70% of all values fall within the range from

one standard deviation (σ) below the mean (m) to one above

• About 95% fall within two standard deviations

Uniform distribution

12

a)-(b variance

2

ba mean

elsewhere 0

bgafor , 1

2

abHUniform

• The gray level values of the noise are evenly distributed across a specific range

Salt-and-pepper

• There are only 2 possible values, a and b, and the probability of each is typically less than 0.2 – with numbers greater than this the noise will swamp out the image

)pepper"("a gfor )salt"(" b gfor &

ABpapersaltH

Rayleigh

4

) - (4 nce varia

4

mean

: where

2g /2

gRayleigh eH

Original image without noise, and its histogram

image with added Gaussian noise with mean = 0 and variance = 600, and its

histogram

image with added uniform noise with mean = 0 and variance = 600, and its

histogram

image with added salt-and-pepper noise with the probability of each 0.08, and its

histogram

Noise Removal Using Spatial Filters

• Spatial filters can be effectively used to remove various types of noise

• Operate on small neighborhoods, 3x3 to 11x11• Will use the degradation model with the

assumption that h(r,c) causes no degradation where the only corruption to the image is caused by additive noise

d(r,c) = I(r,c) + n(r,c)where

d(r,c) = degraded imageI(r,c) = original imagen(r,c) = additive noise function

• Two primary categories; order filters and mean filters

• Order filters – implemented by arranging the neighborhood pixels in order from smallest to largest gray level value, and using this ordering to select the “correct” value

• Mean filters determine, in one sense or another, an average value

• Mean filters work best with Gaussian or uniform noise

• Order filters work best with salt-and-pepper, negative exponential, or Rayleigh noise

• Mean filters have disad of blurring the image edges, or details

• Order filters such as mean can be used to smooth images

Order Filters

• Operate on small subimages, windows, and replace the center pixel value (similar to convolution process)

• Given an N x N wondow, W, the pixel values can be ordered as follows

valueslevel)(gray Intensity theare ,.......,I,I,I

where

......,

2321

2321

N

N

I

IIII

• (85, 88, 95, 100, 104, 104, 110, 110, 114)• Min = 85, Med = 104, max = 114 (will be replaced

at the center value)• Median filter is most useful• Max & min filters can eliminate salt or pepper

noise

85 88 95

104 104 100

114 110 110

a) Image with added salt-and-pepper noise, the probability for salt = probability for pepper = 0.10, b) after median

filtering with a 3x3 window, all the noise is not removed

a) b)

c) after median filtering with a 5x5 window, all the noise is removed, but the image is blurry acquiring the “painted”

effect

c)

• Two order filters are midpoint and alpha-trimmed mean filters – both order and mean filters since they rely on ordering the pixels values, but are then calculated by an averaging process

• Midpoint filter – the average of max & min within the window;

• Most useful for Gaussian & uniform noise2

II Midpoint

......,set Ordered

2N1

2321

NIIII

• Alpha-trimmed mean is the average of pixel values within the window, but with some of the endpoint ranked excluded

• Useful for images containing multiple types of noise, Gaussian and salt-and-pepper noise

where T is the number of pixel values excluded at each end of the ordered set, and can range from 0 to (N2 – 1)/2

TN

Tii

N

ITN

IIII2

12

2321

2

1 mean trimmed-Alpha

......,set Ordered

• Alpha-trimmed mean filter ranges from a mean to median filter, depending on the value selected for the T parameter

Figure 9.3-5 Alpha-Trimmed Mean. This filter can vary between a mean filter and a median filter. a) Image with added noise: zero-mean Gaussian noise with a variance of 200, and salt-

and-pepper noise with probability of each = 0.03, b) result of alpha-trimmed mean filter, mask size = 3x3, T = 1, c) result of alpha-trimmed mean filter, mask size = 3x3, T = 2, d) result of alpha-

trimmed mean filter, mask size = 3x3, T = 4. As the T parameter increases the filter becomes more like a median filter, so becomes more effective at removing the salt-and pepper noise.

a) b)

c) d)

Mean Filters• Function by finding some form of an average within the

NxN window, using sliding window concept to process entire image

• The most basic – arithmetic mean filter which finds the arithmetic average of pixel values ;

where N2 = the number of pixels in the NxN window, W

• Smooths out local variations & work best with Gaussian, gamma and uniform noise

Wcr

crdN ),(

2),(

1 mean Arithmetic

• Contra-harmonic mean filter works well for images containing salt OR pepper type noise, depending on the filter order, R:

where W is the NxN window under consideration

• Negative values of R, eliminates salt-type noise• Positive values, eliminates pepper-type noise

Wcr

Wcr

),(

R

),(

1R

c)d(r,

c)d(r,

mean harmonic-Contra

• Geometric mean filter works best with Gaussian noise, & retains detail information better than an arithmetic mean filter

• Defined as the product of pixel values within window, raised to the 1/N2 power:

Wcr

N

),(

12c)I(r, mean Geometric

• Harmonic mean filter also fails with pepper noise but works well for salt noise;

• Retaining detail information better than the arithmetic mean filter

Wcr ),(

2

c)d(r,

1N

H Purata

• Yp mean filter is defined as follows:

p

Wcr

P

p N

crdY

/1

),(2

),(mean

Geometric Transforms

• Images that have been spatially, or geometrically, distorted

• Used to modify the location of pixel values within an image, typically to correct images that have been spatially warped

• Often referred as rubber-sheet transforms - image is modeled as a sheet of rubber and stretched and shrunk

• Because of defective optics in image acquisition system, distortion in image display devices, or 2D imaging of 3D surfaces

• This methods are used in map making, image registration, image morphing, and other applications requiring spatial modification

• Simplest – translate, rotate, zoom & shrink• More sophisticated – 1) spatial transform & 2) gray level

interpolation

Input Image Spatial Transform

Spatial Transform

Gray Level Interpolation

Gray Level Interpolation Output Image

Spatial Transforms

• Used to map the input image location to a location in the output image; it defines how the pixel values in output image are to be arranged

• The original, undistorted image, I(r,c), and distorted (or degraded) image is

• Primary idea is to find a mathematical model for the geometric distortion process –and apply the inverse process to find restored image

I(r,c)

Geometric Distortion

)ˆ,ˆ( crd),(ˆˆ

),(ˆˆ

crCc

crRr

)ˆ,ˆ( crd

image distorted for the coordinatecolumn thedefines ),,(ˆ ˆ

image distorted for the coordinate row thedefines ),,(ˆ ˆ

crCc

crRr

),(ˆ and ),(ˆ crCcrR

• Different equations for different portions of the image

• To determine the necessary equations, need to identify a set of points in the original image that match points in the distorted image

• These sets of points are called tiepoints, used to define the equations ),(ˆ and ),(ˆ crCcrR

• Method to restore a geometrically distorted image consists of 3 steps:1. Define quadriterals (4 sided polygons) with known,

or best-guessed tiepoints for the entire image2. Find the equations for each set of

tiepoints,3. Remap all the pixels within each quadrilateral

subimage using the equations corresponding to those tiepoints

),(ˆ and ),(ˆ crCcrR

1. 2 images are divided into subimages, defined by tiepoints (fig. 9.6.3 a&b)

2. Using bilinear model for the mapping equations, these 4 points to generate the equations :

3. Involves application of the mapping equations, , to all the (r,c) pairs

ckrckckrkcrC

rkrckckrkcrR

ˆ),(ˆ

ˆ),(ˆ

8765

4321

),(ˆ and ),(ˆ crCcrR

Exercise

• Example 9.6.1 & 9.6.2• The difficulty in above example arises when

we try to determine the value of d(41.4, 20.6)– Since digital images are defined only at integer

values for Î(r,c) as an estimate to the original image I(r,c) to represent the restored image

Gray Level Interpolation• The simplest – nearest neighbor method, where the pixel is

assigned the value of the closest pixel in the distorted image– Î(2,3) is set to the value of d(41,21), the row and column values

determined by rounding – Easy to implement and computationally fast

• More advance is to interpolate the value– More computationally extensive but more visually pleasing results– Easiest - neighborhood average. Provide smoother object edges

but slightly blurry

Gray Level Interpolation• Better results- uses bilinear interpolation with the

equation:where = the gray level interpolating

equation• Example 9.6.3 & 9.6.4

4321 ˆˆˆˆ)ˆ,ˆ( kcrkckrkcrg

)ˆ,ˆ( crg