What is an image? Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 1. A digital image can be written as a matrix

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1 What is an image? Definition: An image is a 2-dimensional light intensity function, f(x,y), where x and y are spatial coordinates, and f at (x,y) is related to the brightness of the image at that point. Definition: A digital image is the representation of a continuous image f(x,y) by a 2-d array of discrete samples. The amplitude of each sample is quantized to be represented by a finite number of bits. Definition: Each element of the 2-d array of samples is called a pixel or pel (from picture element ) Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 1 A digital image can be written as a matrix f (0,0) f (0,1) L f (0, N 1) f (1,0) f (1,1) L f (1, N 1) f( x, y)= M M M f (L 1,0) f (L 1,1) L f (L 1, N 1) Note: For a color image, f(x,y) might be one of the components. Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 2

2 Image Size and Resolution 200x x100 50x50 25x25 These images were produced by simply picking every n-th sample horizontally and vertically and replicating that value nxn times. Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 3 Color Components Monochrome image R(x,y) = G(x,y) = B(x,y) Red R(x,y) Green G(x,y) Blue B(x,y) Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 4

3 Different numbers of gray levels Contouring Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 5 How many gray levels are required? Contouring is most visible for a ramp 32 levels 64 levels 128 levels 256 levels Digital images typically are quantized to 256 gray levels. Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 6

4 Storage requirements for digital images Image LxN pixels, 2 B gray levels, c color components Size = LxNxBxc Example: L=N=512, B=8, c=1 (i.e., monochrome) Size = 2,097,152 bits (or 256 kbyte) Example: LxN=1024x1280, B=8, c=3 (24 bit RGB image) Size = 31,457,280 bits (or 3.75 MByte) Much less with (lossy) compression! Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 7 Brightness discrimination experiment Can you see the circle? I + I I Note: I is luminance, measured in cd m 2 Visibility threshold I I const. 1K2% Weber fraction Weber s Law Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 8

5 Contrast with 8 Bits According to Weber s Law Assume that the luminance difference between two successive representative levels is just at visibility threshold I max = ( 1+ const. ) 255 I min For const. = 0.01L0.02 I max I min = 13L156 Typical display contrast Cathode ray tube 100:1 Print on paper 10:1 Suggests uniform quantization in the log(i) domain Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 9 Gamma characteristic Cathode ray tubes (CRT) are nonlinear Luminance I I ~ U γ γ = Voltage U, rep. level f Cameras contain γ -predistortion circuit U ~ I 1 γ Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 10

6 log vs. γ-predistortion U U ~ log(i) U ~ I 1 γ I max I min = 100 Similar enough for most practical applications I Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 11 Image Scaling Original image Scaled image f( x, y) a f( x,y) Scaling in the γ-domain is equivalent to scaling the linear luminance domain ( ) γ = a γ ( f( x, y) ) γ I ~ a f( x, y)... same effect as adjusting camera exposure time. Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 12

7 Adjusting γ Original image γ increased by 50% ( ) a ( f( x,y) ) γ with γ = 1.5 f x, y... same effect as using a different photographic film... Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 13 Photographic film density d d 0 Hurter & Driffield curve (H&D curve) for photographic negative toe slope -γ linear region shoulder log E E is exposure γ measures film contrast General purpose films: γ = High-contrast films: γ = Luminance I = I 0 10 d = I 0 10 γ ( log E+ d 0 ) = I 0 10 d0 E γ Lower speed films tend to have higher absolute γ Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 14

8 Changing gradation by γ-adjustment Scaled ramp 2γ 0 Original ramp γ 0 Scaling chosen to approximately preserve brightness of mid-gray Scaled ramp 0.5γ 0 Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 15 Histograms Distribution of gray-levels can be judged by measuring a histogram: For B-bit image, initialize 2 B counters with 0 Loop over all pixels x,y When encountering gray level f(x,y)=i, increment counter #ι Histogram can be interpreted as an estimate of the probability density function (pdf) of the underlying random process. You can also use fewer, larger bins to trade off amplitude resolution against sample size. Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 16

9 Example histogram #pixels gray level Cameraman image Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 17 Example histogram #pixels gray level Pout image Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 18

10 Histogram equalization Idea: find a non-linear transformation ( ) g = T f to be applied to each pixel of the input image f(x,y), such that a uniform distribution of gray levels in the entire range results for the output image g(x,y). Analyse ideal, continuous case first, assuming 0 f 1 0 g 1 T(f) is strictly monotonically increasing, i.e., there exists f = T 1 ( g) Goal: pdf p g (g) = const. over the range 0 g 1 Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 19 Histogram equalization for continuous case From basic probability theory Consider the transformation function Then... p g p g g = T f ()= g p f () f df dg ()= g p f () f df dg ()= p f ( α) f =T 1 ( g ) f dα 0 f 1 0 dg df = p f () f f =T 1 ( g ) = p f f () p f 1 f () f =T 1 ( g) = 1 0 g 1 Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 20

11 Histogram equalization for discrete case Now, f only assumes discrete amplitude values with probabilities f 0, f 1,L, f L 1 P 0 = n 0 n P 1 = n 1 n Discrete approximation of The resulting values g k are in the range [0,1] and need to be scaled and rounded appropriately. L g = T f g k = T( f k )= P i k i= 0 P L 1 = n L 1 n ()= p f ( α) 0 f dα Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 21 Histogram equalization example Original image Pout Pout after histogram equalization Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 22

12 Histogram equalization example Original image Pout... after histogram equalization #pixels #pixels gray level gray level Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 23 Histogram equalization example Original image Cameraman Cameraman after histogram equalization Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 24

13 Histogram equalization example Original image Cameraman... after histogram equalization #pixels #pixels gray level gray level Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 25 Histogram equalization example Original image Moon Moon after histogram equalization Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 26

14 Histogram equalization example Original image Moon... after histogram equalization #pixels #pixels gray level gray level Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 27 Luminance-based segmentation Holes could be filled by morphological image processing algorithms Original image Peter f(x,y) Thresholded Peter m(x,y) const. f (x,y) m(x,y) Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 28

15 Chroma key Color is more powerful for pixel-wise segmentation: 3-d vs. 1-d space Take picture in front of a blue screen (or green, or orange) Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 29 Soft chroma key Extract blueness for each pixel α α 1 α Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 30

16 Landsat image processing Original Landsat image false color picture out of bands 4,5,6 Water area segmented and enhanced to show sediments Source: US Geological Survey USGS, Bernd Girod: EE368 Digital Image Processing Pixel Operations no. 31

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