basics of filtering - mahesh goyani

10/19/2016

1

© Mahesh Goyani

Digital Image Processing

Mahesh GoyaniAssistant Professor

Department of Computer Engineering,

Government Engineering College

Modasa - 383315.

E-Mail ID: [email protected]

Web Site: www.maheshgoyani.in

10/19/2016 1

LECTURE NOTES ON

FILTERING IN SPATIAL DOMAIN

© Mahesh Goyani10/19/2016 2

KEY STEPS OF DIGITAL IMAGE PROCESSING

Image Acquisition

Image Restoration

Morphological Processing

Segmentation

Representation & Description

Image Enhancement

Object Recognition

Problem Domain

Colour Image Processing

Image Compression


DIGITAL IMAGE PROCESSING

Point Processing

>>imadjust

>>histeq

Spatial filtering

>>filter2

SpatialDomain

FrequencyDomain

Filtering

>>fft2/ifft2

>>fftshift

Enhancement Restoration

Inverse filtering

Wiener filtering

IMAGE ENHANCEMENT


Basics of Filtering


FUNDAMENTAL OF SPATIAL PROCESSING

Enhancement at any point in image depends only on gray level at that point.

These techniques are known as point processing.

Another approach in which to use a function of the values of f in predefined

neighborhood of (x,y) to determine the value of g at (x,y). In this approach filter

or mask (also known as kernel, window or template) is used.

Size of mask determine size of neighborhood and mask coefficient

determine the nature of process.

Enhancement techniques based on this approaches are known as mask

processing or filtering


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MECHANICS OF SPATIAL FILTERING

mn

i

ii fw1

a

as

b

bt

tysxftswyxg ),(),(),(


MECHANICS OF SPATIAL FILTERING

mn

i

ii fw1

a

as

b

bt

tysxftswyxg ),(),(),(

f(x-1,y-1) f(x-1,y) f(x-1,y+1)

f(x,y-1) f(x,y) f(x,y+1)

f(x+1,y-1) f(x+1,y) f(x+1,y+1)

w(-1,-1) w(-1,0) w(-1,1)

w(0,-1) w(0,0) w(0,1)

w(1,-1) w(1,0) w(1,1)

w(-1,-1) w(-1,0) w(-1,1)

w(0,-1) w(0,0) w(0,1)

w(1,-1) w(1,0) w(1,1)

)1,1()1,1(),1()0,1()1,1()1,1(

)1,()1,0(),()0,0()1,()1,0(

)1,1()1,1(),1()0,1()1,1()1,1(

yxfwyxfwyxfw

yxfwyxfwyxfw

yxfwyxfwyxfw),( yxg


OPERATION STEP

1. Select a single pixel.

2. Determine the pixel’s neighborhood.

3. Apply a function to the values of the pixels in the neighborhood.

This function must return a scalar.

4. Find the pixel in the output image whose position corresponds

to that of the center pixel in the input image. Set this output

pixel to the value returned by the function.

5. Repeat steps 1 through 4 for each pixel in the input image


NEIGHBORHOOD

Neighbourhood of a pixel p at position x, y is a set N(p) of pixels defined

relative to p.

Example 1:

N(p) = {(x,y): |x-xP|=1, |y-yP| = 1}

P

Q


MORE EXAMPLES OF NEIGHBORHOOD

P

P

P

P

P

P

P


WHAT IS FILTERING ?

Modify the pixels in an image based on some function of a local

neighborhood of the pixels

10 5 3

4 5 1

1 1 7

10

Some function(MAX)


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CORRELATION VS. CONVOLUTION



F

(0,0)

(N,N)

H4

1 2

3

Matlab:filter2

imfilter



F

(0,0)

(N,N)

H4

1 2

3

H4

12

3

Matlab:conv2




FILTERING - EXAMPLE

1 -1 -1

1 2 -1

1 1 1

2 2 2 3

2 1 3 3

2 2 1 2

1 3 2 2Rotate

1-1-1

12-1

111


FILTERING - EXAMPLE

3

2

1

2

2

1

3

2

32

21

22

32 5

3

2

1

2

2

1

3

2

32

21

22

32

1-2-1

24-1

111

1-1-1

12-1

111


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FILTERING - EXAMPLE

3

2

1

2

2

1

3

2

32

21

22

32 45

3

2

1

2

2

1

3

2

32

21

22

32

3-1-2

24-2

111

1-1-1

12-1

111


FILTERING - EXAMPLE

3

2

1

2

2

1

3

2

32

21

22

32 4 45

3

2

1

2

2

1

3

2

32

21

22

32

3-3-1

34-2

111

1-1-1

12-1

111


FILTERING - EXAMPLE

3

2

1

2

2

1

3

2

32

21

22

32 4 4 -25

3

2

1

2

2

1

3

2

32

21

22

32

1-3-3

16-2

111

1-1-1

12-1

111


FILTERING - EXAMPLE

3

2

1

2

2

1

3

2

32

21

22

32 4 4

9

-25

3

2

1

2

2

1

3

2

32

21

22

32

2-2-1

14-1

221

1-1-1

12-1

111


FILTERING - EXAMPLE

3

2

1

2

2

1

3

2

32

21

22

32

6

4 4

9

-25

3

2

1

2

2

1

3

2

32

21

22

32

1-2-2

32-2

222

1-1-1

12-1

111


FILTERING - EXAMPLE

12

7

6

4

8

6

14

4

59

59

511

-25


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TEMPLATE MATCHING

Input image

Template

Output

The brighter the value in the output, the better the match


MASK ELEMENT SELECTION

For smoothing (Low pass filters), elements of mask must be

positive

For sharpening (High pass filters), elements of mask contains

both positive and negative wrights.

Normalize the mask element


FILTER TYPES

Smoothing Filters

Low Pass (Linear) Filters

Box Filter

Weighted Averaging Filter

Order Statistics (Non Linear) Filters

Median Filter

Max Filter

Min Filter

Sharpening Filters

Laplacian

Unsharp Masking & high boost Filtering

Gradient


Smoothing Filters


LOW PASS (LINEAR) SPATIAL FILTERS

Used in preprocessing steps such as removal of small details from an image.

The response of linear spatial filtering is given by a sum of products of the

filter coefficients and the corresponding image pixels in the area spanned by the

filter mask.

Linear Filtering of an image f of size M X N with a filter mask of size m X n is

given by:

a=(m-1) /2 and b= (n-1)/2

a

as

b

bt

tysxftsw ),(),(

a

as

b

bt

tsw ),(

),( yxg



Low Pass Filters

Size of mask determines the degree of smoothing and loss of details

Box filter Weighted average (Tent )filter .

- To reduce blurring effect.


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LOW PASS (LINEAR) SPATIAL FILTERS - EXAMPLE

Averaging filter with masks of sizes n=3

,5,9,15 and 35 respectively

Small detailed is removed using different size

mask

When mask size is larger than smaller object

is removed

Intensity of smaller objects blend with the

background and larger objects become blob like

and easy to detect.


LOW PASS (LINEAR) SPATIAL FILTERS - EXAMPLE

Image Taken by Hubble Telescope

Filtered Image with15 X 15 Average Mask

Thresholded Image


LINEAR FILTER – EXAMPLE

000

010

000

Original Identical image

* =



000

100

000

Shifted leftBy 1 pixel

* =Original



111

111

111

Blur (with a average filter)

* =Original



Advantages:

Reduced sharp transitions in gray levels

Remove noise.

Blur High frequency

Do averaging

Disadvantages:

Edges are blurred


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GAUSSIAN SMOOTHING

Gaussian kernel :

Rotationally symmetric

Weights nearby pixels more than distant ones

This makes sense as ‘probabilistic’ inference

about the signal

A Gaussian gives a good model of a fuzzy blob


GAUSSIAN SMOOTHING FILTER

Weights are samples from Gaussian Function

The value of σ determines the degree of smoothing.


GAUSSIAN SMOOTHING FILTER

As σ increases, more samples must be obtained to represent the

Gaussian function accurately.


GAUSSIAN SMOOTHING

What parameters matter here?

Variance of Gaussian Determines extent

of smoothing

σ = 5 with 3030 kernel

σ = 2 with 3030 kernel


GAUSSIAN SMOOTHING IN MATLAB

>> hsize = 10;

>> sigma = 5;

>> h = fspecial(‘gaussian’ hsize, sigma);

>> outim = imfilter(im, h);

>> imshow(outim);

>> mesh(h);

>> imagesc(h);

outim


SMOOTHING BY AVERAGING

depicts box filter: white = high value, black = low value

Original Filtered“Ringing” artifacts!


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SMOOTHING WITH A GAUSSIAN

Original Filtered


EFFICIENT IMPLEMENTATION

Both, the BOX filter and the Gaussian filter are separable:

First convolve each row with a 1D filter

Then convolve each column with a 1D filter

Remember:

Convolution is linear – associative and commutative

x y x y x yg g I g g I g g I

I

gx

I’gy


FILTERING – BOUNDARY ISSUES

What is the size of the output?

MATLAB: filter2(g,f,shape)

shape = ‘full’: output size is sum of sizes of f and g

shape = ‘same’: output size is same as f

shape = ‘valid’: output size is difference of sizes of f and g

f

gg

gg

full

f

gg

gg

same

f

gg

gg

valid


FILTERING – BOUNDARY ISSUES

How should the filter behave near the image boundary?

The filter window falls off the edge of the image

Need to extrapolate

Methods:

• Clip filter (black)

• Wrap around

• Copy edge

• Reflect across edge

imfilter(f,g,0)

imfilter(f,g,‘circular’)

imfilter(f,g,‘replicate’)

imfilter(f,g,‘symmetric’)


FILTER SHAPES

Box Filter Tent Filter

Gaussian Filter


ORDER STATISTIC (NON LINEAR) FILTER

Median Filter : Sort the values of pixels in question and its neighbors

Replace the value of centre pixel with median

Median of a set of values is such that half values in the set are less than or

equal to median and half are greater than or equal to median.

In 3X3 neighborhood the median is the 5th largest value

They provide good result in case of random (impulse) noise. Salt and paper

noise they give good result with less blurring

Forces points with distinct gray levels to be more like their neighbors


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MEDIAN FILTER: HOW IT WORKS ?

A median filter is good for removing impulse, isolated noise

Degraded image

Salt noise

Pepper noise

Sorted array

Salt noisePepper noise

Median

Filter output

Normally, impulse noise has high magnitude and is isolated. When we sort pixels in

the moving window, noise pixels are usually at the ends of the array.

Therefore, it’s rare that the noise pixel will be a median value.

Movingwindow


MEDIAN FILTER - EXAMPLE


MEDIAN VS. AVERAGING FILTER - EXAMPLE

Image of Circuit board corrupted by salt and

pepper noise

Noise reduction with3 X 3 Averaging Filter

Noise reduction with3 X 3 Median Filter


LOW PASS AND MEDIAN FILTER

Original image Averaging filter Median filter

Edge

Noise pulse

1 1 1

1 1 1

The low-pass filter can provide image smoothing and noise reduction, but

subdues and blurs sharp edges.

Median filters can provide noise filtering without blurring.


LOW PASS AND MEDIAN FILTER

Low pass spatial filtering

Linear Filter

Neighborhood averaging

Can use different size of masks

Sum of the mask coefficients is 1

Drawbacks: blur edges and other sharp features

Median filtering

Nonlinear

The gray level of each pixel is replaced by the

median of its neighbors

Good at denoising pepper & Salt noise


MAX AND MIN FILTER

Max filter: ),(max),(ˆ),(

tsgyxfxySts

It determines brightest point in image.

Useful in eliminating pepper noise.

It determines darkest point in image.

Useful for eliminating salt noise.

Min filter: ),(min),(ˆ),(

tsgyxfxySts

4

5

8

7

2

4

2

3

3

3

3

3

3 3 3 3

MIN

MAX


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MAX AND MIN FILTERS - EXAMPLE

Image corrupted by pepper noise with prob. = 0.1

Image corrupted

by salt noise with prob. = 0.1

Image obtained

using a 3x3max filter

Image obtained

using a 3x3min filter


ORDER STATISTICS FILTER

sort

f4 f3 f7 f6 f8 f2 f1 f9 f5

medianfilter

increasing order

maxfilter

minfilter

f1 f2 f3

f4 f5 f6

f7 f8 f9


Sharpening Filters


SHARPENING (HIGH PASS) SPATIAL FILTERS

Since averaging is analogous to integration.

Sharpening could be accomplished by spatial differentiation.

We are interested in the behavior of these derivatives in areas

of constant gray level, at the onset and end of discontinuities(step

and ramp discontinuities), and along gray-level ramps.

These types of discontinuities can be noise points, lines, and

edges.


FUNDAMENTALS

Highlight fine details or enhance details that have been blurred.

Sharpening can be accomplished by spatial differentiation.

Strength of response of derivative operator is proportional to

degree of intensity discontinuity of the image at the point at which

operator is applied.

Image differentiation enhances edges and other discontinuity,

and deemphasize areas with slowly varying intensities.


FOUNDATION

First order derivative

Must be zero in constant areas

Must be nonzero at the onset of a gray level step or ramp

Must be nonzero along ramps

Second order derivative

Must be zero in constant areas

Must be nonzero at the onset & end of a gray level step or ramp

Must be zero along ramps of constant slope.


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SPATIAL DERIVATIVES

First-order derivative :

Second-order derivative :

)(2)1()1(2

2

xfxfxfx

f

)()1( xfxfx

f


SHARPENING SPATIAL FILTERS – IN DEPTH


SHARPENING SPATIAL FILTERS – IN DEPTH


1ST ORDER DERIVATIVE

The 1st-order derivative is nonzero along the entire ramp

First order derivatives are use for edge extraction

First order derivatives have a stronger response to a gray level

step

First order derivatives produce thicker edges.


2ND ORDER DERIVATIVE

2nd order derivative produces make thin edge

2nd-order derivative is nonzero only at the onset and end of the

ramp.

Second order derivative is useful to enhance fine detail

(including noise) much more than a first order derivative

Second order derivatives produce a double response at step

changes in gray level

Second order derivatives response is stronger to a line than to a

step and to a point than to a line


LAPLACIAN

Extensively used in segmentation

Derivative of any order are linear operations, so the Laplacian

is a linear operator.

Laplacian are isotropic (rotation invariant) for rotation

increments of 90o and 45o.

The Laplacian is often applied to an image that has first been

smoothed with something approximating a Gaussian smoothing

filter in order to reduce its sensitivity to noise.


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DISCRETE FORM OF DERIVATIVES

),(2),1(),1(2

2

yxfyxfyxfx

f

f(x+1,y)f(x,y)f(x-1,y)

f(x,y+1)

f(x,y)

f(x,y-1)

),(2)1,()1,(2

2

yxfyxfyxfy

f


2D LAPLACIAN

The digital implementation of the 2-Dimensional Laplacian is

obtained by summing 2 components :

2

2

2

22

x

f

x

ff

),(4)1,()1,(),1(),1(2 yxfyxfyxfyxfyxff

1

1

-4 1

1


2D LAPLACIAN

1

1

-4 1

1

0 0

0 0

0

0

-4 0

0

1 1

1 1

1

1

-8 1

1

1 1

1 1


2D LAPLACIAN

-1

-1

4 -1

-1

0 0

0 0

0

0

4 0

0

-1 -1

-1 -1

-1

-1

8 -1

-1

-1 -1

-1 -1

90o Rotation Invariant

45o Rotation Invariant


IMPLEMENTATION

-1 2 -1

2 -4 2

-1 2 -1

0 1 0

1 -4 1

0 1 0

1 1 1

1 -8 1

1 1 1

1 -2 1

-2 4 -2

1 -2 1

0 -1 0

-1 4 -1

0 -1 0

-1 -1 -1

-1 8 -1

-1 -1 -1

),(),(

),(),(),(

2

2

yxfyxf

yxfyxfyxg

If the center coefficient is negative

If the center coefficient is positive

Where f(x,y) is the original image

is Laplacian filtered image

g(x,y) is the sharpen image

),(2 yxf


LAPLACIAN OPERATOR

Since the sum of all the weights is zero, the resulting signal will have a zero DC

value

Produce noisier result

It is used to highlights gray level discontinuities and deemphasizes regions

with slowly varying gray levels

Background features can be recovered while still preserving the sharpening

effect of the Laplacian operator by adding original and Laplacian images

Unsharp masking and high-boost filtering


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SHARPENING USING LAPLACIAN

-1 4 -1 1

0

0

+ =

0 -1 0

0 -1 0 0

0

0

0

0

0

5

-1

-1

0

-1

0

0

-1

0

LAPLACIANOriginal Image Sharpened Image


IMPLEMENTATION

-1 -1 -1

-1 8 -1

-1 -1 -1

Filtered = Conv(image,mask)

filtered = filtered - Min(filtered)

filtered = filtered * (255.0/Max(filtered))

sharpened = image + filtered

sharpened = sharpened - Min(sharpened )

sharpened = sharpened * (255.0/Max(sharpened ))


LAPLACIAN OPERATOR

+

Edges detected by the Laplacian

can be used to sharpen the image !


LAPLACIAN OPERATOR – IMAGE SHARPENING

),(),(

),(),(),(

2

2

yxfyxf

yxfyxfyxg


LAPLACIAN OF GAUSSIAN (LOG)

Convolution is associative, we can convolve the Gaussian

smoothing filter with the Laplacian filter first of all, and then

convolve this hybrid filter with the image to achieve the required

result.

Advantages:

Fewer arithmetic operations.

The LoG mask can be precalculated in advance so only one

convolution needs to be performed at run-time on the image.


DERIVATIVE OF GAUSSIAN


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The 2D Laplacian of Gaussian (LoG) function

Matlab: fspecial(‘log’,…)



Areas where the image has a constant intensity (i.e. where the intensity

gradient is zero), the LoG response will be zero.

LoG response will be positive on the darker side, and negative on the lighter

side.

This means that at a reasonably sharp edge between two regions of uniform

but different intensities, the LoG response will be:

Zero at a long distance from the edge.

Positive just to one side of the edge.

Negative just to the other side of the edge.

Zero at some point in between, on the edge itself.


UNSHARP MASKING

Used in printing industries

STEPS:

Blur the original image

Subtracting a blurred image from its original (Difference is

called MASK)

Add this mask to original image


UNSHARP MASKING & HIGH BOOST FILTERING

),(),(),( yxfyxfyxgmask

),(*),(),( yxgkyxfyxg mask

If k = 1; Unsharp masking

If k > 1; High boost filtering

If k < 1; Deemphasize contribution of

Unsharp masking



High-boosting filtering

Highpass = Original – Lowpass

High boost = Original – K*Lowpass = Original + K*Highpass

Visual effect : brighten while sharpening on the original image

Fsharp = a*F – Fblurred , a>=1

1),(),(),( AwhereyxfyxAfyxfhb

),(),(),()1(),( yxfyxfyxfAyxfhb

),(),()1(),( yxfyxfAyxf shb


HIGH BOOST FILTERING

0 -1 0

-1 A+4 -1

0 -1 0

-1 -1 -1

-1 A+8 -1

-1 -1 -1

),(),(

),(),(

),(2

2

yxfyxAf

yxfyxAf

yxfhb

If the center coefficient is negative

If the center coefficient is positive


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Original Image

Gaussian Smoothing (size: 5 X 5, Sigma = 3)

Unsharp Mask

Sharpened image (k=1)

High boost filtering (K = 4.5)



A = 0

A = 1 A = 1.7

1 1 1

1 8 1

1 1 1



1 1 1

1 8 1

1 1 1

A high-pass filter A high-boost filter

1 1 1

1 9 1

1 1 1

Original Image



Input Image Lowpass Filter

Histogram Shrink

Subtract Images

Histogram Stretch

Sharpened Image



The subtraction has the visual effect of causing overshoot

and undershoot at the edges, which will emphasize the edges.

By scaling the low passed image with a histogram shrink,

we can control the amount of edge emphasis desired.

To get more sharpening effect, shrink the histogram less.


1ST DERIVATIVE - GRADIENT

The gradient of the image, gives the direction of the largest

possible increase from light to dark and the rate of change in that

direction.

The result therefore shows how "abruptly" or "smoothly" the

image changes at that point, and therefore how likely it is that, it

represents an edge, as well as how that edge is likely to be oriented.

In practice, the magnitude (likelihood of an edge) calculation is

more reliable and easier to interpret than the direction calculation.


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SHARPENING FILTERS BASED ON FIRST DERIVATIVES

Magnitude: provides information about edge strength.

Direction: perpendicular to the direction of the edge.

x

f

y

f

Direction (grad(f)) = tan-1

yxyx GGGGfmag 22)( Cheaper Approximation

Gradient =

y

fx

f

G

G

y

xf


ENHANCEMENT USING 1ST DERIVATIVE - THE GRADIENT

Used to enhance prominent edges.

It has stronger response in area of gray level transitions (gray

level ramps and steps)

It always point the direction of maximum rate of change of f at

coordinates (x,y)

Magnitude give rate of increase of f(x,y) per unit distance in

the direction of Δf


EDGE CAUSES

Depth discontinuity

Surface orientation discontinuity

Reflectance discontinuity (i.e., change in surface

material properties)

Illumination discontinuity (e.g., shadow)


ROBERTS CROSS GRADIENT OPERATORS

X direction and y direction gradient can be found using

following mask

The primary disadvantage of the Roberts operator is its high

sensitivity to noise, because very few pixels are used to

approximate the gradient.

-1 0

0 1

0 -1

1 0

z1 z2 z3

z4 z5 z6

z7 z8 z9

Gx = (z9-z5) and Gy = (z8-z6)

2

68

2

59 )()( zzzzf

6859 zzzzf


SOBEL OPERATORS

Mask of even size are awkward to apply.

Weight of value 2 is to achieve some smoothing by giving more

importance to the center point

Sum of all coefficients in masks sum to zero indicating that

they would give zero response of 0 in an area of constant gray

level

-1 -2 -1

0 0 0

1 2 1

-1 0 1

-2 0 2

-1 0 1

)2()2()2()2( 741963321987 zzzzzzzzzzzzf


SOBEL OPERATOR - EXAMPLE

The Sobel filter extracts all of the edges in an image,

regardless of direction

It is implemented as the sum of two directional edge

enhancement operators


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DIFFERENT TYPE OF KERNELS

1

111

1

-1 -1

Prewitt 1

-2

-1

-3

555

-3

-3 -3 -3

Kirsch

0

0

111

0

-1 -1 -1

Prewitt 2

0 0

121

0

-1 -2 -1

Sobel

0

-1 -1

11

0 0 0

- 2

2

Frei & Chen


COMBINING SPATIAL ENHANCEMENT METHOD

Goal:

want to sharpen the original image and bring out

more skeletal detail.

problems: narrow dynamic range of gray level and

high noise content makes the image difficult to

enhance

Solution:

1. Laplacian to highlight fine detail

2. gradient to enhance prominent edges

3. gray-level transformation to increase the

dynamic range of gray levels



Laplacian filter of bone scan (a)

Sharpened version of bone scan achieved by subtracting (a) and (b) Sobel filter of bone

scan (a)

(a)

(b)

(c)

(d)



The product of (c) and (e) which will be used as a mask

Sharpened image which is sum of (a) and (f)

Result of applying a power-law trans. to (g)

(e)

(f)

(g)

(h)

Image (d) smoothed with a 5*5 averaging filter



Compare the original and final images


THANKS…!!!

?


10/19/2016

18

AUTHOR’S PROFILE

Mahesh Goyani has completed his graduation in Computer Engineering from SCET, VNSGU,

Surat in 2005 with distinction. He received his Master Degree in field of Computer

Engineering with 9.38 CPI (81.03 %) from BVM College, SPU, Anand in 2009. He has secured

1st rank twice in university during his master degree. His area of interest is Image

Processing, Computer Graphics, Computer Algorithms and Artificial Intelligence.

Publication: He has published many research papers in national and international journals and conferences. He

was invited as a SESSION CHAIR in International Conference on Engineering, Science and Information Technology,

Tirunelveli, Tamilnadu, Sept - 2011. He has published 11 books in filed of Computer Graphics and Computer

Algorithms for various universities.

He is the member of technical review committee of International Journal of Computer Science & Issues (IJCSE,

Mauritius), Electronics & Telecommunication Research Institute (ETRI, Korea), International journal of

Engineering & Technology (IJET, Singapore), International journal of Computer Science & Information Security

(IJCSIS, Pittsburg, USA). He has worked as a program committee member and reviewer in many International

Conferences. He is also a life time member of ISTE technical society.

Web Site: www.maheshgoyani.in


basics of filtering - mahesh goyani

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