spatial filtering - national institute of technology calicut · figure:spatial filtering with a 3 3...

ReviewSmoothing Spatial FiltersSharpening Spatial Filters

Spatial Filtering

Dr. Praveen Sankaran

Department of ECE

NIT Calicut

January 7, 2013

Dr. Praveen Sankaran DIP Winter 2013


Outline

1 Review

2 Smoothing Spatial FiltersLinearNonlinear

3 Sharpening Spatial FiltersGradients and their use in Sharpening



Spatial Domain

Refers to the image plane itself.↓

Direct manipulation of image pixels.

Figure: Spatial Filtering with a 3×3 mask (kernel, template or window)



Spatial Filter



Correlation and Convolution - Representations

Correlation

w (m,n)�g (m,n) =a

∑s=−a

b

∑t=−b

w (s, t)g (m+ s, y + t)

Convolution

w (m,n)?g (m,n) =a

∑s=−a

b

∑t=−b

w (s, t)g (m− s, y − t)



Generating a Gaussian Mask

Creating a �lter essentially boils down to specifying the valuesof mask coe�cients.

Basic form →h (x ,y) = e

− x2+y2

2σ2

sample,quantize

⇓

h (m,n) = e−m

2+n2

2σ2

Sample the continuous function about its center.w1 w2 w3

w4 w5 w6

w7 w8 w9

=

h (−1,−1) h (0,−1) h (1,−1)h (−1,0) h (0,0) h (1,0)h (−1,1) h(0,1) h (1,1)



LinearNonlinear

Outline

1 Review





LinearNonlinear

Averaging Filters

Objective → Noise reduction by reducing sharp transitions inintensities.

We have seen an averaging �lter before. These are alsoreferred to as low pass �lters.

Creating an averaging �lter - replace pixel with average intensity inneighborhood

Average = 19

9

∑i=1

gi =9

∑i=1

19gi ,

19

19

19

19

19

19

19

19

19

→averaging mask

Note that we are giving equal weight to the pixel underconsideration and the pixel values around it.



LinearNonlinear

Averaging Filters - Weighted

Idea → replace pixel value underconsideration with a weightedaverage of values in mask region.

Multiply pixel values withdi�erent mask coe�cients.

Done to reduce blurring.

116×

1 2 12 4 21 2 1

i (m,n) =

a

∑s=−a

b

∑t=−b

w (s, t)g (m± s, y ± t)

∑as=−a ∑

bt=−bw (s, t)

,

m = 0,1, · · ·M−1, n = 0,1,2, · · ·N−1.

Note that we need not worry about our operation beingconvolution or correlation with these masks.



LinearNonlinear

Filter Size Matters

Figure: E�ect of varying �lter sizes on an Image. Original image withsize 500×500. Filter sizes with m = 3,5,9,15,35



LinearNonlinear

Outline

1 Review





LinearNonlinear

Order-Statistic (Nonlinear) Filters - Median Filter

Replace value of a pixel by the median of the pixel values inthe mask region or region-of-interest (ROI).

Typical application → remove salt-&-pepper (?) noise.



Gradients and their use in SharpeningSummary

Outline

1 Review






First Derivative

1 Must be zero in areas of constant intensity;

2 Must be non-zero at the onset of an intensity step or ramp;

3 Must be non-zero along ramps.

4 ∂ f∂x

= f (x+1,y)− f (x ,y), remember the problem we talkedabout earlier?

From Taylor's theorem,

f (x±ξ ,y) = f (x ,y)+ ∂ f (x ,y)∂x

(±ξ )+ 12!

∂2f (x ,y)∂x2

ξ 2+ · · ·Ignoring higher order terms we end up with three di�erent scenarios.




Di�ering Approximations

1∂ f (x ,y)

∂xw f (x+ξ ,y)−f (x ,y)

ξ

2∂ f (x ,y)

∂xw f (x ,y)−f (x−ξ ,y)

ξ

3∂ f (x ,y)

∂xw f (x+ξ ,y)−f (x−ξ ,y)

2ξ

Substituting ξ = 1, three separate, corresponding operators can beformed for a digital image.

Which one do we choose?

g [m+1,n]−g [m,n]g [m,n]−g [m−1,n]g [m+1,n]−g [m−1,n]

2




Second Derivative

1 Must be zero in areas of constant intensity;

2 Must be non-zero at the onset and end of an intensity step orramp;

3 Must be zero along ramps of constant slope.




The Laplacian

Laplacian

∇2f (x ,y) = ∂2f (x ,y)∂x2

+ ∂2f (x ,y)∂y2

From Taylor's theorem,

f (x±ξ ,y) = f (x ,y)+ ∂ f (x ,y)∂x

(±ξ )+ 12!

∂2f (x ,y)∂x2

ξ 2+ · · · , and take

∂ f (x ,y)∂x

w f (x+ξ ,y)−f (x−ξ ,y)2ξ

, ξ = 1

∂2f∂x2≈ f (x+1,y)+ f (x−1,y)−2f (x ,y) and,

∂2f∂y2≈ f (x ,y +1)+ f (x ,y −1)−2f (x ,y)




The Laplacian (Final Approximated Form)

Laplacian Form in digital image

∇2g ≈g [m+1,n]+g [m−1,n]+g [m,n+1]+g [m,n−1]−4g [m,n].

Mask Values0 1 01 −4 10 1 0

Note that any di�erent approximation of the Taylor seriesexpansion could have given a di�erent form for the Laplacian.




Using Laplacian to Sharpen an Image

i (m,n) =g [m,n]+ c

[∇2g [m,n]

]Add Laplacian image to theoriginal.




Unsharp Masking and Highboost Filtering

1 Blur the original image;

2 Subtract the blurred image from the original (this forms themask);

3 Add mask to the original.

imask [m,n] = g [m,n]−gavg [m,n],

g [m,n] = g [m,n]+k× imask [m,n]

If k = 1, process = unsharp masking.

If k > 1, process = highboost �ltering.




First Gradients - Robert's

We de�ne an image j [m,n] =

√(∂g∂m

)2+(

∂g∂n

)2→ gradient

image.

Roberts [1965] →(

∂g∂m

)= (w9−w5),(

∂g∂n

= (w8−w6))

gradient image =

i [m,n] =

√[(w9−w5)

2+(w8−w6)2]

w1 w2 w3

w4 w5 w6

w7 w8 w9

[−1 00 1

] [0 −11 0

]




Sobel Operator

Roberts had a mask of even size → no center of symmetry.

Approximate →(∂g∂m

)= (w7+2w8+w9)−

(w1+2w2+w3),(∂g∂n

)= (w3+2w6+w9)−

(w1+2w4+w7)

w1 w2 w3

w4 w5 w6

w7 w8 w9

−1 −2 −1

0 0 01 2 1

−1 0 1−2 0 2−1 0 1




Summary

Application of Spatial approach to:

Low pass �ltering.Sharpening.

First gradient, Roberts, Sobel.

Second gradient, Laplacian

Application of gradient to sharpening.




Questions

3.1, 3.2, 3.3, 3.4, 3.5

3.6, 3.7, 3.11

3.13, 3.14

3.15, 3.16, 3.17,3.18, 3.26, 3.27, 3.28


spatial filtering - national institute of technology calicut · figure:spatial filtering with a 3 3...

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