digital image processing, 2nd ed. © 2002 r. c. gonzalez & r. e. woods chapter 3 image...

Digital Image Processing, 2nd ed.Digital Image Processing, 2nd ed.www.imageprocessingbook.com

© 2002 R. C. Gonzalez & R. E. Woods

Chapter 3Image Enhancementin the Spatial Domain

Chapter 3Image Enhancementin the Spatial Domain



Image Enhancement in theSpatial Domain

Image Enhancement in theSpatial Domain

• The spatial domain: – The image plane– For a digital image is a Cartesian coordinate system of discrete rows

and columns. At the intersection of each row and column is a pixel. Each pixel has a value, which we will call intensity.

• The frequency domain :– A (2-dimensional) discrete Fourier transform of the spatial domain– We will discuss it in chapter 4.

• Enhancement :– To “improve” the usefulness of an image by using some transformation

on the image. – Often the improvement is to help make the image “better” looking,

such as increasing the intensity or contrast.



BackgroundBackground

• A mathematical representation of spatial domain enhancement:

where f(x, y): the input image

g(x, y): the processed image

T: an operator on f, defined over some neighborhood of (x, y)

)],([),( yxfTyxg



Gray-level TransformationGray-level Transformation



Some Basic Gray Level TransformationsSome Basic Gray Level Transformations



Image NegativesImage Negatives

rLs 1

• Let the range of gray level be [0, L-1], then



Log TransformationsLog Transformations

)1log( rcs where c : constant

0r



Power-Law TransformationPower-Law Transformation

crs

where c, : positive constants



Power-Law TransformationExample 1: Gamma Correction

Power-Law TransformationExample 1: Gamma Correction

4.0



Power-Law TransformationExample 2: Gamma CorrectionPower-Law TransformationExample 2: Gamma Correction

4.0 3.0

6.01



Power-Law TransformationExample 3: Gamma CorrectionPower-Law TransformationExample 3: Gamma Correction

0.3

0.50.4

1



Piecewise-Linear Transformation FunctionsCase 1: Contrast Stretching

Piecewise-Linear Transformation FunctionsCase 1: Contrast Stretching



Piecewise-Linear Transformation FunctionsCase 2:Gray-level Slicing

Piecewise-Linear Transformation FunctionsCase 2:Gray-level Slicing

An image Result of using the transformation in (a)



Piecewise-Linear Transformation FunctionsCase 3:Bit-plane Slicing

Piecewise-Linear Transformation FunctionsCase 3:Bit-plane Slicing

• Bit-plane slicing: – It can highlight the contribution made to total image appearance by

specific bits.

– Each pixel in an image represented by 8 bits.

– Image is composed of eight 1-bit planes, ranging from bit-plane 0 for the least significant bit to bit plane 7 for the most significant bit.



Piecewise-Linear Transformation FunctionsBit-plane Slicing: A Fractal Image






2

7 6

45 3

1 0



Histogram ProcessingHistogram Processing



Histogram EqualizationHistogram Equalization

• Histogram equalization:– To improve the contrast of an image

– To transform an image in such a way that the transformed image has a nearly uniform distribution of pixel values

• Transformation:– Assume r has been normalized to the interval [0,1], with r = 0

representing black and r = 1 representing white

– The transformation function satisfies the following conditions:• T(r) is single-valued and monotonically increasing in the interval

•

10 r

10for 1)(0 rrT

10 )( rrTs




• For example:




• Histogram equalization is based on a transformation of the probability density function of a random variable.

• Let pr(r) and ps(s) denote the probability density function of random variable r and s, respectively.

• If pr(r) and T(r) are known, then the probability density function ps(s) of the transformed variable s can be obtained

• Define a transformation function where w is a dummy variable of integration and the right side of this equation is the cumulative distribution

function of random variable r.

r

r dwwprTs0

)()(ds

drrpsp rs )()(




• Given transformation function T(r),

ps(s) now is a uniform probability density function.

• T(r) depends on pr(r), but the resulting ps(s) always is uniform.

10 1)(

1)()()( s

rprp

ds

drrpsp

rrrs

)()()(

0rpdwwp

dr

d

dr

rdT

ds

drr

r

r

r

r dwwprT0

)()(




• In discrete version:– The probability of occurrence of gray level rk in an image is

n : the total number of pixels in the image

nk : the number of pixels that have gray level rk

L : the total number of possible gray levels in the image– The transformation function is

– Thus, an output image is obtained by mapping each pixel with level rk in the input image into a corresponding pixel with level sk.

1,...,2,1,0 )()(0 0

Lkn

nrprTs

k

j

k

j

jjrkk

1,...,2,1,0 )( Lkn

nrp k

r




• Transformation functions (1) through (4) were obtained form the histograms of the images in Fig 3.17(1), using Eq. (3.3-8).



Histogram MatchingHistogram Matching

• Histogram matching is similar to histogram equalization, except that instead of trying to make the output image have a flat histogram, we would like it to have a histogram of a specified shape, say pz(z).

• We skip the details of implementation.



Local EnhancementLocal Enhancement

• The histogram processing methods discussed above are global, in the sense that pixels are modified by a transformation function based on the gray-level content of an entire image.

• However, there are cases in which it is necessary to enhance details over small areas in an image.

original global local



Use of Histogram Statistics for Image EnhancementUse of Histogram Statistics for Image Enhancement

• Moments can be determined directly from a histogram much faster than they can from the pixels directly.

• Let r denote a discrete random variable representing discrete gray-levels in the range [0,L-1], and p(ri) denote the normalized histogram component corresponding to the ith value of r, then the nth moment of r about its mean is defined as

where m is the mean value of r

• For example, the second moment (also the variance of r) is

1

0

)(L

iii rprm

1

0

22 )()()(

L

iii rpmrr

1

0

)()()(L

ii

nin rpmrr



Use of Histogram Statistics for Image EnhancementUse of Histogram Statistics for Image Enhancement

• Two uses of the mean and variance for enhancement purposes:– The global mean and variance (global means for the entire

image) are useful for adjusting overall contrast and intensity.

– The mean and standard deviation for a local region are useful for correcting for large-scale changes in intensity and contrast. ( See equations 3.3-21 and 3.3-22.)



Use of Histogram Statistics for Image EnhancementExample: Enhancement based on local statistics

Use of Histogram Statistics for Image EnhancementExample: Enhancement based on local statistics



Enhancement Using Arithmetic/Logic OperationsEnhancement Using Arithmetic/Logic Operations

• Two images of the same size can be combined using operations of addition, subtraction, multiplication, division, logical AND, OR, XOR and NOT. Such operations are done on pairs of their corresponding pixels.

• Often only one of the images is a real picture while the other is a machine generated mask. The mask often is a binary image

consisting only of pixel values 0 and 1. • Example: Figure 3.27



Enhancement Using Arithmetic/Logic OperationsEnhancement Using Arithmetic/Logic Operations

AND

OR



Image SubtractionExample 1






• When subtracting two images, negative pixel values can result. So, if you want to display the result it may be necessary to readjust the dynamic range by scaling.



Image AveragingImage Averaging

• When taking pictures in reduced lighting (i.e., low illumination), image noise becomes apparent.

• A noisy image g(x,y) can be defined by

where f (x, y): an original image : the addition of noise• One simple way to reduce this granular noise is to take

several identical pictures and average them, thus smoothing out the randomness.

),(),(),( yxyxfyxg

),( yx



Noise Reduction by Image AveragingExample: Adding Gaussian Noise

Noise Reduction by Image AveragingExample: Adding Gaussian Noise

• Figure 3.30 (a): An image of Galaxy Pair NGC3314.

• Figure 3.30 (b): Image corrupted by additive Gaussian noise with zero mean and a standard deviation of 64 gray levels.

• Figure 3.30 (c)-(f): Results of averaging K=8,16,64, and 128 noisy images.



Noise Reduction by Image AveragingExample: Adding Gaussian NoiseNoise Reduction by Image AveragingExample: Adding Gaussian Noise

• Figure 3.31 (a):

From top to bottom: Difference images between Fig. 3.30 (a) and the four images in Figs. 3.30 (c) through (f), respectively.

• Figure 3.31 (b): Corresponding histogram.



Basics of Spatial FilteringBasics of Spatial Filtering

• In spatial filtering (vs. frequency domain filtering), the output image is computed directly by simple calculations on the pixels of the input image.

• Spatial filtering can be either linear or non-linear.

• For each output pixel, some neighborhood of input pixels is used in the computation.

• In general, linear filtering of an image f of size MXN with a filter mask of size mxn is given by

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

• This concept called convolution. Filter masks are sometimes called convolution masks or convolution kernels.

a

as

b

bt

tysxftswyxg ),(),(),(



• Nonlinear spatial filtering usually uses a neighborhood too, but some other mathematical operations are use. These can include conditional operations (if …, then…), statistical (sorting pixel values in the neighborhood), etc.

• Because the neighborhood includes pixels on all sides of the center pixel, some special procedure must be used along the top, bottom, left and right sides of the image so that the processing does not try to use pixels that do not exist.




Smoothing Spatial FiltersSmoothing Spatial Filters

• Smoothing linear filters– Averaging filters (Lowpass filters in Chapter 4))

• Box filter

• Weighted average filter

Box filter Weighted average



Smoothing Spatial FiltersSmoothing Spatial Filters

• The general implementation for filtering an MXN image with a weighted averaging filter of size mxn is given by

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

a

as

b

bt

a

as

b

bt

tsw

tysxftswyxg

),(

),(),(),(



Smoothing Spatial FiltersImage smoothing with masks of various sizes

Smoothing Spatial FiltersImage smoothing with masks of various sizes



Smoothing Spatial FiltersAnother Example

Smoothing Spatial FiltersAnother Example



Order-Statistic FiltersOrder-Statistic Filters

• Order-statistic filters– Median filter: to reduce impulse noise (salt-and-

pepper noise)



Sharpening Spatial FiltersSharpening Spatial Filters

• Sharpening filters are based on computing spatial derivatives of an image.

• The first-order derivative of a one-dimensional function f(x) is

• The second-order derivative of a one-dimensional function f(x) is

)()1( xfxfx

f

)(2)1()1(2

2

xfxfxfx

f



Sharpening Spatial FiltersAn Example

Sharpening Spatial FiltersAn Example



Use of Second Derivatives for EnhancementThe Laplacian


• Development of the Laplacian method– The two dimensional Laplacian operator for continuous

functions:

– The Laplacian is a linear operator.

2

2

2

22

y

f

x

ff

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

2

yxfyxfyxfx

f

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

2

yxfyxfyxfy

f

)(4)]1,()1,(),1(),1([2 xfyxfyxfyxfyxff





• To sharpen an image, the Laplacian of the image is subtracted from the original image.

• Example: Figure 3.40

positive. ismask Laplacian theoft coefficiencenter theif ),(

negative. ismask Laplacian theoft coefficiencenter theif ),(),(

2

2

fyxf

fyxfyxg



Use of Second Derivatives for EnhancementThe Laplacian: Simplifications

Use of Second Derivatives for EnhancementThe Laplacian: Simplifications

The g(x,y) maskNot only f2



Use of First Derivatives for EnhancementThe Gradient


• Development of the Gradient method– The gradient of function f at coordinates (x,y) is defined as

the two-dimensional column vector:

– The magnitude of this vector is given by

y

fx

f

G

G

y

xf

2

122

2

122)(mag

y

f

x

fGGf yxf

yx GGf





Roberts cross-gradientoperators

Sobeloperators



Use of First Derivatives for EnhancementThe Gradient: Using Sobel Operators

Use of First Derivatives for EnhancementThe Gradient: Using Sobel Operators



Combining Spatial Enhancement Methods

Combining Spatial Enhancement Methods

digital image processing, 2nd ed. © 2002 r. c. gonzalez & r. e. woods chapter 3 image...

Documents

digital image processing

woods image enhancement

image plane

image result

woods image negatives

spatial domain slide

processed image t

input image gx