Lecture #5. Point transformations (cont.) Histogram transformations. Intro to neighborhoods and spatial filtering

Transcription

1 Lecture #5 Point transformations (cont.) Histogram transformations Equalization Specification Local vs. global operations Intro to neighborhoods and spatial filtering

2 Brightness & Contrast 2002 R. C. Gonzalez & R. E. Woods

3 Point Ops via Functional Mappings Image: I, point operator J J = [] I Pixel: Input I(r,c) function, f Output J(r,c) The transformation of image I into image J is accomplished by replacing each input intensity, g, with a specific output intensity, k, at every location (r,c) where I(r,c) = g. If I and then ( r, c) = g f ( g) = k J( r, c) = k. The rule that associates k with g is usually specified with a function, f, so that f (g) = k.

4 Point Ops via Functional Mappings One-band Image J ( r, c) = f ( I( r, c) ), for all pixels locations ( r, c). Three-band Image J( r, c, b) = f ( I( r, c, b ), J( r, c, b) = f ( I( r, c, b) ), b for b = 1,2,3 or and all ( r, c).

5 Point Ops via Functional Mappings One-band Image Either all 3 bands are mapped through the same function, f, or ( r, c) = f ( I( r, c) ), J for all pixels locations ( r, c). Three-band Image J( r, c, b) = f ( I( r, c, b ), J( r, c, b) = f ( I( r, c, b) ), for b b = 1,2,3 or and all ( r, c). each band is mapped through a separate function, f b.

6 Point Ops via Functional Mappings A look-up table (LUT) implements a functional mapping. If k = for g f ( g), = 0,,255, andif k takes on values in{ 0,,255}, To remap a 1-band image, I, to J : then the LUT that implements f is a 256x1 array whose (g +1) th value is k = f (g). J = LUT I ( +1)

7 Point Ops via Functional Mappings If I is 3-band, then a) each band is mapped separately using the same LUT for each band or b) each band is mapped using different LUTs one for each band. J = LUT( I + 1), (:,:, b) = LUT ( I( :,:, b) + 1) for b = 1,2,3. a) J or b) b

8 Point Ops via Functional Mappings output value input value E.g.: index input value output

9 Point Ops via Functional Mappings input a pixel with this value cell index contents output is mapped to this value

10 How to Generate a Look-Up Table For example: Let a = 2. Let x { 0,,255} e ( x; a) = a( x 127) / 32 Or in Matlab: a = 2; x = 0:255; LUT = 255./ (1+exp(-a*(x-127)/32)); This is just one example.

11 Point Processes: Increase Brightness I = R + G + B 3 J k ( r, c) ( r, c) g, if I k( r, c) + g < 256, if I k( r, c) + g > 255 { 1, 2 } is the bandindex. I k + = 255 g 0 and k, g transform mapping

12 Point Processes: Decrease Brightness J k ( r, c) g 0 and k, 3 if I k( r, c) g 0 ( r, c) g, if I k( r, c) { 1, 2 } is the bandindex. 0, < = I k transform mapping 255-g

13 Point Processes: Increase Contrast Let J k Tk ( r, c) = a[ I k ( r, c) 127] + 127, where a > , if Tk ( r, c) < 0, ( r, c) = Tk ( r, c), if 0 Tk ( r, c) 255, 255, if T ( r, c) > 255. k { 1, 2, 3} k transform mapping

14 Point Processes: Decrease Contrast T k ( r, c) where = a 0 [ I ( r, c) 127] k a < 1.0 and + 127, k { 1,2,3 } transform mapping

15 Point Processes: Contrast Stretch Let Then, J m m J I = min = min [ I( r, c) ], M I = max[ I( r, c) ], [ J( r, c) ], M = max[ J( r, c) ]. I r, c m M m ( ) ( ) ( ) I r, c = M m + m. J J I J I J M J m J m I 127 M I 255 transform mapping

16 Information Loss from Contrast Adjustment orig lo-c hi-c histograms

17 Information Loss from Contrast Adjustment orig orig lo-c lo-c hi-c lo-c abbreviations: original low-contrast high-contrast restored difference rest diff difference between original and restored low-contrast orig hi-c hi-c rest diff difference between original and restored high-contrast

18 Point Processes: Increased Gamma J ( r, c) ( r, c) 1/ 255 I = for > transform mapping

19 Point Processes: Decreased Gamma J ( r, c) ( r, c) 1/ 255 I = for < m M transform mapping

20 Gamma Correction: Effect on Histogram

21 The Probability Density Function of an Image pdf [lower case] Let A = 255 g = 0 Note that since h I k R rows by C h I ( g + 1). k ( g +1) (the k th color band of image I ) with value g, A is the number of pixels in I. I k columns then is the number A = That is if R C. of I pixels in Then, 1 pi ( g+ 1) = h ( 1) k I g+ k A is the graylevel probability density function of I. is This is the probability that an arbitrary pixel from I k has value g. k

22 The Probability Density Function of an Image p band (g+1) is the fraction of pixels in (a specific band of) an image that have intensity value g. p band (g+1) is the probability that a pixel randomly selected from the given band has intensity value g. Whereas the sum of the histogram h band (g+1) over all g from 1 to 256 is equal to the number of pixels in the image, the sum of p band (g+1) over all g is 1. p band is the normalized histogram of the band.

23 The Probability Distribution Function of an Image Let q = [q 1 q 2 q 3 ] = I(r,c) be the value of a randomly selected pixel from I. Let g be a specific gray level. The probability that q k g is given by PDF [upper case] P I k ( g + ) = p ( + 1) = h ( + 1) ( + 1) g g Ik 1 = 0 1 I = k Ik 255 = 0 A = 0 hi + k = 0 g h ( ), 1 where h Ik ( +1) is the histogram of the kth band of I. This is the probability that any given pixel from I k has value less than or equal to g.

24 The Probability Distribution Function of an Image Let q = [q 1 q 2 q 3 ] = I(r,c) be the value of a randomly selected pixel from I. Let g be a specific gray level. The probability that q k g is given by Also called CDF for Cumulative Distribution Function. P I k ( g + ) = p ( + 1) = h ( + 1) ( + 1) g g Ik 1 = 0 1 I = k Ik 255 = 0 A = 0 hi + k = 0 g h ( ), 1 where h Ik ( +1) is the histogram of the kth band of I. This is the probability that any given pixel from I k has value less than or equal to g.

25 The Probability Distribution Function of an Image A.k.a. Cumulative Distribution Function. P band (g+1) is the fraction of pixels in (a specific band of) an image that have intensity values less than or equal to g. P band (g+1) is the probability that a pixel randomly selected from the given band has an intensity value less than or equal to g. P band (g+1) is the cumulative (or running) sum of p band (g+1) from 0 through g inclusive. P band (1) = p band (1) and P band (256) = 1; P band (g+1) is nondecreasing. Note: the Probability Distribution Function (PDF, capital letters) and the Cumulative Distribution Function (CDF) are exactly the same things. Both PDF and CDF will refer to it. However, pdf (small letters) is the density function.

26 Intensity Transform Requirements a) A monotonically increasing intensity transformation prevents an intensity reversal which could cause artifacts in the transformed image. b) A strictly monotonically increasing intensity transformation guarantees that the inverse transformation (from s=t(r) back to r) will be 1:1 preventing ambiguities 2002 R. C. Gonzalez & R. E. Woods

27 Uniform Histogram Transform These are pdf s. Gonzalez uses PDF instead of pdf. s = T( r)= L 1 r 0 ( ) p r w ( )dw L is the number of gray levels. Cumulative Distribution Function (CDF) 2002 R. C. Gonzalez & R. E. Woods

28 Histogram Equalization Example Note: this same 3-bit (8 gray level) 64x64 pixel image will be used for several examples 2002 R. C. Gonzalez & R. E. Woods

29 Histogram Equalization Example pdf Computed PDF 0 ( ) s 0 = T( r 0 )= ( 8 1) p r r j = 7 p r ( r 0 )= 7( 0.19)= 1.33 j 1=0 s 1 = T( r 1 )= ( 8 1) p r ( r j ) = 7 p r ( r 0 )+ 7 p r ( r 1 )= 7( 0.19)+ 7( 0.25)= = 3.08 s 2 = T( r 2 )= 8 1 j =0 2 ( ) = 7( 0.19)+ 7( 0.25)+ 7( 0.21)= = 4.55 ( ) p r r j j =0 s 3 = 5.67 s 4 = 6.23 s 5 = 6.65 s 6 = 6.86 s 7 = 7.00 Discrete CDF (transformation) Input Output Discrete Output R. C. Gonzalez & R. E. Woods

30 Histogram Equalization Resulting histogram when trying to achieve uniform* equalized histogram s k = T( r k )= L 1 ( ) ( ) p r r j k j =0 Must round off to nearest integer value since only L integer levels in output *Resulting histogram is usually not uniform due to discrete nature of transform. In this case we only have five output levels R. C. Gonzalez & R. E. Woods

31 Histogram Equalization dark light Low contrast high contrast 2002 R. C. Gonzalez & R. E. Woods

32 Histogram Equalization s k = T( r k )= L 1 ( ) ( ) p r r j k j =0

33 Histogram Specification G(z) would be a straight line for equalization. 3. Compute CDF v q for desired histogram 1. Compute pdf (histogram) for input image 2. Compute CDF s k for this PDF Inverting a discrete function is far easier than inverting a continuous function 2002 R. C. Gonzalez & R. E. Woods 4. Match CDF s for each s k to create mapping table by inverting G

34 Histogram Specification Original pdf Specified pdf CDF of specified pdf Actual transformed image pdf

35 Histogram Specification Example 1. Compute the PDF of the image to be transformed 0 ( ) s 0 = T( r 0 )= ( 8 1) p r r j = 7 p r ( r 0 )= 7( 0.19)= 1.33 j 1=0 s 1 = T( r 1 )= ( 8 1) p r ( r j ) = 7 p r ( r 0 )+ 7 p r ( r 1 )= 7( 0.19)+ 7( 0.25)= = 3.08 s 2 = T( r 2 )= 8 1 j =0 2 ( ) = 7( 0.19)+ 7( 0.25)+ 7( 0.21)= = 4.55 ( ) p r r j j =0 s 3 = 5.67 s 4 = 6.23 s 5 = 6.65 s 6 = 6.86 s 7 = 7.00 Output CDF (not discrete) Discrete CDF of input histogram Input Output Discrete Output Rounded off 2002 R. C. Gonzalez & R. E. Woods

36 Histogram Specification Example 2. Compute the PDF for the specified histogram 0 ( ) G( z 0 )= ( 8 1) p z z j = 7 p z ( z 0 )= 70 ( )= 0 j 1=0 G( z 1 )= ( 8 1) p z ( z j ) = 7 p z ( z 0 )+ 7 p z ( z 1 )= 70 ( )+ 70 ( )= 0 j =0 2 ( ) G( z 2 )= ( 8 1) p z z j = 70 ( )+ 70 ( )+ 70 ( )= 0 j =0 G( z 3 )= 1.05 G( z 4 )= 2.45 G( z 5 )= 4.55 G( z 6 )= 5.95 G( z 7 )= 7.00 Discrete CDF of desired histogram Input Output Discrete Output Rounded off 2002 R. C. Gonzalez & R. E. Woods

37 Histogram Specification Example 3. Match the PDFs to get the transformation Discrete CDF of input histogram Input CDF Discrete CDF Discrete CDF desired Pixel CDF Discrete CDF Discrete CDF of input histogram Input CDF input CDF desired New Value 0 1 (1.33) 1 (1.05) (3.08) 2 (2.45) (4.55) 5 (4.55) (5.67) 6 (5.95) (6.23) 6 (5.95) (6.65) 7 (7.00) (6.86) 7 (7.00) (7.00) 7 (7.00) 7

38 Histogram Specification Example 4. Put mapping in a LUT Input Transformed Value 0 --> > > > > > > > 7

39 MATLAB: Histogram Equalization HISTOGRAM EQUALIZATION IN MATLAB >> f=imread( fig10.10(a).jpg ); % load in figure 3.15(a) >> imshow(f) % show image in a window >> figure, imhist(f) % show histogram in a new window % This histogram is not normalized. >> ylim( auto ) % set histogram tick marks and % axis limits automatically >> g=histeq(f,256); % Create histogram equalized image % you can also do this with % the cumsum function >> hnorm=imhist(f)./numel(f); % compute normalized histogram >> figure, imshow(g) % show this figure in a new window >> figure, imhist(g) % generate a histogram of % the equalized image >> ylim( auto ) % set limits again SEE GWE, Section for a discussion of histogram specification using MATLAB

40 MATLAB: Useful Transformations THRESHOLDING >> f=imread( fig10.10(a).jpg ); % load in an image >> g=(f>100)*255 % show figure in a new window (f>100) evaluates to 1 (true) if the pixel is >100, or 0 (false) if the pixel is 100 Multiplying by 255 is necessary to have a uint8 255 level image. ROTATION >> f=imread( fig10.10(a).jpg ); % load in an image >> g=imrotate(f,angle, method ); % rotate it by angle bilinear %Supports methods: nearest, bilinear, bicubic

41 MATLAB: Getting Picture Coordinates INPUT CURSOR VALUES %(Very useful for specifying reference points in an image) >> [x,y]=ginput %Displays the graph window, displays a cross hair. Will input %coordinate pairs from the graph window until you type return. %Position cursor using mouse and click to input each coordinate pair. x = y = >> [x,y]=ginput(n) Will input n coordinate pairs until you press return

42 Examples of Point Processing - gamma - brightness original + brightness + gamma histogram mod - contrast original + contrast histogram EQ

43 Histogram

44 Histogram Equalization

45 Histogram Specification Manually specified histogram h (1) CDF s k =G(z q ) corresponding to h (2) Inverse transform z=g -1 (s k ) for intensity mapping Resulting histogram

46 Local Histogram Operations

47 Local Histogram Operations g( x, y)= ( ) if m S xy k 0 m g AND k 1 G Sxy k 2 G ( ) otherwise Ei f x, y f x, y compare to global mean m G and standard deviation G

48 Spatial Neighborhoods

49 Spatial Filtering Let I and J be images such that J = T [I]. T [ ] represents a transformation, such that, J ( r, c) = T[]( I r, c) = f I( u, v) u { r s,..., r,... r + s}, v { c d,..., c,... c + d} ({ }). That is, the value of the transformed image, J, at pixel location (r,c) is a function of the values of the original image, I, in a 2s+1 2d+1 rectangular neighborhood centered on pixel location (r,c).

50 Moving Windows The value, J(r,c) = T[I](r,c), is a function of a rectangular neighborhood centered on pixel location (r,c) in I. There is a different neighborhood for each pixel location, but if the dimensions of the neighbor-hood are the same for each location, then trans-form T is sometimes called a moving window transform.

51 Spatial Filtering 2002 R. C. Gonzalez & R. E. Woods

52 Moving-Window Transformations Neutral Buoyancy Facility at NASA Johnson Space Center We ll take a section of this image to demonstrate the MWT photo: R.A.Peters II, 1999