3.4& Fundamentals& mechanics of spatial filtering(page 166) Spatial filter(mask) Filter coefficients Filter response

Transcription

1 Image enhancement in the spatial domain( ) SLIDE 1/21 3.4& Fundamentals& mechanics of spatial filtering(page 166) Spatial filter(mask) Filter coefficients Filter response Example: 3 3mask Linear spatial filtering g(x,y)=w( 1, 1)f(x 1,y 1)+ +w(1,1)f(x+1,y+1)

2 Image enhancement in the spatial domain( ) SLIDE 2/ Spatial correlation and convolution(page 168) (1-dimensional)

3 Image enhancement in the spatial domain( ) SLIDE 3/21 (2-dimensional)

4 Image enhancement in the spatial domain( ) SLIDE 4/21 Convolvingafunctionwithaunitimpulseyieldsacopyofthefunctionat the location of the impulse(chapter 4) If the filter mask is symmetric, correlation and convolution yield the same result If f contains a region identical to w, the value of the correlation function is a maximum when w is centered on that region. Application in pattern matching(chapter 12) Equationsforanimagef(x,y)andageneralm nfiltermaskw(x,y): a b (Correlation) w(x,y) f(x,y)= w(s,t)f(x+s,y+t) (Convolution) w(x, y) f(x, y) = s= at= b a b s= at= b w(s,t)f(x s,y t) Common terms used interchangeably: convolution filter/mask/kernel

5 Image enhancement in the spatial domain( ) SLIDE 5/ Vector representation of linear filtering(page 172) Generallinearfilter: (Response) R = w 1 z 1 +w 2 z w mn z mn mn = w k z k k=1 = w T z 3x3example: (Response) R = w 1 z 1 +w 2 z w 9 z 9 9 = w k z k k=1 = w T z Generating spatial filter masks(page 173) (Read) Based on what we want to accomplish

6 Image enhancement in the spatial domain( ) SLIDE 6/ Smoothing spatial filters Smoothing linear filters 1x1 3x3 R= z i 5x5 9x9 i=1 15x15 35x35

7 Image enhancement in the spatial domain( ) SLIDE 7/21 Example

8 Image enhancement in the spatial domain( ) SLIDE 8/ Order-statistic nonlinear filters Non-linear filters: Order(rank) pixels, e.g. median filter Noise reduction: Salt-and-pepper noise

9 Image enhancement in the spatial domain( ) SLIDE 9/ Sharpening spatial filters(page 179) Purpose: highlight fine detail Smoothing: pixel averaging: integration Sharpening: differentiation: enhances edges and deemphasizes slowly varying areas Foundation One dimensional derivatives Derivatives of digital functions: differences Requirements for definitions: First-order derivative Second-order derivative (1)Zeroinflatsegments (1)Zeroinflatsegments (2)Nonzeroatonsetofsteporramp (2)Nonzeroatonsetofsteporramp (3) Nonzero along ramp (3) Zero along ramp of constant slope First-order derivative: Second-order derivative: f x =f(x+1) f(x) 2 f x 2 =f(x+1)+f(x 1) 2f(x)

10 Image enhancement in the spatial domain( ) SLIDE 10/21 First derivative edge detection Second derivative sharpening

11 Image enhancement in the spatial domain( ) SLIDE 11/ : Using the second derivative for image sharpening- the Laplacian Isotropic filters: Rotation invariant Development of the method Leplacian derivative operator Continuous form: 2 f = 2 f x 2+ 2 f y 2 Discrete form: x-direction Discrete form: y-direction 2 f x 2=f(x+1,y)+f(x 1,y) 2f(x,y) 2 f y 2 =f(x,y+1)+f(x,y 1) 2f(x,y)

12 Image enhancement in the spatial domain( ) SLIDE 12/21 Discrete form: 2-D Laplacian- sum of the two components 2 f(x,y)=f(x+1,y)+f(x 1,y)+f(x,y+1)+f(x,y 1) 4f(x,y) Implementation (a)and(c): Isotropicresultsforincrementsof90 o (b)and(d): Isotropicresultsforincrementsof45 o

13 Image enhancement in the spatial domain( ) SLIDE 13/21 In order to recover background features, while still preserving the sharpening effect... g(x,y)=f(x,y)+c [ 2 f(x,y) ] Theconstantisc= 1ifFig3.37(a)or(b)isused,andc=+1if(c)or(d) is used Example 3.15 Image sharpening

14 Image enhancement in the spatial domain( ) SLIDE 14/ : Unsharp masking and highboost filtering Unsharp masking Used in printing industry (a) Blur original image (b) Subtract blurred image from original(difference is called mask) (c)addmasktooriginal Unsharpmasking: Choosek=1 Highboost filtering: Choose k > 1 g mask (x,y)=f(x,y) f(x,y) }{{} Blurred g(x,y)=f(x,y)+k g mask (x,y)

15 Image enhancement in the spatial domain( ) SLIDE 15/21

16 Image enhancement in the spatial domain( ) SLIDE 16/21 Example 3.16

17 Image enhancement in the spatial domain( ) SLIDE 17/ : First-order derivatives for(nonlinear) image sharpening- the gradient Continuous form: Gradient Magnitude of gradient f =grad(f)= ( gx g y ) M(x,y)=mag( f)= Approximation: M(x,y) g x + g y = f x f y g 2 x+g 2 y Discreteform:Roberts: M(x,y) z 9 z 5 + z 8 z 6 Sobel: M(x,y) (z 7 +2z 8 +z 9 ) (z 1 +2z 2 +z 3 ) + (z 3 +2z 6 +z 9 ) (z 1 +2z 4 +z 7 )

18 Image enhancement in the spatial domain( ) SLIDE 18/21

19 Image enhancement in the spatial domain( ) SLIDE 19/21 Example : Combining spatial enhancement methods...

20 Image enhancement in the spatial domain( ) SLIDE 20/21

21 Image enhancement in the spatial domain( ) SLIDE 21/21