Chapter 3 Image Enhancement in the Spatial Domain
- Slides: 131
Chapter 3: Image Enhancement in the Spatial Domain
Image Preprocessing Restoration Enhancement Spectral Domain Spatial Domain Point Processing • ex. Histogram equalization Spatial filtering • Ex. Laplacian • Inverse filtering • Wiener filtering Filtering • Ex. Fourier tras. 2
Principle Objective of Enhancement n n Process an image so that the result will be more suitable than the original image for a specific application. A method which is quite useful for enhancing an image may not necessarily be the best approach for enhancing another images 3
2 domains n Spatial Domain : (image plane) n n Frequency Domain : n n Techniques are based on direct manipulation of pixels in an image Techniques are based on modifying the Fourier transform of an image There are some enhancement techniques based on various combinations of methods from these two categories. 4
Good images n For human visual n n n For machine perception n The visual evaluation of image quality is a highly subjective process. It is hard to standardize the definition of a good image. The evaluation task is easier. A good image is one which gives the best machine recognition results. A certain amount of trial and error usually is required before a particular image enhancement approach is selected. 5
Spatial Domain n Procedures that operate directly on pixels. g(x, y) = T[f(x, y)] where n f(x, y) is the input image n g(x, y) is the processed image n T is an operator on f defined over some neighborhood of (x, y) 6
Mask/Filter n (x, y) • n Neighborhood of a point (x, y) can be defined by using a square/rectangular (common used) or circular subimage area centered at (x, y) The center of the subimage is moved from pixel to pixel starting at the top of the corner 7
Point Processing n Neighborhood = 1 x 1 pixel g depends on only the value of f at (x, y) n T = gray level (or intensity or mapping) n n transformation function s = T(r) Where n r = gray level of f(x, y) n s = gray level of g(x, y) 8
Contrast Stretching n Produce higher contrast than the original by n n darkening the levels below m in the original image Brightening the levels above m in the original image 9
Thresholding n Produce a two-level (binary) image 10
Mask Processing or Filter n n Neighborhood is bigger than 1 x 1 pixel Use a function of the values of f in a predefined neighborhood of (x, y) to determine the value of g at (x, y) The value of the mask coefficients determine the nature of the process Used in techniques n n Image Sharpening Image Smoothing 11
3 basic gray-level transformation functions n Output gray level, s Negative n nth root Log nth power Linear function n Logarithm function n n Identity Inverse Log Negative and identity transformations Log and inverse-log transformation Power-law function n nth power and nth root transformations Input gray level, r 12
Output gray level, s Image Negatives n Negative nth root n Log nth power n n Identity Inverse Log Input gray level, r An image with gray level in the range [0, L-1] where L = 2 n ; n = 1, 2… Negative transformation : s = L – 1 –r Reversing the intensity levels of an image. Suitable for enhancing white or gray detail embedded in dark regions of an image, especially when the black area dominant in size. 13
Log Transformations Output gray level, s Negative n nth root Log Identity n nth power Inverse Log Input gray level, r n s = c log (1+r) c is a constant and r 0 Log curve maps a narrow range of low gray-level values in the input image into a wider range of output levels. Used to expand the values of dark pixels in an image while compressing the higherlevel values. 14
Inverse Logarithm Transformations n n Do opposite to the Log Transformations Used to expand the values of high pixels in an image while compressing the darkerlevel values. 15
Power-Law Transformations s = cr Output gray level, s n n Plots of s Input gray level, r = cr for various values of (c = 1 in all cases) n c and are positive constants Power-law curves with fractional values of map a narrow range of dark input values into a wider range of output values, with the opposite being true for higher values of input levels. c = = 1 Identity function 16
Histogram Processing n n Histogram of a digital image with gray levels in the range [0, L-1] is a discrete function Where n n n h(rk) = nk rk : the kth gray level nk : the number of pixels in the image having gray level rk h(rk) : histogram of a digital image with gray levels rk 17
Normalized Histogram n dividing each of histogram at gray level rk by the total number of pixels in the image, n p(rk) = nk / n n For k = 0, 1, …, L-1 p(rk) gives an estimate of the probability of occurrence of gray level rk The sum of all components of a normalized histogram is equal to 1 18
Histogram Processing n n n Basic for numerous spatial domain processing techniques Used effectively for image enhancement Information inherent in histograms also is useful in image compression and segmentation 19
h(rk) or p(rk) Example rk Dark image Components of histogram are concentrated on the low side of the gray scale. Bright image Components of histogram are concentrated on the high side of the gray scale. 20
Example Low-contrast image histogram is narrow and centered toward the middle of the gray scale High-contrast image histogram covers broad range of the gray scale and the distribution of pixels is not too far from uniform, with very few vertical lines being much higher than the others 21
Histogram Equalization n n As the low-contrast image’s histogram is narrow and centered toward the middle of the gray scale, if we distribute the histogram to a wider range the quality of the image will be improved. We can do it by adjusting the probability density function of the original histogram of the image so that the probability spread equally 22
Histogram transformation s = T(r) s n n Where 0 r 1 T(r) satisfies n sk= T(rk) T(r) n 0 rk 1 (a). T(r) is singlevalued and monotonically increasingly in the interval 0 r 1 (b). 0 T(r) 1 for 0 r 1 r 23
2 Conditions of T(r) n n n Single-valued (one-to-one relationship) guarantees that the inverse transformation will exist 0 T(r) 1 for 0 r 1 guarantees that the output gray levels will be in the same range as the input levels. The inverse transformation from s back to r is r = T -1(s) ; 0 s 1 24
Probability Density Function n n The gray levels in an image may be viewed as random variables in the interval [0, 1] PDF is one of the fundamental descriptors of a random variable 25
Applied to Image n Let n n n pr(r) denote the PDF of random variable r ps (s) denote the PDF of random variable s If pr(r) and T(r) are known and T-1(s) satisfies condition (a) then ps(s) can be obtained using a formula : 26
Applied to Image The PDF of the transformed variable s is determined by the gray-level PDF of the input image and by the chosen transformation function 27
Transformation function n A transformation function is a cumulative distribution function (CDF) of random variable r : where w is a dummy variable of integration Note: T(r) depends on pr(r) 28
Cumulative Distribution function n n CDF is an integral of a probability function (always positive) is the area under the function Thus, CDF is always single valued and monotonically increasing Thus, CDF satisfies the condition (a) We can use CDF as a transformation function 29
Finding ps(s) from given T(r) Substitute and yield 30
ps(s) n n n As ps(s) is a probability function, it must be zero outside the interval [0, 1] in this case because its integral over all values of s must equal 1. Called ps(s) as a uniform probability density function ps(s) is always a uniform, independent of the form of pr(r) 31
yields Ps(s) a random variable s characterized by a uniform probability function 1 0 s 32
Discrete transformation function n n The probability of occurrence of gray level in an image is approximated by The discrete version of transformation 33
Histogram Equalization n n Thus, an output image is obtained by mapping each pixel with level rk in the input image into a corresponding pixel with level sk in the output image In discrete space, it cannot be proved in general that this discrete transformation will produce the discrete equivalent of a uniform probability density function, which would be a uniform histogram 34
Example before after Histogram equalization 35
Example before after Histogram equalization The quality is not improved much because the original image already has a broaden gray-level scale 36
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Example No. of pixels 6 2 3 3 2 4 3 4 3 2 3 5 3 4 2 2 4 x 4 image Gray scale = [0, 9] 5 1 Gray level 0 1 2 3 4 5 6 7 8 9 histogram 38
Gray Level(j) 0 1 2 3 4 5 6 7 8 9 No. of pixels 0 0 6 5 4 1 0 0 0 6 11 15 16 16 16 6 0 sx 9 0 0 0 11 15 16 16 16 / / / / 16 16 3. 3 3 6. 1 6 8. 4 8 9 9 9 39
Example No. of pixels 6 3 6 6 3 8 6 4 6 3 6 9 3 8 2 3 8 3 Output image Gray scale = [0, 9] 5 1 0 1 2 3 4 5 6 7 8 9 Gray level Histogram equalization 40
Note n It is clearly seen that n n n Histogram equalization distributes the gray level to reach the maximum gray level (white) because the cumulative distribution function equals 1 when 0 r L-1 If the cumulative numbers of gray levels are slightly different, they will be mapped to little different or same gray levels as we may have to approximate the processed gray level of the output image to integer number Thus the discrete transformation function can’t guarantee the one to one mapping relationship 41
Histogram Matching (Specification) n n n Histogram equalization has a disadvantage which is that it can generate only one type of output image. With Histogram Specification, we can specify the shape of the histogram that we wish the output image to have. It doesn’t have to be a uniform histogram 42
Consider the continuous domain Let pr(r) denote continuous probability density function of gray-level of input image, r Let pz(z) denote desired (specified) continuous probability density function of gray-level of output image, z Let s be a random variable with the property Histogram equalization Where w is a dummy variable of integration 43
Next, we define a random variable z with the property Histogram equalization Where t is a dummy variable of integration thus s = T(r) = G(z) Therefore, z must satisfy the condition z = G-1(s) = G-1[T(r)] Assume G-1 exists and satisfies the condition (a) and (b) We can map an input gray level r to output gray level z 44
Procedure Conclusion 1. 2. Obtain the transformation function T(r) by calculating the histogram equalization of the input image Obtain the transformation function G(z) by calculating histogram equalization of the desired density function 45
Procedure Conclusion 3. Obtain the inversed transformation function G-1 z = G-1(s) = G-1[T(r)] 4. Obtain the output image by applying the processed gray-level from the inversed transformation function to all the pixels in the input image 46
Example Assume an image has a gray level probability density function pr(r) as shown. Pr(r) 2 1 0 1 2 r 47
Example We would like to apply the histogram specification with the desired probability density function pz(z) as shown. Pz(z) 2 1 0 1 2 z 48
Step 1: Obtain the transformation function T(r) s=T(r) 1 One to one mapping function 0 1 r 49
Step 2: Obtain the transformation function G(z) 50
Step 3: Obtain the inversed transformation function G-1 We can guarantee that 0 z 1 when 0 r 1 51
Discrete formulation 52
Image Equalization Result image after histogram equalization Transformation function Histogram of the result image for histogram equalization The histogram equalization doesn’t make the result image look better than the original image. Consider the histogram of the result image, the net effect of this method is to map a very narrow interval of dark pixels into the upper end of the gray scale of the output image. As a consequence, the output image is light and has a washed-out appearance. 53
Solve the problem Since the problem with the transformation function of the histogram equalization was caused by a large concentration of pixels in the original image with levels near 0 Histogram Equalization Histogram Specification a reasonable approach is to modify the histogram of that image so that it does not have this property 54
Histogram Specification n n (1) the transformation function G(z) obtained from (2) the inverse transformation G-1(s) 55
Result image and its histogram The output image’s histogram Original image After applied the histogram equalization Notice that the output histogram’s low end has shifted right toward the lighter region of the gray scale as desired. 56
Note n n Histogram specification is a trial-anderror process There are no rules for specifying histograms, and one must resort to analysis on a case-by-case basis for any given enhancement task. 57
Note n n Histogram processing methods are global processing, in the sense that pixels are modified by a transformation function based on the gray-level content of an entire image. Sometimes, we may need to enhance details over small areas in an image, which is called a local enhancement. 58
a) Local Enhancement b) c) (a) n n n (b) (c) Original image (slightly blurred to reduce noise) global histogram equalization (enhance noise & slightly increase contrast but the construction is not changed) local histogram equalization using 7 x 7 neighborhood (reveals the small squares inside larger ones of the original image. define a square or rectangular neighborhood and move the center of this area from pixel to pixel. at each location, the histogram of the points in the neighborhood is computed and either histogram equalization or histogram specification transformation function is obtained. another approach used to reduce computation is to utilize nonoverlapping regions, but it usually produces an undesirable checkerboard effect. 59
Explain the result in c) n n Basically, the original image consists of many small squares inside the larger dark ones. However, the small squares were too close in gray level to the larger ones, and their sizes were too small to influence global histogram equalization significantly. So, when we use the local enhancement technique, it reveals the small areas. Note also the finer noise texture is resulted by the local processing using relatively small neighborhoods. 60
Enhancement using Arithmetic/Logic Operations n n Arithmetic/Logic operations perform on pixel by pixel basis between two or more images except NOT operation which perform only on a single image 61
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Logic Operations n n n Logic operation performs on gray-level images, the pixel values are processed as binary numbers light represents a binary 1, and dark represents a binary 0 NOT operation = negative transformation 63
Example of AND Operation original image AND image mask result of AND operation 64
Example of OR Operation original image OR image mask result of OR operation 65
Image Subtraction g(x, y) = f(x, y) – h(x, y) n enhancement of the differences between images 66
Image Subtraction n n n b d a). original fractal image b). result of setting the four lower-order bit planes to zero n n a c refer to the bit-plane slicing the higher planes contribute significant detail the lower planes contribute more to fine detail image b). is nearly identical visually to image a), with a very slightly drop in overall contrast due to less variability of the gray -level values in the image. c). difference between a). and b). (nearly black) d). histogram equalization of c). (perform contrast stretching transformation) 67
Image Averaging n consider a noisy image g(x, y) formed by the addition of noise (x, y) to an original image f(x, y) g(x, y) = f(x, y) + (x, y) 68
Image Averaging n if noise has zero mean and be uncorrelated then it can be shown that if = image formed by averaging K different noisy images 69
Image Averaging n then = variances of g and if K increase, it indicates that the variability (noise) of the pixel at each location (x, y) decreases. 70
Image Averaging n thus = expected value of g (output after averaging) = original image f(x, y) 71
Image Averaging n Note: the images gi(x, y) (noisy images) must be registered (aligned) in order to avoid the introduction of blurring and other artifacts in the output image. 72
Spatial Filtering n n n use filter (can also be called as mask/kernel/template or window) the values in a filter subimage are referred to as coefficients, rather than pixel. our focus will be on masks of odd sizes, e. g. 3 x 3, 5 x 5, … 73
Spatial Filtering Process n n simply move the filter mask from point to point in an image. at each point (x, y), the response of the filter at that point is calculated using a predefined relationship. 74
Linear Filtering n Linear Filtering of an image f of size Mx. N filter mask of size mxn is given by the expression where a = (m-1)/2 and b = (n-1)/2 To generate a complete filtered image this equation must be applied for x = 0, 1, 2, … , M-1 and y = 0, 1, 2, … , N-1 75
Smoothing Spatial Filters n n used for blurring and for noise reduction blurring is used in preprocessing steps, such as n n n removal of small details from an image prior to object extraction bridging of small gaps in lines or curves noise reduction can be accomplished by blurring with a linear filter and also by a nonlinear filter 76
Smoothing Linear Filters n n output is simply the average of the pixels contained in the neighborhood of the filter mask. called averaging filters or lowpass filters. 77
Smoothing Linear Filters n n replacing the value of every pixel in an image by the average of the gray levels in the neighborhood will reduce the “sharp” transitions in gray levels. sharp transitions n n n random noise in the image edges of objects in the image thus, smoothing can reduce noises (desirable) and blur edges (undesirable) 78
3 x 3 Smoothing Linear Filters box filter weighted average the center is the most important and other pixels are inversely weighted as a function of their distance from the center of the mask 79
Weighted average filter n the basic strategy behind weighting the center point the highest and then reducing the value of the coefficients as a function of increasing distance from the origin is simply an attempt to reduce blurring in the smoothing process. 80
General form : smoothing mask n filter of size mxn (m and n odd) summation of all coefficient of the mask 81
a c e Example n n n b d f a). original image 500 x 500 pixel b). - f). results of smoothing with square averaging filter masks of size n = 3, 5, 9, 15 and 35, respectively. Note: n n big mask is used to eliminate small objects from an image. the size of the mask establishes the relative size of the objects that will be blended with the background. 82
Example original image result after smoothing result of thresholding with 15 x 15 averaging mask we can see that the result after smoothing and thresholding, the remains are the largest and brightest objects in the image. 83
Order-Statistics Filters (Nonlinear Filters) n n the response is based on ordering (ranking) the pixels contained in the image area encompassed by the filter example n n median filter : R = median{zk |k = 1, 2, …, n x n} max filter : R = max{zk |k = 1, 2, …, n x n} min filter : R = min{zk |k = 1, 2, …, n x n} note: n x n is the size of the mask 84
Median Filters n n replaces the value of a pixel by the median of the gray levels in the neighborhood of that pixel (the original value of the pixel is included in the computation of the median) quite popular because for certain types of random noise (impulse noise salt and pepper noise) noise , they provide excellent noise-reduction capabilities, capabilities with considering less blurring than linear smoothing filters of similar size. 85
Median Filters n n forces the points with distinct gray levels to be more like their neighbors. isolated clusters of pixels that are light or dark with respect to their neighbors, and whose area is less than n 2/2 (one-half the filter area), are eliminated by an n x n median filter. eliminated = forced to have the value equal the median intensity of the neighbors. larger clusters are affected considerably less 86
Example : Median Filters 87
Sharpening Spatial Filters n n to highlight fine detail in an image or to enhance detail that has been blurred, either in error or as a natural effect of a particular method of image acquisition. 88
Blurring vs. Sharpening n n n as we know that blurring can be done in spatial domain by pixel averaging in a neighbors since averaging is analogous to integration thus, we can guess that the sharpening must be accomplished by spatial differentiation. 89
Derivative operator n n 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. thus, image differentiation n n enhances edges and other discontinuities (noise) deemphasizes area with slowly varying gray-level values. 90
First-order derivative n a basic definition of the first-order derivative of a one-dimensional function f(x) is the difference 91
Second-order derivative n similarly, we define the second-order derivative of a one-dimensional function f(x) is the difference 92
First and Second-order derivative of f(x, y) n when we consider an image function of two variables, f(x, y), at which time we will dealing with partial derivatives along the two spatial axes. Gradient operator Laplacian operator (linear operator) 93
Discrete Form of Laplacian from yield, 94
Result Laplacian mask 95
Laplacian mask implemented an extension of diagonal neighbors 96
Other implementation of Laplacian masks give the same result, but we have to keep in mind that when combining (add / subtract) a Laplacian-filtered image with another image. 97
Effect of Laplacian Operator n as it is a derivative operator, n n n it highlights gray-level discontinuities in an image it deemphasizes regions with slowly varying gray levels tends to produce images that have n n grayish edge lines and other discontinuities, all superimposed on a dark, featureless background. 98
Correct the effect of featureless background n n easily by adding the original and Laplacian image. be careful with the Laplacian filter used if the center coefficient of the Laplacian mask is negative if the center coefficient of the Laplacian mask is positive 99
Example n n a). image of the North pole of the moon b). Laplacian-filtered image with 1 1 -8 1 1 c). Laplacian image scaled for display purposes d). image enhanced by addition with original image 100
Mask of Laplacian + addition n to simply the computation, we can create a mask which do both operations, Laplacian Filter and Addition the original image. 101
Laplacian MATLAB Example >> f=imread(‘Fig_Moon. jpg’); >> w 4=fspecial(‘laplacian’, 0) >> w 8=[1 1 1; 1 -8 1; 1 1 1] >>f=im 2 double(f); >> >> >> %load in lunar north pole image % creates 3 x 3 laplacian, alpha=0 [0: 1] % create a Laplacian that fspecial can’t % output same as input unit 8 so % negative values are truncated. % Convert to double to keep negative values. g 4=f-imfilter(f, w 4, ’replicate’); % filter using default values g 8=f-imfilter(f, w 8, ’replicate’); % filter using default values imshow(f) % display original image imshow(g 4) % display g 4 processed image imshow(g 8) % display g 8 processed image 102
Mask of Laplacian + addition 0 0 1 5 1 1 103
Example 104
Note 0 -1 5 -1 0 0 -1 9 -1 0 = 0 0 0 0 1 0 0 0 0 + 0 -1 4 -1 0 + 0 -1 8 -1 0 105
Unsharp masking sharpened image = original image – blurred image n to subtract a blurred version of an image produces sharpening output image. 106
High-boost filtering n n generalized form of Unsharp masking A 1 107
High-boost filtering n if we use Laplacian filter to create sharpen image fs(x, y) with addition of original image 108
High-boost filtering n yields if the center coefficient of the Laplacian mask is negative if the center coefficient of the Laplacian mask is positive 109
High-boost Masks n n A 1 if A = 1, it becomes “standard” Laplacian sharpening 110
Example 111
Gradient Operator n first derivatives are implemented using the magnitude of the gradient commonly approx. the magnitude becomes nonlinear 112
Gradient Mask n z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 simplest approximation, 2 x 2 113
Gradient Mask n z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 Roberts cross-gradient operators, 2 x 2 114
Gradient Mask n z 1 z 2 z 3 z 4 z 5 z 6 z 7 z 8 z 9 Sobel operators, 3 x 3 the weight value 2 is to achieve smoothing by giving more important to the center point 115
Note n the summation of coefficients in all masks equals 0, indicating that they would give a response of 0 in an area of constant gray level. 116
Example 117
Example of Combining Spatial Enhancement Methods n n 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 118
Example of Combining Spatial Enhancement Methods n 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 119
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- Combining spatial enhancement methods
- Image enhancement in spatial domain
- Image enhancement in spatial domain
- Image enhancement in spatial domain
- What is enhancement in the spatial domain?
- Combining spatial enhancement methods
- Enhancement using arithmetic/logic operations
- Combining spatial enhancement methods
- Combining spatial enhancement methods
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- Objective of image enhancement
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- Spatial domain
- Spatial operations in image processing
- Digital image processing
- Intensity transformations and spatial filtering
- Compression in digital image processing
- Spatial resolution in digital image processing
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