Comments on Sigma Filter Degradation of a smoothed image is due to - - PowerPoint PPT Presentation

Comments on Sigma Filter

• Degradation of a smoothed image is due to blurring of
• bject boundaries
• Here boundaries are better preserved by limiting the

smoothing only to a homogeneous subset of pixels in the neighborhood

• The selected subset comprises those pixels that have

similar intensities

• Pixels with very different intensities are excluded by

making corresponding weights equal to 0 IIT Bombay Slide 31a GNR607 Lecture 13-16 B. Krishna Mohan

Lee filter

Simple Lee filter

• gij = fmean + k.(fij – fmean)
• k varies between 0 and 2

k = 0, gij = fmean  simple averaging k = 1, gij = fij  no smoothing at all k = 2, gij = fij + (fij – fmean) Interpretation of (fij – fmean) ???

IIT Bombay Slide 32 GNR607 Lecture 13-16 B. Krishna Mohan

Lee filter

IIT Bombay Slide 33 GNR607 Lecture 13-16 B. Krishna Mohan

• a. Original

image

• b. Wallis filter
• c. K=2
• d. K=3
• e. K=0.5

f. K=0

General form of Lee filter

• The general form of Lee filter is given by
• kij is given by
• Greater noise, smaller kij, hence more smoothing

( )

ij mean ij ij mean

g f k f f = + −

2 2 2 2 ij ij mean v ij

k f σ σ σ = +

IIT Bombay Slide 34 GNR607 Lecture 13-16 B. Krishna Mohan

Comments on General form of Lee filter

• Noise variance has to be estimated from

homogeneous areas

• Unless noise variance is very low, this filter

smoothes the image like average filter

• Greater noise, smaller kij, hence more smoothing

2 2 2 2 ij ij mean v ij

k f σ σ σ = +

IIT Bombay Slide 34a GNR607 Lecture 13-16 B. Krishna Mohan

Gradient Inverse Filter

• The gradient inverse filter applies weights to the

neighbors in an inverse proportion to their difference to the central pixel (i,j)’s gray level

• Let u(i,j,k,l)=
• Else, u(i,j,k,l) = 2.0

1 , ( , ) ( , ) | ( , ) ( , ) | if f i k j l f i j f i k j l f i j + + ≠ + + −

IIT Bombay Slide 35 GNR607 Lecture 13-16 B. Krishna Mohan

Gradient Inverse Filter

• The gradient inverse filter is defined by
• hi,j,k,l = 0.5 (weight for the centre pixel)
• hi,j,k,l = 0.5[ui,j,k,l / ]

∑

i,j,k,l k,l

u

IIT Bombay Slide 36 GNR607 Lecture 13-16 B. Krishna Mohan

, , , , , k w l w i j i j k l i k j l k w l w

g h f

=+ =+ − − =− =−

= ∑ ∑

K-Nearest Neighbor algorithm

• K-nearest neighbor average: compute equally

weighted average of k-nearest neighbors – k neighbors whose gray levels are closest to the central pixel in the neighborhood

• Sort the neighbors on the basis of similarity of

gray level to the central pixel

• Compute the average of K neighbors whose

gray levels are closest

IIT Bombay Slide 37 GNR607 Lecture 13-16 B. Krishna Mohan

Example

Consider the neighborhood 33 41 37 32 46 39 30 29 28 K = 4 Closest 4 gray levels to 46 are 41, 39, 37, 33 Including the central pixel, the average is (1/5)(46 + 41 + 39 + 37 + 33) = 39.20 ~ 39

IIT Bombay Slide 38 GNR607 Lecture 13-16 B. Krishna Mohan

Non-linear filtering

• Nonlinear filters have certain advantages
• ver linear filters when dealing with noise
• Common examples are the rank order

filters

• A typical rank order filter is of the form
• gij = H[fi,j,k,l], where H represents a user-

specified rank criterion

IIT Bombay Slide 39 GNR607 Lecture 13-16 B. Krishna Mohan

Rank filtering

• Modal filter
• Central pixel is assigned the gray level that
• ccurs most frequently in the neighborhood
• gij = mode {fi-k,j-l | k,l=-w, …, o, …, w}
• e.g., fn = 11 12 14 15 12 16 11 15 15
• 12 is replaced by 15, which occurs most

frequently in the neighborhood

IIT Bombay Slide 40 GNR607 Lecture 13-16 B. Krishna Mohan

Median Filter

• Median filter is the most commonly used

non-linear filter for image smoothing

• When the image is corrupted by random

salt-and-pepper noise, median operation is very effective in removing the noise, without degrading the input image

• gij = median {fi-k,j-l | k,l=-w, …, o, …, w}

IIT Bombay Slide 41 GNR607 Lecture 13-16 B. Krishna Mohan

Mean v/s Median filter

• Consider an example:
• 15 17 16

15 17 17

• 18 17 15

157 18 15

• 17 14 16

17 14 16

• Case 1

Case 2

• Mean=16

Mean=32

• Median=16

Median=17

• In arithmetic averaging, noise is distributed over the

neighbours

• In median filtering, the extreme values are pushed to one

end of the sequence after sorting, hence ignored when filtered IIT Bombay Slide 42 GNR607 Lecture 13-16 B. Krishna Mohan

Algorithm

• Consider the size of the window around the pixel
• Collect all the pixels in the window and sort them

in ascending / descending order

• Select the gray level after sorting, according to

the rank criterion

• It can easily be verified that median and mode

filters are nonlinear, according to the definition of linearity

IIT Bombay Slide 43 GNR607 Lecture 13-16 B. Krishna Mohan

Example

Median filtering Example here is over 7x7 neighborhood IIT Bombay Slide 44 GNR607 Lecture 13-16 B. Krishna Mohan

Trimmed Mean Filter

• Trimmed-Mean Operator:
• trimmed-mean: first k and last k gray levels

not used

• trimmed-mean: equal weighted average of

central N-2k elements

∑

− + =

− =

k N k n n mean trimmed

x k N z

1 ) (

2 1

IIT Bombay Slide 45 GNR607 Lecture 13-16 B. Krishna Mohan

Some Comments

• Shift variant filters can adapt to the image

conditions better

• More computations are involved in shift

variant filtering

• Gaussian smoothing has some optimal

properties for which it is popular

• Degree of smoothing can be controlled by

varying the width σ of the Gaussian filter

IIT Bombay Slide 46 GNR607 Lecture 13-16 B. Krishna Mohan

Comments contd…

• Simple averaging type filters fare poorly in

case of signal dependent noise

• Particularly with SAR images noise

suppression is challenging

• Noise filtering is performed in case of SAR

prior to image formation or after image formation

• Shift variant and nonlinear filters more

successful with SAR images

IIT Bombay Slide 47 GNR607 Lecture 13-16 B. Krishna Mohan

Comments contd…

• An important requirement of image smoothing:

the sharpness in the image should be least affected

• Many comparative studies to evaluate

methods

• Estimating noise statistics key to improving

quality of data like SAR images

• Additional techniques – based on

mathematical morphology

IIT Bombay Slide 48 GNR607 Lecture 13-16 B. Krishna Mohan

Edge Enhancement Methods

Edge

• Edge:

boundary where brightness values significantly differ among neighbors edge: brightness value appears to abruptly jump up (or down)

IIT Bombay Slide 49 GNR607 Lecture 13-16 B. Krishna Mohan

IIT Bombay Slide 50 GNR607 Lecture 13-16 B. Krishna Mohan

Original image (left), Sharpened Image (right)

Edge Detection

Essential to mark the boundaries of objects Area, shape, size, perimeter, etc. can be computed from clearly identified object boundaries Intensity / color / texture / surface orientation gradient employed to detect edges Gradient magnitude denotes the strength of edge Gradient direction relates to direction of change of intensity / color IIT Bombay Slide 51 GNR607 Lecture 13-16 B. Krishna Mohan

Different Edges

A

Different colors

Different brightness IIT Bombay Slide 53 GNR607 Lecture 13-16 B. Krishna Mohan

Different Intensities

Different Edges

Different textures Different surfaces IIT Bombay Slide 54 GNR607 Lecture 13-16 B. Krishna Mohan

Types Of Edges

Gray level profile derivatives

• Step edge:
• Ramp edge:
• Peak edge:

2nd 1st 1st IIT Bombay Slide 55 GNR607 Lecture 13-16 B. Krishna Mohan