Chapter 3: Image Enhancement in the Spatial Domain

3.1 Background 3.2 Some basic intensity transformation functions 3.3 Histogram processing 3.4 Fundamentals of spatial filtering 3.5 Smoothing spatial filters 3.6 Sharpening spatial filters

Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 1

Preview

• Spatial domain processing: direct manipulation of pixels in an image • Two categories of spatial (domain) processing • Intensity transformation: - Operate on single pixels - Contrast manipulation, image thresholding • Spatial filtering - Work in a neighborhood of every pixel in an image - Image smoothing, image sharpening Dept. of Internet & Multimedia Eng., Changhoon Yim 2 R.C. Gonzalez & R.E. Woods

3.1 Background

• Spatial domain methods: operate directly on pixels • Spatial domain processing g(x,y) = T[f(x, y)] f(x, y) : input image g(x, y) : output (processed) image T : operator • Operator T is defined over a neighborhood of point (x, y) R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 3

3.1 Background

Spatial filtering • For any location (x,y) , output image g(x,y) is equal to the result of applying T to the neighborhood of (x,y) in f • Filter: mask, kernel, template, window Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 4

3.1 Background

• The simplest form of T: g depends only on the value of f at (x, y) T becomes intensity (gray-level) transformation function s = T(r) r: intensity of f(x,y) s: intensity of g(x,y) • Point processing: enhancement at any point depends only on the gray level at that point Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 5

3.1 Background

Point processing • (a) Contrast stretching • Values of r below k are compressed into a narrow range of s • (b) Thresholding Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 6

3.2 Some Basic Intensity Transformation Functions

• r : pixel value before processing s : pixel value after processing T : transformation s = T(r) • 3 types • Linear (identity and negative transformations) • Logarithmic (log and inverse-log transformations) • Power-law (nth power and nth root transformations) Dept. of Internet & Multimedia Eng., Changhoon Yim 7 R.C. Gonzalez & R.E. Woods

3.2 Some Basic Gray Level Transformations

R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 8

3.2.1 Image Negatives • Negative of an image with gray level [0, L-1] s = L – 1 – r • Enhancing white or gray detail embedded in dark regions of an image Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 9

3.2.2 Log Transformations • General form of log transformation s = c log(1+r) c: constant, r ≥ 0 • This transformation maps a narrow range of low gray-level values in the input image into a wider range of output levels • Classical application of log transformation: Display of Fourier spectrum R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 10

3.2.2 Log Transformations • (a) Original Fourier spectrum: 0 ~ 1,500,000 range scaled to 0 ~ 255 • (b) Result of log transformation: 0 ~ 6.2 range scaled to 0 ~ 255 Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 11

3.2.3 Power-Law Transformations R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 12

3.2.3 Power-Law Transformations • Basic form of power-law transformations s = c r γ c, γ : positive constants • Gamma correction: process of correcting this power-law response • Example: cathode ray tube (CRT) Intensity to voltage response is power function with exponent ( γ ) 1.8 to 2.5

Solution: preprocess the input image by performing transformation s = r 1/2.5

= r 0.4

Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 13

3.2.3 Power-Law Transformations CRT monitor gamma correction example R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 14

original 3.2.3 Power-Law Transformations MRI gamma correction example γ = 0.6

γ = 0.4

R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim γ = 0.3

15

original 3.2.3 Power-Law Transformations Arial image gamma correction example γ = 3.0

γ = 4.0

R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim γ = 5.0

16

3.2.4 Piecewise-Linear Transformation Functions Contrast stretching Low contrast image Piecewise linear function Result of contrast stretching R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim Result of thresholding 17

3.2.4 Piecewise-Linear Transformation Functions Contrast stretching (c) Contrast stretching (r 1 , s 1 ) = (r min , 0)

,

(r 2 , s 2 ) = (r max , L-1) r min , r max : minimum, maxmum level of image (d) Thresholding: r 1 = r 2 = m, s 1 = 0, s 2 m: mean gray level = L-1 R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 18

3.2.4 Piecewise-Linear Transformation Functions Intensity-level slicing R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 19

3.2.4 Piecewise-Linear Transformation Functions Intensity-level slicing (a) Display a high value for all gray levels in the range of interest, and a low value for all other images - produces binary image (b) Brightens the desired range of gray levels but preserves the background and other parts R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 20

3.2.4 Piecewise-Linear Transformation Functions Intensity-level slicing R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 21

3.2.4 Piecewise-Linear Transformation Functions Bit-plane slicing R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 22

3.2.4 Piecewise-Linear Transformation Functions Bit-plane slicing R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 23

3.2.4 Piecewise-Linear Transformation Functions Bit-plane slicing (a) Multiply bit plane 8 by 128 Multiply bit plane 7 by 64 Add the results of two planes Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 24

3.3 Histogram Processing

• The histogram of digital image with gray levels in the range [0, L-1] is a discrete function • h(r k ) = n k r k : kth gray level n k : number of pixels in image having gray levels r k • Normalized histogram p(r k ) = n k /n n : total number of pixels in image n = MN (M: row dimension, N: column dimension) Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 25

3.3 Histogram Processing

R.C. Gonzalez & R.E. Woods Histogram • horizonal axis: r k ( k th intensity value) • vertical axis: n k : number of pixels, or n k /n : normalized number Dept. of Internet & Multimedia Eng., Changhoon Yim 26

• • 3.3.1 Histogram Equalization r : intensities of the image to be enhanced r is in the range [0, L-1] r = 0: black, r = L-1: white s : processed gray levels for every pixel value r s = T(r), 0 ≤ r ≤ L-1 • Requirements of transformation function T (a) T(r) is a (strictly) monotonically increasing in the interval 0 ≤ r ≤ L-1 • (b) 0 ≤ T(r) ≤ L-1 for 0 ≤ r ≤ L-1 Inverse transformation r = T -1 (s), 0 ≤s ≤L-1 Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 27

3.3.1 Histogram Equalization Intensity transformation function R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 28

3.3.1 Histogram Equalization Intensity levels: random variable in interval [0, L-1]

s



r dr ds

probability density function (PDF)

s

  (

L

 1)  0

r r

cumulative distribution function (CDF)

ds dr



dr

 (

L

 1)

d dr

[  0

r r

]  (

L

 1)

r s



r dr ds



r

(

L

1  1)

r

 Uniform probability density function

L

1  1 0 Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods

L

1 29

3.3.1 Histogram Equalization R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 30

3.3.1 Histogram Equalization 

n k MN k

 0,1, 2,...,

L

 1 MN : total number of pixels in image n k : number of pixels having gray level r k L : total number of possible gray levels

s k



T r

 (

L

 1)

j k

  0 

L

 1

MN j k

  0

n j k

 0,1, 2,..,

L

 1 • histogram equalization (histogram linearization): Processed image is obtained by mapping each pixel r k (input image) into corresponding level s k (output image) Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 31

3.3.1 Histogram Equalization R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 32

3.3.1 Histogram Equalization Histogram equalization from dark image (1) Histogram equalization from light image (2) Histogram equalization from low contrast image (3) Histogram equalization from high contrast image (4) Dept. of Internet & Multimedia Eng., Changhoon Yim 33 R.C. Gonzalez & R.E. Woods

3.3.1 Histogram Equalization Transformation functions R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 34

3.3.3 Local Histogram Processing • Histogram processing methods in previous section are global • Global methods are suitable for overall enhancement • Histogram processing techniques are easily adapted to local enhancement • Example (b) Global histogram equalization Considerable enhancement of noise (c) Local histogram equalization using 7x7 neighborhood Reveals (enhances) the small squares inside the dark squares Contains finer noise texture Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 35

3.3.3 Local Histogram Processing Original image Global histogram equalized image Local histogram equalized image R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 36

3.3.4 Use of Histogram Statistics for Image Enhancement r : discrete random variable representing intensity values in the range [0, L-1] p(r i ): normalized histogram component corresponding to value r i 

n



i L

 1   0 (

r i



m

)

n p r

n th moment of r  2

m



i L

  1  0

i



i L

  1  0 (

r i



m

) 2

p r

mean (average) value of r variance of r, б 2 (r)

m

 1

MN M x

  1

N

  0

y

  1 0 sample mean  2  1

MN x M

 1

N

 1    0

y

 0 

m

] 2 sample variance Dept. of Internet & Multimedia Eng., Changhoon Yim 37 R.C. Gonzalez & R.E. Woods

3.3.4 Use of Histogram Statistics for Image Enhancement • Global mean and variance are measured over entire image Used for gross adjustment of overall intensity and contrast • Local mean and variance are measured locally Used for local adjustment of local intensity and contrast (x,y): coordinate of a pixel S xy : neighborhood (subimage), centered on (x,y) r 0 , …, r L-1 : L possible intensity values p Sxy : histogram of pixels in region S xy

m S xy



i L

 1   0

S xy r

local mean  2

S xy



i L

  1  0 (

r i



m S xy

) 2

p S xy r

local variance Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 38

3.3.4 Use of Histogram Statistics for Image Enhancement • A measure whether an area is relatively light or dark at (x,y) Compare the local average gray level m Sxy to the global mean m G (x,y) is a candidate for enhancement if m Sxy ≤ k 0 m G • Enhance areas that have low contrast Compare the local standard deviation б Sxy to the global standard deviation б G (x,y) is a candidate for enhancement if б Sxy ≤ k 2 б G • Restrict lowest values of contrast (x,y) is a candidate for enhancement if k 1 б G ≤ б Sxy • Enhancement is processed simply multiplying the gray level by a constant E  ( , ) if

m S xy



G

1 

G

 

S xy



k

2 

G

Dept. of Internet & Multimedia Eng., Changhoon Yim R.C. Gonzalez & R.E. Woods 39

3.3.4 Use of Histogram Statistics for Image Enhancement Problem: enhance dark areas while leaving the light area as unchanged as possible E = 4.0, k 0 = 0.4, k 1 = 0.02, k 2 = 0.4, Local region (neighborhood) size: 3x3 R.C. Gonzalez & R.E. Woods Dept. of Internet & Multimedia Eng., Changhoon Yim 40

3.4 Fundamentals of Spatial Filtering

• Operations with the values of the image pixels in the neighborhood and the corresponding values of subimage • Subimage: filter, mask, kernel, template, window • Values in the filter subimage: coefficient • Spatial filtering operations are performed directly on the pixels of an image • One-to-one correspondence between linear spatial filters and filters in the frequency domain Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 41

3.4.1 Mechanics of Spatial Filtering • A spatial filter consists of (1) a neighborhood (typically a small rectangle) (2) a predefined operation • A processed (filtered) image is generated as the center of the filter visits each pixel in the input image • Linear spatial filtering using 3x3 neighborhood • At any point (x,y), the response, g(x,y), of the filter g(x,y) = w(-1,-1)f(x-1,y-1) + w(-1,0)f(x-1,y) + … + w(0,0)f(x,y) + …+ w(1,0)f(x+1,y) + w(1,1)f(x+1,y+1) Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 42

3.4.1 Mechanics of Spatial Filtering Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 43

3.4.1 Mechanics of Spatial Filtering • Filtering of an image f with a filter w of size m x n a = (m-1) / 2, b = (n-1) / 2 or m = 2a+1, n = 2b+1 (a, b: positive integer) 

s a b

 

b

 

t

) Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 44

3.4.2 Spatial Correlation and Convolution Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 45

3.4.2 Spatial Correlation and Convolution Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 46

3.4.2 Spatial Correlation and Convolution • Correlation of a filter w(x,y) of size m x n with an image f(x,y) 

s a b

 

b

 

t

) • Convolution of w(x,y) and f(x,y) 

s a b

 

b

 

t

) Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 47

3.4.3 Vector Representation of Linear Filtering • Linear spatial filtering by m x n filter

R

    

i mn

  1 

T

w z

• Linear spatial filtering by 3 x 3 filter

R



w z

1 1 

w z

2 2  

w z

9 9 

i

9   1

w z i i

 Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 48

3.4.3 Vector Representation of Linear Filtering • Spatial filtering at the border of an image • Limit the center of the mask no less than (n-1)/2 pixels from the border -> Smaller filtered image • Padding -> Effect near the border • Adding rows and columns of 0’s • Replicating rows and columns Dept. of Internet & Multimedia Eng., Changhoon Yim Chap.3 Intensity Transformations and Spatial Filtering 49

3.4.4 Generating Spatial Filter Masks • Linear spatial filtering by 3 x 3 filter

R

    

i

9   1 • Average value in 3 x 3 neighborhood

R

 1 9

i

9   1

z i

• Gaussian function 

e



x

2 

y

2 2  2 Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 50

3.5 Smoothing Spatial Filters

• Linear spatial filters for smoothing: averaging filters, lowpass filters • Noise reduction • Undesirable side effect: blur edges standard average weighted average Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 51

3.5.1 Smoothing Linear Filters • Standard averaging by 3x3 filter

R

 1 9

i

9   1

z i

• Weighted averaging Reduce blurring compared to standard averaging • General implementation for filtering with a weighted averaging filter of size m x n (m=2a+1, n=2b+1) 

s a b

 

b s a b

 

b

 

t

) Dept. of Internet & Multimedia Eng., Changhoon Yim Chap.3 Intensity Transformations and Spatial Filtering 52

3.5.1 Smoothing Linear Filters Result of smoothing with square averaging filter masks Original n=3 n=5 n=9 n=15 Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim n=35 53

3.5.1 Smoothing Linear Filters Application example of spatial averaging Original 15x15 averaging Result of thresholding Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 54

3.5.2 Order-Statistic (Nonlinear) Filters • Order-statistic filters are nonlinear spatial filters whose response is based on ordering (ranking) the pixels • Median filter • Replaces the pixel value by the median of the gray levels in the neighborhood of that pixel • Effective for impulse noise (salt-and-pepper noise) • Isolated clusters of pixels that are light or dark with respect to their neighbors, and whose area is less than n 2 /2, are eliminated by an n x n median filter • Median • 3x3 neighborhood: 5 th largest value • 5x5 neighborhood: 13 th largest value • Max filter: select maximum value in the neighborhood • Min filter: select minimum value in the neighborhood Dept. of Internet & Multimedia Eng., Changhoon Yim Chap.3 Intensity Transformations and Spatial Filtering 55

3.5.2 Order-Statistic Filters Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 56

3.6 Sharpening Spatial Filters

• Objective of sharpening : highlight fine detail to enhance detail that has been blurred • Image blurring can be accomplished by digital averaging Digital averaging is similar to spatial integration • Image sharpening can be done by digital differentiation Digital differentiation is similar to spatial derivative • Image differentiation enhances edges and other discontinuities Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 57

3.6.1 Foundation • • • • Image sharpening by first- and second-order derivatives Derivatives are defined in terms of differences Requirement of first derivative 1) Must be zero in flat areas 2) Must be nonzero at the onset (start) of step and ramp 3) Must be nonzero along ramps Requirement of second derivative 1) Must be zero in flat areas 2) Must be nonzero at the onset (start) of step and ramp 3) Must be zero along ramps of constant slope Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 58

3.6.1 Foundation 

f



x



f



x

(

x

  2

f



x

2    1) first-order derivative second-order derivative Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 59

3.6.1 Foundation Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 60

3.6.1 Foundation • At the ramp • First-order derivative is nonzero along the ramp • Second-order derivative is zero along the ramp • Second-order derivative is nonzero only at the onset and end of the ramp •At the step • Both the first- and second-order derivatives are nonzero • Second-order derivative has a transition from positive to negative ( zero crossing ) • Some conclusions • First-order derivatives generally produce thicker edges • Second-order derivatives have stronger response to fine detail • First-order derivatives generally produce stronger response to gray-level step • Second-order derivatives produce a double response at step Dept. of Internet & Multimedia Eng., Changhoon Yim 61 Chap.3 Intensity Transformations and Spatial Filtering

3.6.2 Use of Second Derivatives for Enhancement • Isotropic filters: rotation invariant • Simplest isotropic second-order derivative operator: Laplacian  2

f

  2 

x

2

f

  2 

y

2

f

2-D Laplacian operation  2 

x

2

f

  2

f



y

2      x-direction y-direction  2       Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 62

3.6.2 Use of Second Derivatives for Enhancement 4 neighbors negative center coefficient 8 neighbors negative center coefficient 4 neighbors positive center coefficient 8 neighbors positive center coefficient Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 63

3.6.2 Use of Second Derivatives for Enhancement • Image enhancement (sharpening) by Laplacian operation       2   2 ( , ) if the center coefficient of the Laplacian mask is negative ( , ) if the center coefficient of the Laplacian mask is positive • Simplification    

f x y f x y

 1)] 4 ( , )  

f x



f x y

 1, )    1)]     1) Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 64

3.6.2 Use of Second Derivatives for Enhancement Original image Laplacian image Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim Enhanced image (original-Laplacian) 65

3.6.2 Use of Second Derivatives for Enhancement Original image Enhanced image 4 neighbors Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim Enhanced image 8 neighbors 66

3.6.3 Unsharp Masking and Highboost Filtering

g mask

  

mask

• When k=1, unsharp masking • When k > 1, highboost filtering original image – blurred image Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 67

3.6.3 Unsharp Masking and Highboost Filtering Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 68

3.6.3 Unsharp Masking and Highboost Filtering Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 69

3.6.4 Using First-Order Derivatives for Image Sharpening • First derivatives in image processing is implemented using the magnitude of the gradient 

g g x y

   

f x

gradient of f at (x,y)  mag(

f

)    

f



x

 2 [

g

2

x

   

f y

  2 

g

 1/ 2  

y



g x



g y

magnitude of gradient Approximation of magnitude of gradient by absolute values Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 70

3.6.4 Using First-Order Derivatives for Image Sharpening 3x3 region Roberts operators Sobel operators Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 71

3.6.4 Using First-Order Derivatives for Image Sharpening • Simplest approximation to first-order derivative

g x

 (

z

8 

z

5 ) and

g y

 (

z

6 

z

5 ) • Roberts cross-gradient operators

g x

 (

z

9 

z

5 ) and

g y

 (

z

8 

z

6 )  [

g

2

x



g y

 [(

z

9 

z

5 ) 2  (

z

8 

z

6 

g x



g y



z

9 

z

5 

z

8 

z

6 Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 72

3.6.4 Using First-Order Derivatives for Image Sharpening • Sobel operators

g x g y

  ( (

z

7

z

3   2 2

z

8

z

6  

z z

9 9 ) )   ( (

z

1

z

1   2 2

z

2

z

4  

z

3

z

7 ) and )   (

g z x

7 + (

z

3   

g y

2

z

8 2

z

6  

z

9

z

9 ) )   ( (

z

1

z

1   2 2

z

2

z

4  

z

3 )

z

7 ) Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 73

3.7.3 Use of First Derivatives for Enhancement Optical image of contact lens Sobel gradient Chap.3 Intensity Transformations and Spatial Filtering Dept. of Internet & Multimedia Eng., Changhoon Yim 74