Cp467 12 lecture 6 sharpening PDF

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useful ppt on dip, has some intuitive information...

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Lecture 6 Sharpening Filters 1. 2 2. 3. 4. 5. 6. 7.

The concept of sharpening filter First and second order derivatives Laplace filter Unsharp mask High boost filter Gradient mask Sharpening image with MatLab

Sharpening Spatial Filters • To highlight fine detail in an image or to enhance detail that has been blurred, either in error or as a natural effect ff t off a particular ti l method th d off image i acquisition. i iti • Blurring vs vs. Sharpening • Blurring/smooth is done in spatial domain by pixel averaging in a neighbors, i hb it iis a process off integration i t ti

• Sh Sharpening i iis an inverse i process, to t find fi d th the difference diff by b the th neighborhood, done by spatial differentiation.

2

Derivative operator • The strength of the response of a derivative operator is proportional to the degree of discontinuity of the image at the point at which the operator is applied. • Image differentiation – enhances edges and other discontinuities (noise) – deemphasizes area with slowly varying gray-level values. values

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Sharpening edge by First and second order derivatives • Intensity function f =

f’ f

• First derivative f’ =

• Second-order derivative f’’

f f’’ f-f

f’’ =

• f- f’’ =

4

First and second order difference of 1D • The basic definition of the first-order derivative of a onedimensional function f(x) is the difference

∂f = f ( x + 1) − f ( x) ∂x • The second-order derivative of a one-dimensional function f(x) is the difference

∂2 f = f ( x + 1) + f ( x − 1) − 2 f ( x) 2 ∂x

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First and Second-order derivative of 2D • when we consider an image function of two variables, f(x, y), at which time we will dealing with partial d i ti derivatives along l th the ttwo spatial ti l axes.

Gradient operator (linear operator)

∂f ( x , y ) ∂f ( x, y ) ∂f ( x, y ) ∇f = = + ∂x∂y ∂x ∂y

Laplacian operator (non-linear)

2 2 ∂ ∂ f ( x , y ) f ( x, y ) 2 ∇ f = + 2 ∂x ∂y 2

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Discrete form of Laplacian from

∂2 f = f ( x + 1, y) + f ( x − 1, y ) − 2 f ( x, y ) 2 ∂x ∂2 f = f ( x, y + 1) + f ( x, y − 1) − 2 f ( x, y ) 2 ∂y

∇ 2 f = [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) + f ( x, y − 1) − 4 f ( x, y)] 8

Result Laplacian mask

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Laplacian mask implemented an extension of diagonal neighbors

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Other implementation of Laplacian masks

give the same result, but we have to keep in mind that when combining (add / subtract) a Laplacian-filtered i image with ith another th image. i 11

Effect of Laplacian Operator • as it is a derivative operator, – it highlights gray-level discontinuities in an image – it deemphasizes regions with with slowly slowly varying varying gray gray levels levels

• tends to produce images that have – grayish edge lines and other discontinuities, all superimposed on a dark, dark – featureless background.

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Correct the effect of featureless background • easily by adding the original and Laplacian image. • be careful with the Laplacian filter used

⎧ f ( x, y ) − ∇ 2 f ( x, y ) g ( x, y ) = ⎨ 2 + ∇ f ( x , y ) f ( x, y ) ⎩

if the h center coefficient ffi i of the Laplacian mask is negative

if the center coefficient of the Laplacian mask is positive

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Example • a). image of the North pole of the moon • b). Laplacian-filtered image with 1

1

1

1

-8

1

1

1

1

• c). Laplacian image scaled for display purposes • d). image enhanced by addition with original image 14

Mask of Laplacian + addition • to simply the computation, we can create a mask which do both operations, Laplacian Filter and Addition the original i i l image. i

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Mask of Laplacian + addition

g ( x, y ) = f ( x, y ) − [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) + f ( x, y − 1) + 4 f ( x, y)] = 5 f ( x, y) − [ f ( x + 1, y) + f ( x − 1, y) + f ( x , y + 1) + f ( x, y − 1)] 0

-1

0

-1

5

-1

0

-1

0

16

Example

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Note

⎧ f ( x, y ) − ∇ 2 f ( x, y ) g ( x, y) = ⎨ 2 + ∇ f ( x , y ) f ( x, y ) ⎩ 0

0

0

0

1

0

0

0

0

-1

0

0

-1

9

-1

0

-1

0

0

-1 1

0

-1

5

-1

0

-1

0

=

=

0

-1

0

-1

4

-1

0

0

-1

0

0

0

0

-1

0

0

1

0

-1

8

-1

0

0

0

0

-1 1

0

+

+

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Unsharp masking

f s ( x, y ) = f ( x, y ) − f ( x, y ) sharpened image = original image – blurred image • to subtract a blurred version of an image produces sharpening output image.

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Unsharp mask

High-boost filtering

f hb ( x, y ) = Af ( x , y ) − f ( x, y ) fhb(x, y) = (A−1) f (x, y)− f (x, y) f (x, y) = (A−1) f (x, y) − fs (x, y) • generalized form of Unsharp masking • A≥ 1

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High-boost filtering

f hb ( x, y ) = ( A − 1) f ( x, y ) − f s ( x, y) • if we use Laplacian filter to create sharpen image fs(x,y) with addition of original image

⎧ f ( x , y ) − ∇2 f ( x , y ) f s ( x, y ) = ⎨ 2 + ∇ ( , ) f x y f ( x, y ) ⎩

⎧ Af ( x , y ) − ∇ 2 f ( x , y ) f hb ( x, y) = ⎨ 2 Af x y + ∇ f ( x, y ) ⎩ ( , ) 23

High-boost Masks

 

A≥1 if A = 1, 1 it becomes “standard” standard Laplacian sharpening

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Example

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Gradient Operator • first derivatives are implemented using the magnitude of the gradient. gradient ⎡ ∂∂ff ⎤ ⎡Gx ⎤ ⎢ ∂x ⎥ ∇f = ⎢ ⎥ = ⎢ ∂ f ⎥ Gy ⎦ ⎢ ⎥ ⎣ 1 2 2 2 ⎢⎣ ∂ y ⎦⎥ ∇ f = mag (∇f ) = [G + G ] x

⎡⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎤ = ⎢⎜ ⎟ + ⎜⎜ ⎟⎟ ⎥ ⎢⎣⎝ ∂x ⎠ ⎝ ∂y ⎠ ⎥⎦ 2

2

y

1 2

commonly approx.

∇f ≈ Gx + G y

the magnitude becomes nonlinear 26

Gradient Mask • simplest approximation, 2x2

Gx = ( z8 − z5 ) ∇f = [G + G ] 2 x

2 y

1

and 2

z1

z2

z3

z4

z5

z6

z7

z8

z9

G y = ( z 6 − z5 )

= [( z8 − z 5 ) + ( z 6 − z 5 ) ] 2

2

1

2

∇f ≈ z8 − z5 + z6 − z5 27

Gradient Mask

z1 z4

z2 z5

z3 z6

• Roberts cross-gradient operators, 2x2

z7

z8

z9

Gx = ( z9 − z5 ) ∇f = [G + G ] 2 x

2 y

1

and 2

Gy = ( z8 − z6 )

= [( z 9 − z5 ) + ( z 8 − z6 ) ] 2

2

1

2

∇f ≈ z9 − z 5 + z8 − z 6

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Gradient Mask

z1 z4

z2 z5

z3 z6

• Sobel operators, 3x3

z7

z8

z9

Gx = ( z7 + 2 z8 + z9 ) − ( z1 + 2 z 2 + z3 ) G y = ( z3 + 2 z6 + z9 ) − ( z1 + 2 z 4 + z 7 ) ∇f ≈ Gx + G y the weight value 2 is to achieve smoothing by giving more important to the center point 29

Example

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Example of Combining Spatial Enhancement Methods

• 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

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Example of Combining Spatial Enhancement Methods •

solve : 1. Laplacian to highlight fine detail 2. gradient to enhance prominent edges 3. gray-level transformation to increase the dynamic range of gray levels

32...