Digital Halftoning. Sasan Gooran. PhD Course May 2013

Transcription

1 Digital Halftoning Sasan Gooran PhD Course May 2013

2 DIGITAL IMAGES (pixel based) Scanning Photo Digital image ppi (pixels per inch): Number of samples per inch

3 ppi (pixels per inch) ppi (scanning resolution): Number of samples per inch The higher ppi the better the representation of the con-tone image (Photo) Higher ppi requires more memory ppi should not be unncessarily high Choice of ppi????

4 ppi = 72

5 ppi = 36

6 ppi = 18

7 DIGITAL IMAGES Memory bits/pixel Grayscale tones RGB 3*8=24 256^3=16.7 million colors

8 DIGITAL HALFTONING Since most printing devices are not able to reproduce different shadows of gray the original digital image has to be transformed into an image containing white (0 s) and black (1 s)

9 Halftoning

10 DIGITAL HALFTONING Con-tone Halftoned Prepress Halftoning Print Image Image

11 DIGITAL HALFTONING Example Periodic and clustered dots (AM)

12 DIGITAL HALFTONING Example Non-periodic and dispersed dots (FM)

13 HALFTONE CELL Pixel (/a number of pixels) Halftone cell The fractional area covered by the ink corresponds to the value of the pixel (or the area)

14 HALFTONE CELL Halftone cell Original image Halftoned image

15 SCREEN RULING/ FREQUENCY lpi (lines per inch): Number of halftone cells per inch The higher lpi the better the print (?!) High lpi requires more stable print press etc. Does a higher lpi always lead to a better print? (to be answered later)

16 RULE OF THUMB ppi = D esired s ize*2 lpi Original size * Ex. A 10 x 15 cm 2 photo that is supposed to be 20 x 30 cm 2 when printed at 150 lpi has to be scanned with a ppi about 2*2*150 = 600.

17 Micro dot HALFTONE CELL dpi: Number of micro dots per inch This halftone cell represents at most = 65 gray tones

18 HALFTONE CELL Micro dot Screen ruling: number of halftone cells per inch (lpi) Halftone cell In this case: 17 gray tones Resolution: number of micro dots per inch (dpi)

19 lpi & dpi lpi: Number of halftone cells per inch A halftone cell consists of micro dots dpi: Number of micro dots per inch The ratio dpi/lpi decides the size of the halftone cell

20 lpi & dpi dpi lpi 2 + 1= number of gray tones

21 lpi & dpi (Example) Assume that dpi is fixed at 600 lpi = 150 only gives 17 gray tones lpi = 100 only gives 37 gray tones lpi = 50 gives 145 gray tones Does a higher lpi always lead to a better print? Not necessarily!

22 High lpi, few gray tones

23 Lower lpi, more gray tones

24 Low lpi, more gray tones but large halftone dots, (not satisfying)

25 AM & FM HALFTONING AM (Amplitude Modulated) The size of the dots is variable, their frequency is constant FM (Frequency Modulated) 1st generation The size of the dots is constant, their frequency varies FM (Frequency Modulated) 2nd generation The size of the dots and their frequency vary

26 AM & FM (1st & 2nd Generation) Halftone AM FM, 1st FM, 2nd

27 AM & FM Halftone AM FM

28 FM Halftone, 1st and 2nd generation First Second

29 Hybrid Halftoning FM_1 FM_2 AM

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34 THRESHOLDING b( m, n) = 1, 0, if if g( m, n) t( m, n) g( m, n) < t( m, n) g and b are the original and the halftoned image, respectively. t is the threshold matrix.

35 THRESHOLDING Original Originalbild image Threshold Tröskelmatris matrix Halftoned Rastrerad image bild This threshold matrix represents 10 gray tones

36 THRESHOLD MATRIX Example: Line

37 THRESHOLD MATRIX Example: Spiral

38 THRESHOLD MATRIX Clustered & Dispersed, 45 degrees Clustered Dispersed

39 TABLE HALFTONING Mean Original image Halftoned image

40 TABLE HALFTONING Clustered Dispersed

41 FM HALFTONING Error Diffusion Original image Error filter Halftoned image

42 Error Diffusion The threshold value is 0.5 Suffers from artifacts, See specially the highlights and shadows and also the mid-tone regions

43 Error Diffusion The threshold value is a random number between 0.25 and 0.75 Better?

44 Iterative Method Controlling Dot Placement (IMCDP) Assumptions: The original continuous-tone image is scaled between 0 and 1 0 and 1 represent white and black respectively The binary/halftoned image is totally white to begin with

45 IMCDP The mean of the density values of the original image corresponds to the area of the inked regions Original Image Binary Image The first dot is placed where the original image has its largest density value

46 IMCDP The impact of the placed dot is fed back to the original image by a filter Original Image Binary Image The next dot is placed where the modified image has its largest density value

47 Iterative Halftoning, IMCDP Original IMCDP

48 IMCDP(filter) A Gaussian filter is used Experiments show that an 11 x 11 Gaussian filter leads to satisfactory results in most cases The size of the filter should be changing for the light and dark parts of the original image

49 IMCDP(filter) For halftoning of a constant image with a coverage of p% the size of the filter is decided by: a = 100 / p The size of the filter is (2a + 1) x (2a + 1) rounded

50 IMCDP(filter) 11 x 11 filter 21 x 21 filter

51 IMCDP

52 Models of Visual Perception { 1.1 } (0.114 ) H( f ) = 2.6( f )exp f f is the frequency in cycles/ degree The spacing between the dots is given by: 1 f = τ = 2arctan( ) = degrees 2Rd Rd π R is the printer resolution and d is the viewing distance.

53 Models of Visual Perception Viewing distance, d = 30 inches Printer resolution, R = 300 dpi

54 A simple Printer Model (Dot overlap Model) β γ T α β α α 2β α β β 2 α γ 1 1 2α 1 α α 1 2α γ α 2β α β β α β b(i,j) b p(i,j) p( i, j) = 1 if b( i, f1α + f2β f3γ if b( i, j) j) = 1 = 0

55 Least Square Model Based Algorithm g(original) EYE MODEL z b(binary) PRINTER MODEL EYE MODEL w 2, j i, j ) ε = ( z i w i j The squared error One way: Start with an initial binary image b. For each pixel (i,j) find the binary value b(i,j) that minimizes ε.

56 Objective Quality Measures

57 Objective Quality Measure (Halftone Images) Why difficult? A method that works well for certain kinds of images, might produce results of low quality for other images The definition of a good halftoning method may vary from application to application There might be a number of requests that cannot be formulated by a simple objective measure And so on

58 Objective Quality Measure (Halftone Images) A number of criteria The original grayscale image and the binary image should be as similar as possible (How to define this similarity?) The black dots in the highlights (and the white dots in the shadows) should be placed homogeneously. In color case, the color should also be reproduced as accurate as possible And so on

59 A simple measure e = ( b( i, j) g( i, j)) 2 i, j g is the original image and b is the resulting binary image Which image b gives the lowest error e?

60 SNR (Signal-to-Noise ratio) g( i, j) 2 SNR( db) = 10 log 10 ( i, j ( g( i, j) b( i, j)) 2 ) i, j

61 SNR These kinds of measures are very easy to apply but they assume that the distortion is only caused by additive noise. These measures don t correlate well with our perceived visual quality

62 Quantization Noise Spectrum (QNS) The quantization noise is defined as: q( i, j) = g( i, j) b( i, j) The quantization noise spectrum (QNS) is defined as: Q( k, l ) 2 Q is the 2-dimensional Fourier transform of q The smaller the quantization noise spectrum, the more similar b and g are.

63 Similarity By similarity we mean the perceptual similarity. Since the eye acts as a low-pass filter it is desirable that the QNS is is small in the low pass region, that means: e = Ω Q( k, l ) 2 is small Ω denotes a low-pass region.

64 QNS (Example) g = 1/32 Error diffusion IMCDP

65 QNS The error e has been calculated for the images shown in previous slide when W is a circular low-pass region that occupy 12.5% of the image. The error is slightly smaller for the image halftoned by ED than the one by IMCDP!!!! Therefore: It is not only the magnitude of the QNS in the low-pass region that is important. The shape of QNS also plays a significant role. Desirable: A more or less circularly symmetric QNS with small magnitude in the low pass region

66 QNS (Example) Error diffusion IMCDP

67 QNS (Example) Error diffusion IMCDP

68 Homogeneousness One way of studying the characteristic of a halftoning method is to study the halftone patterns (tints) produced by the method. By a halftone pattern we mean the result of halftoning a constant image. We want the dots in the halftone pattern to be placed as homogeneously as possible over the entire image The set of distances from each dot to its closest dot gives a good picture of how close/far the dots in the halftone pattern are placed. The couple mean value and standard deviation of the data in this set can be used as a measure for homogeneousness of the pattern. (NOTE: Useful for very light and dark tones only) Desirable: Big mean value and small standard deviation

69 Homogeneousness 11 x 11 filter 21 x 21 filter (Mean value, standard deviation)=(7.28, 1.19) for the image to the left and (8.76, 0.82) for the image to the right

70 Frequency Response Original ED (Floyd & Steinberg filter) ED (Jarvis-Judice-Ninke filter) IMCDP The frequency is increased from left to right

71 Frequency Gain Use the original image in the previous page as the input image and Compute the frequency gain: G ( f ) = I I out in I out and I in are the Fourier transform of the output and the input Image, respectively. Desirable: G(f) is close to 1 at low frequencies.

72 Frequency Gain ED (F & S) ED (J & J & N) IMCDP

73 Frequency Gain From the previous diagrams we see that error diffusion methods have a tendency of high-pass filtering (edge enhancement) the original image The frequency gain for the image halftoned by IMCDP is very close to 1 at low frequencies The gain at higher frequencies are not of any particular interest because the eye is less sensitive there

74 Halftone Image Quality A method that works well for certain images, might produce results of low quality for other images. An image with two gray levels (0.49 in the left half and 0.5 in the right half) is halftoned by Floyd-Steinberg error diffusion Original image Error diffusion While the border between these two gray levels are hardly detected by the eye, it is emphasized by error diffusion because of a sudden change of pattern structure