Printer Model + Genetic Algorithm = Halftone Masks

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2 Printer Model + Genetic Algorithm = Halftone Masks Peter G. Anderson, Jonathan S. Arney, Sunadi Gunawan, Kenneth Stephens Laboratory for Applied Computing Rochester Institute of Technology Rochester, New York, USA April 20, 2004 pg. 1

3 Abstract We present a genetic algorithm that automatically generates halftone masks optimized for use in specific printing systems. The search is guided by a single figure of merit based on a model of the printing process and the human visual system. Our experiments show that genetic algorithms are effective in finding improved halftone masks and that two methods of reducing the search space to particular subsets of possible halftone masks greatly enhance the search performance. pg. 2

4 Outline 1. Genetic Algorithms (GAs) 2. Screen-independent printer model (our GA s fitness function) 3. Halftone Masks 4. Halftone Mask Fitness (from the printer model) 5. Permutation Masks (a) Full Permutations (b) Gear Wheels (c) Hybrid Masks 6. Evolutionary Improvement 7. Proof of the Pudding pg. 3

5 Genetic Algorithms Mimic Evolutionary Processes GAs use selective breeding to find good solutions to hard problems indirectly. A GA works with a population of problem solutions and iteratively removes poor ones by selection, combines the pieces of superior ones by selection and breeding, and occasionally slightly modifying some by mutation. pg. 4

6 What is a Genetic Algorithm? A GA requires a fitness criterion for the solutions. Our present fitness is the figure of merit for a permutation array when used as a halftoning mask. This fitness expresses how well a mask renders a constant gray image from the point of view of the human visual system and the ability of a given printer to render small dots. pg. 5

7 Genetic Algorithm Pseudocode Randomly initialize population[gen:=0] Repeat until a halting criterion is met: Compute the fitnesses of members of population[gen] Fill population[gen+1] with mutated children of selected parents in population[gen] gen := gen+1 pg. 6

8 We use Various Halting Criteria Halt after a sufficiently fit individual appears. Halt after a given amount of time has elapsed. Halt when progress appears stalled. pg. 7

9 Select Parent with 2-Tournaments Pick two arbitrary members in the population. The better one will be a parent. Alternatives (problematical): Use arbitrary members of the population s best half as parents. Select parents with probability proportional to their fitnesses. pg. 8

10 Children Are Parents Crossed Over The crossover operation passes traits of two parents onto two children. In case each parent has high fitness for a different reason, a child may combine those reasons (genes) and have higher fitness than both parents. (On the other hand...) Crossing over bit strings is easy (swap pieces). Crossing over permutations (our case) is trickier, but textbook. pg. 9

11 Mutation Promotes Continued Exploration Mutation keeps genetic diversity. Without mutation, we evolve a population of identical copies ( clones ) at a local optimum. Mutation must be done sparingly, or the GA degenerates into random search. pg. 10

12 Exploration & Exploitation GAs combine exploration of the search space, using a large population and mutation, with exploitation of a promising region of the search space, using selection and crossover. Extreme examples of exploration and exploitation are random search and hill-climbing, respectively. Neither of these techniques is well suited to hard problems with enormous search spaces. pg. 11

13 Halftoning Masks A halftoning mask, M, is an array of threshold values used to convert a continuous tone image, I, to a bi-level image B. Values of I and M are numbers in the same range, typically The values in B are 0 or 1. pg. 12

14 Halftoning with a Mask Treat the mask as covering the entire image by tiling. Mask halftoning converts I to B using M according to: I pq < M pq B pq = 0 I pq M pq B pq = 1 When B is printed on a high-resolution printer, it should look the same as the continuous tone image I. pg. 13

15 Halftone Mask Fitness Good halftoning masks do not introduce visible artifacts (unwanted textures), and use dot patterns that the printer can reproduce faithfully. One unwanted visual artifact is high power at low frequency from halftoning with white noise or halftoning with large clustered dots. pg. 14

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18 Good Masks Avoid Printer Instabilities Isolated pixels and small clusters are hard to print. Here are several nominally 25% coverage patterns. pg. 17

19 25% Coverage with 1 1, 2 2, 3 3, 4 4 Clusters pg. 18

20 25% Coverage with 4 4, 5 5, 7 7, 9 9 Clusters pg. 19

21 Our Printer Simulation Tool pg. 20

22 Fitness of a Mask = Visual Smoothness of Printed Gray Threshold a solid gray (= g) image with mask, M, to obtain bi-level image. Simulate the printer s spread and transfer functions. Add noise. Filter through a human visual system. The fitness of mask, M, is the smoothness of the final image. We use a different gray value g for each generation. pg. 21

23 The First Spread Function: Physical Dot Gain A convolution (smoothing or low pass filter) of the bi-level image to model: laser s response to ON & OFF, laser s beam profile, scatter of toner onto photoconductor, transfer of toner particles to paper, fusing (heat and pressure). The convolution kernel was determined experimentally. pg. 22

24 Nonlinear Transfer Function: Mass Coverage C(x, y) photoconductor voltage as function of exposure, toner sticking as function of voltage, toner to paper transfer as function of toner on photoconductor. The specific function was determined experimentally. Noise increases monotonically with the input to the transfer function. pg. 23

25 A Second Spread Function: Optical Dot Gain C T R T (x, y) = 10 ɛc(x,y) C(x, y) is the mass function. T (x, y) is the transmittance. R(x, y) is the reflectance. R(x, y) = T (x, y) [T (x, y) PSF (x, y)] PSF (x, y) is from the literature [Gustafson & Kruse]. pg. 24

26 The Human Visual System: dpi at 18 Another low pass filter to blur fine grained material. The final fitness criterion: minimize perceived reflectance RMS variation. σ 2 V = W (ν, ω)csf (ν, ω)dνdω W is the noise power spectrum of reflectance R. CSF is the human visual system s contrast sensitivity. (CSF comes from the literature.) pg. 25

27 GA Search for Good Permutation Masks A genetic algorithm will necessarily perform better if some knowledge of the problem is built into the search procedure other than just the black box of the fitness evaluation. We limit our search to H W halftone masks containing all the values 0 HW 1, i.e., permutation matrices. For most the present work, H W 64. Modify the comparison details of the halftoning threshold by scaling i.e., comparing 256 M pq and H W I pq. We know that certain families of permutations perform better than arbitrary ones, so we further restrict our searches, as discussed below. pg. 26

28 The Space of Unrestricted, Full Permutations The largest search space: (HW )! = 64 2! possible elements. Randomly chosen masks in this search space will behave like Roberts s Method, producing highly mottled bi-level images. pg. 27

29 Slow, Steady Progress with Unrestricted Permutations pg. 28

30 Gear Wheels For this type of mask we require that H and W have no common factors (H and W are relatively prime ), so we choose H = 63, W = 64. Imagine two meshing gear wheels with H and W teeth. Label the teeth on each wheel with a permutation of {0, 1, 2,, H 1} and {0, 1, 2,, W 1}, respectively. pg. 29

31 Gear Wheels, Continued The relatively-prime criterion guarantees that every pair of teeth from the two wheels will meet exactly once in W revolution of the H-tooth wheel (and H revolutions of the W -tooth wheel). Starting at time t = 0 and rotating the gears one tooth at every moment, we have the tooth pair (p t, q t ) meeting at time t. The permutation matrix is defined by M pt,q t = t. There are H! W! possible gear tooth permutation matrices. The size of this search space is H! W! pg. 30

32 Gear Wheels, Restated The above description of how to create a gear-wheel permutation matrix is not particularly suitable to our chosen programming language, MatLab. An equivalent algorithm works along the following lines. Start with a permutation matrix M constructed using two trivial permutations, (0, 1, 2, H 1) and (0, 1, 2, W 1). Then, rearrange the rows of M according to a chosen H-permutation and the columns of M using the chosen W -permutation. pg. 31

33 Trivial Permutations Gear Wheel Mask pg. 32

34 Better Progress with Gear Wheel Permutations pg. 33

35 Hybrid Masks (The Best) Combine two 8 8 permutation masks, K and L, to form a mask. The size of this search space is 64 2! Combine two 8 8 matrices, K and L like this: Lay out 8 8 copies of 64 K then add L pq to the (p, q)-copy of 64 K: 64 K + L K + L K + L K + L K + L K + L K + L K + L K + L 77 pg. 34

36 Best Progress with Hybrid Permutations pg. 35

37 Images from Evolved Masks pg. 36

38 Acknowledgment We gratefully acknowledge Hewlett Packard s financial support for this research. The computer programming was done by Computer Science graduate students: Samuel A. Inverso, Daniel R. Kunkle, Timothy M. Lebo, and Chadd Merrigan. pg. 37