Digital Halftoning

Sasan Gooran

PhD Course May 2013

DIGITAL IMAGES (pixel based)

Scanning

Photo Digital image

ppi (pixels per inch): Number of samples per inch

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????

ppi = 72

ppi = 36

ppi = 18

DIGITAL IMAGES Memory

•  Grayscale 8 256 tones

•  RGB 3*8=24 256^3=16.7

bits/pixel

million colors

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)

Halftoning

DIGITAL HALFTONING

Prepress Halftoning Print Con-tone

Image Halftoned Image

DIGITAL HALFTONING Example

Periodic and clustered dots (AM)

DIGITAL HALFTONING Example

Non-periodic and dispersed dots (FM)

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)

HALFTONE CELL

Original image Halftoned image

Halftone cell

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)

RULE OF THUMB

lpisizeOriginalsDppi *2* ize esired=

Ex. A 10 x 15 cm2 photo that is supposed to be 20 x 30 cm2

when printed at 150 lpi has to be scanned with a ppi about 2*2*150 = 600.

HALFTONE CELL Micro dot

dpi: Number of micro dots per inch This halftone cell represents at most 82 + 1= 65 gray tones

HALFTONE CELL

Halftone cell Resolution: number of micro dots per inch (dpi)

Micro dot

In this case: 17 gray tones

Screen ruling: number of halftone cells per inch (lpi)

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

lpi & dpi

gray tones of n12

lpidpi umber=+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

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!

High lpi, few gray tones

Lower lpi, more gray tones

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

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

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

AM FM, 1st FM, 2nd

AM & FM Halftone

AM FM

FM Halftone, 1st and 2nd generation

First Second

Hybrid Halftoning

AM FM_1 FM_2

THRESHOLDING

⎩⎨⎧

<

≥=

),(),( if ,0),(),( if ,1

),(nmtnmgnmtnmg

nmb

g and b are the original and the halftoned image, respectively.

t is the threshold matrix.

THRESHOLDING

This threshold matrix represents 10 gray tones

0.6

0.1

1

0.3

0.2 0

Originalbild Rastrerad bildTröskelmatrisOriginal image Halftoned image Threshold matrix

THRESHOLD MATRIX Example: Line

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

THRESHOLD MATRIX Example: Spiral

1 2 3 4 12 13 14 5 11 16 15 6 10 9 8 7

THRESHOLD MATRIX Clustered & Dispersed, 45 degrees

14 12 13 16 19 21 20 17 5 4 3 10 28 29 30 23 6 1 2 11 27 32 31 22 9 7 8 15 24 26 25 18 19 21 20 17 14 12 13 16 28 29 30 23 5 4 3 10 27 32 31 22 6 1 2 11 24 26 25 18 9 7 8 15

1 30 8 28 2 29 7 27 17 9 24 16 18 10 23 15 5 25 3 32 6 26 4 31 21 13 19 11 22 14 20 12 2 29 7 27 1 30 8 28 18 10 23 15 17 9 24 16 6 26 4 31 5 25 3 32 22 14 20 12 21 13 19 11

Clustered Dispersed

TABLE HALFTONING

Original image Halftoned image

Mean

TABLE HALFTONING

Clustered Dispersed

FM HALFTONING Error Diffusion

Original image Halftoned image Error filter

Error Diffusion

The threshold value is 0.5

Suffers from artifacts, See specially the highlights and shadows and also the mid-tone regions

Error Diffusion

The threshold value is a random number between 0.25 and 0.75

Better?

Iterative Method Controlling Dot Placement (IMCDP)

•  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

Assumptions:

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

Original Image

IMCDP The impact of the placed dot is fed back to the original image by a filter

The next dot is placed where the modified image has its largest density value

Binary Image

Iterative Halftoning, IMCDP IMCDP Original

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

IMCDP(filter)

For halftoning of a constant image with a coverage of p% the size of the filter is decided by:

pa /100=

The size of the filter is (2a + 1) x (2a + 1) rounded

IMCDP(filter)

11 x 11 filter 21 x 21 filter

IMCDP

Models of Visual Perception

{ }1.1)114.0(exp)114.00192.0(6.2)( fffH −+=

f is the frequency in cycles/degree The spacing between the dots is given by:

πτ

1801)21arctan(21

RdRdf=== degrees

R is the printer resolution and d is the viewing distance.

Models of Visual Perception

Viewing distance, d = 30 inches

Printer resolution, R = 300 dpi

A simple Printer Model (Dot overlap Model)

α

β

γ T

0 0 0 0 0 0 0

0 0 1 1 0 1 0

0 1 0 0 0 0 0

0 0 0 0 0 0 0

0 β α α 2β α β

β 2α−γ 1 1 2α 1 α

α 1 2α−γ α 2β α β

β α β 0 0 0 0

p(i,j)bb(i,j)

⎩⎨⎧

=

=

−+=

0),(1),(1

),(321 jib

jibifif

fffjip

γβα

Least Square Model Based Algorithm

EYE MODEL

PRINTER MODEL EYE MODEL

g(Original)

b(Binary)

z

w

2,, )(∑∑ −=

i jjiji wzε 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.

Objective Quality Measures

Objective Quality Measure (Halftone Images)

•  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 …

Why difficult?

Objective Quality Measure (Halftone Images)

•  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 …

A number of criteria

A simple measure

2

,

)),(),((∑ −=ji

jigjibe

g is the original image and b is the resulting binary image

Which image b gives the lowest error e?

SNR (Signal-to-Noise ratio)

))),(),((

),((log10)(

,

2,

2

10 ∑

∑−

=

ji

ji

jibjig

jigdBSNR

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

Quantization Noise Spectrum (QNS)

),(),(),( jibjigjiq −=

2),( lkQ

The quantization noise is defined as:

The quantization noise spectrum (QNS) is defined as:

Q is the 2-dimensional Fourier transform of q

The smaller the quantization noise spectrum, the more similar b and g are.

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:

∑Ω

=2),( lkQe

is small

Ω denotes a low-pass region.

QNS (Example)

Error diffusion IMCDP

g = 1/32

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

QNS (Example)

Error diffusion IMCDP

QNS (Example)

Error diffusion IMCDP

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

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

Frequency Response

Original

ED (Floyd & Steinberg filter)

ED (Jarvis-Judice-Ninke filter)

IMCDP

The frequency is increased from left to right

Frequency Gain

Use the original image in the previous page as the input image and Compute the frequency gain:

in

out

II

fG =)(

Iout and Iin are the Fourier transform of the output and the input Image, respectively.

Desirable: G(f) is close to 1 at low frequencies.

Frequency Gain

ED (F & S) ED (J & J & N)

IMCDP

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

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

•  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

Original image Error diffusion

0.49 0.5