Digital Image Analysis and Processing

Transcription

1 Digital Image Analysis and Processing CPE Image Enhancement Part I Intensity Transformation Chapter 3 Sections: Dr. Iyad Jafar

2 Outline What is Image Enhancement? Background Intensity Transformation Functions Negatives Log and Inverse Log Power-Law Piecewise Transformation Graylevel l and Bit Slicingi 2 Histogram Processing Equalization Specification Local Processing

3 What is Image Enhancement? The purpose of image enhancement is to process an image such that it is more suitable than the original image for a specific application The word specific is important, because algorithms developed for some images may not work for others There is no general theory for enhancement and the evaluation of its outcome is highly subjective 3 Enhancement can be performed in Spatial domain: direct operation on the pixel values Frequency domain: modify the image frequency components (Ch. 4)

4 Background 4 Spatial domain of the image is the set of composing the image pixels Enhancement in the spatial domain involves direct operation on the pixel intensities This can be expressed mathematically as g(xy) g(x,y) = T[f(xy)] T[f(x,y)] f(x,y) is the input image g(x,y) is the output image T[ ] is an operator defined over some neighborhood of (x,y) Important Keep in mind that g(x,y) may take any value from the set of available gray levels only. Thus, when mapping we should assign the mapped value to the closest level

5 Background Defining the neighborhood around (x,y) Use a square/rectangular subimage that is centered at (x,y) Operation Move the center of the subimage from pixel to pixel and apply the operator T at each location (x,y) to compute the output t g(x,y) 5

6 Background The simplest form of the operator T is when the neighborhood size is 1x1 pixels. Accordingly, g(x,y) is only dependent on the value of f at (x,y) In this case, T is called the gray-level or intensity transformation function that can be represented as s = T(r) s is a variable denoting g(x,y) r is a variable denoting f(x,y) 6 This is kind of processing is referred as point processing

7 Background Intensity transformation function examples 7 T(r) performs contrast t T(r) reduces the number of stretching by mapping levels levels in the image to two less than k to narrow range while those above k are mapped to wider range

8 Point Processing Example Thresholding Thresholding transformations are particularly useful for segmentation in which we want to isolate an object of interest from a background s = r > threshold 0.0 r <= threshold 8

9 Basic Gray Level Transformations Mapping can be performed by mathematical substitution or lookup tables Some common functions are Linear (negative/identity) Logarithmic (log/inverse log) Power law (n th power/n th root) 9

10 Basic Gray Level Transformations Image Negatives Can be performed by using s = L 1 r where L-1 is the maximum intensity value 10

11 Basic Gray Level Transformations Log and inverse Log Transformations The general form of the log transformation b is the base s = clog b(1+r) Maps narrow range oflowintensity levels to wider range and wide range of high intensity levels to narrower range Usually used to expand the values of dark pixels and compress the higherh levell values 11 The general form of the inverse log cr s = b 1 Its operation is the opposite of the log transformation

12 Basic Gray Level Transformations Log Transformation Example It is very important in mapping wide dynamic ranges into narrow ones Fourier spectrum values in the range [0,1.5x10 6 ] transformed to [0,255] using log transformation s = log(1 + r) 12

13 Basic Gray Level Transformations Inverse Log Transformation Example e cr -1 13

14 Basic Gray Level Transformations Power-Law transformations The general form s = cr Power law is similar to log when gamma < 1 and similar to inverse log when gamma > 1 γ 14

15 Basic Gray Level Transformations Power-Law transformations Gamma-correction Display devices have intensity-to-voltage response that is a power functions.thus, images tend to be darker when displayed. Correction is needed using nth root before feeding the image to the monitor 15

16 Basic Gray Level Transformations Power-Law Transformation The images to the right shows a magnetic resonance (MR) image of a fractured human spine Different curves highlight different detail s = r s = r 0.4 s = r 0.3

17 Piecewise-Linear Transformations y Can represent p arbitrarilyy complex p functions to achieve different results y Contrast stretching y r1 r2 and s1 s2 to preserve the order of ggrayy levels y The result depends on the values of r1, r2, s1, and s2 17

18 Piecewise-Linear Transformations Gray-level Slicing Used to highlight hli ht specific range of gray levelsels Two approaches 18

19 Piecewise-Linear Transformations Bit-plane Slicing Highlight the contribution of specific bits to the appearance of the image Each pixel value is represented by asetof bits Lower bits correspond to fine details while higher bits correspond to the global visual content Useful in image compression! 19

20 Piecewise-Linear Transformations Bit-plane Slicing - example Plane 0 Plane 1 Plane 2 Plane 3 Plane 4 Plane 5 Plane 6 Plane 7 20

21 Piecewise-Linear Transformations Bit-plane Slicing - example Planes 7 & 6 Planes 7,6,5 Planes 7,6,5,4 21

22 Piecewise-Linear Transformations Bit-plane Slicing - example Plane 7 Plane 6 Plane 5 Plane 4 Plane 3 22 Plane 2 Plane 1 Plane 0

23 Histogram Processing For an image with gray levels in [0,L-1] and MxN pixels, the histogram is a discrete function given by h( r ) = n k where r k is the k th intensity value and n k is the number of pixels in the image with intensity r k It is a common practice to normalize the histogram function by the number of pixels in the image by k 23 p( r ) = k The normalized histogram canbeusedasanestimate of the probability bili density function of the image n k MN Histograms are widely used in image processing: enhancement, compression, segmentation

24 Histogram Processing For enhancement, histograms can be used to infer the type of image quality:dark, bright, low or high contrast Dark Image Low Contrast 24 Bright Image High Contrast

25 Histogram Equalization It is quite acceptable that high contrast images have flat histograms (uniform distribution) Histogram equalization attempts to transform the original histogram into a flat one for the goal of better contrast In what follows, we will derive the transformation function that achieves such task 25

26 Histogram Equalization Let r be a continuous variable that represents the intensity values in the range [0,L-1] A valid transformation function for enhancement purposes s = T(r) should satisfy T(r) is monotonically increasing in the interval 0 r L-1 T(r) is bounded by [0,L-1] for all values of r The inverse transformation function that maps s back to r 1 r = T (s) requires that T(r) to be strictly monotonically increasing 26

27 Histogram Equalization Examples of transformation functions Monotonically Increasing Strictly monotonically increasing 27

28 Histogram Equalization Consider the gray level intensity represented by r as a random variable ibl in the interval [0,L-1] We can use the normalized histogram p r (r) as the probability bili density function for r If a transformation function s = T(r) is used to map the pixelsintosinthe range[0,1], then the following relation holds dr p(s) s = p(r) r ds 1 r= T where p s (s) is the normalized histogram of the output image (s) 28

29 Histogram Equalization Now, if we know that the output image has a flat histogram, i.e. 1 p s( s ) =, s [0,L-1] L 1 we can substitute in the equation in the previous slide and solve for s by integrating both sides This gives the desired d transformation ti function s=t(r) thatt performs histogram equalization s = T(r ) = (L 1) p r(w)dw For digital images, the transformation function is simply rk (L-1) s = T(r ) = (L-1) p (r ) = n MN k r w w 29 w= 0 w= 0 r 0 rk

30 Histogram Equalization The function given before maps the image histogram into a flat histogram regardless of its original shape Example 3.1: consider an image with intensity distribution given by 2r p(r r ) =, r [0,L-1] 2 (L 1) show that the applying the histogram equalization function transforms the histogram into a flat one. 30 Check example 3.5 for a simple illustration on performing histogram equalization

31 Histogram Equalization Example

32 Histogram Equalization Example 3.3 Original Equalized Output Histogram 1 2 s 3 r Transformation for the processed images 4 32

33 Notes Histogram Equalization The number of output levels in the equalized image is usually less than the original image due to the finite number of available levels Histogram equalization always produces an image whose mean intensity value is in the middle of [0,L-1], irrespective of the original mean. This might not be acceptable in somesituations ti If the inverse mapping for histogram equalization is needed, it is required that there are no empty bins in the original histogram 33

34 Histogram Equalization The Constrained Variational Histogram Equalization Performs equalization under the constraint that the output image mean is as close as possible to the original mean Original CVHE Image HE Image 34

35 Histogram Specification Histogram equalization might not be useful in all cases Another approach is to specify the desired histogram of the output image (based on some knowledge) and then perform the transformation 35 We can use the method used in deriving the transformation function of histogram equalization to find transformation function for the desired histogram, however This requires the availability of p s (s) in mathematical form and the ability to express s in terms of r Example!

36 Histogram Specification Alternatively Let p r (r) () and p z (z) () denote the original and desired histograms and assume that there exist two transformation functions s = T( r ) and s = G(z) This implies that we can find the mapping from r to zby knowing the inverse of G(z) 1 z = G (s) = G 1 (T(r)) T(r) s k = s q G(z) s k s q G -1 (s) 36 r k r This is a simple operation if G -1 (s) can be obtained!!!! z q z

37 Histogram Specification In discrete images, the situation is simpler since we only have finite set of values, i.e. for each s q weknow the corresponding z q The algorithm Compute s k = T(r k ) using 37 Compute s q = G(z q ) using q sk = T(r k ) = (L-1) p r ( r i ) q r k i= 0 z k s = G(z ) = (L-1) p (z ) q q z j j= 0 For each r k find z q such that minimizes for all i in [0,L-1]. T(r ) G( z ) k i

38 Histogram Specification Illustration T(r) G(z) Which value in G(z q ) is closer to s k? From G(z), we know which z q maps to this s q s k r k r z q z 38

39 Histogram Specification Example

40 Histogram Specification Original Image Histogram Equalization Desired Histogram Histogram Specification 40

41 Local Histogram Processing Both histogram equalization and specification methods discussedd earlier are considered d global l In other words, pixels are modified using a transformation function that is defined using all pixels in the image Such methods are suitable for overall enhancement and may not be suitable in situations where we want to enhance small areas in the image whose pixel count contributes less to the global transformation function 41

42 Local Histogram Processing Solution Consider local processing such that for each pixel (x,y) in the image Define a small neighborhood of size mxn that is centered around the pixel Use pixels inside the neighborhood to construct the transformation function Use the computed function map the pixel at (x,y) Repeat for all pixels 42

43 Local Histogram Processing Example 3.5 Global Histogram Equalization 43 Local histogram processing using 3x3 neighborhood

44 Related Matlab Functions Check Matlab help for the following functions imadjust imhist hist bar histeq 44

45 Readings Read subsection titled Using Histogram Statistics for Image Enhancement 45