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r2290 r2291 40 40 \begin{abstract} 41 41 In this contribution we introduce a littleknown property of error diffusion 42 halftoning algorithms which we call error diffusion displacement.42 halftoning algorithms which we call {\it error diffusion displacement}. 43 43 By accounting for the inherent subpixel displacement caused by the error 44 44 propagation, we correct an important flaw in most metrics used to assess the … … 65 65 address the problem of colour reduction. Comparing two algorithms in terms of 66 66 speed or memory usage is often straightforward, but how exactly a halftoning 67 algorithm performs in terms of quality is a far more complex issue, as it68 highlydepends on the display device and the inner workings of the human eye.67 algorithm performs qualitywise is a far more complex issue, as it highly 68 depends on the display device and the inner workings of the human eye. 69 69 70 70 Though this document focuses on the particular case of bilevel halftoning, … … 82 82 yielding visually pleasing results. 83 83 84 Error diffusion dithering, introduced in 1976 by Floyd and Steinberg84 \medskip Error diffusion dithering, introduced in 1976 by Floyd and Steinberg 85 85 \cite{fstein}, tries to compensate for the thresholding error through the use 86 86 of feedback. Typically applied in raster scan order, it uses an error diffusion 87 matrix such as the following one :87 matrix such as the following one, where $x$ denotes the pixel being processed: 88 88 89 89 \[ \frac{1}{16} \left \begin{array}{ccc} … … 93 93 Though efforts have been made to make error diffusion parallelisable 94 94 \cite{parfstein}, it is generally considered more computationally expensive 95 than screening. 96 97 Modelbased halftoning is the third important algorithm category. It relies 98 on a model of the human visual system (HVS) and attempts to minimise an error 99 value based on that model. One such algorithm is direct binary seach (DBS) 100 \cite{allebach}, also referred to as leastsquares modelbased halftoning 95 than screening, but carefully chosen coefficients yield good visual results 96 \cite{kite}. 97 98 \medskip Modelbased halftoning is the third important algorithm category. It 99 relies on a model of the human visual system (HVS) and attempts to minimise 100 an error value based on that model. One such algorithm is direct binary seach 101 (DBS) \cite{allebach}, also referred to as leastsquares modelbased halftoning 101 102 (LSMB) \cite{lsmb}. 102 103 … … 108 109 DBS yields halftones of impressive quality. However, despite efforts to make 109 110 it more efficient \cite{bhatt}, it suffers from its large computational 110 requirements and error diffusion remains a widely used technique.111 requirements and error diffusion remains a more widely used technique. 111 112 112 113 \section{Error diffusion displacement} … … 121 122 bottomright of each pixel (Fig. \ref{fig:direction}), one may expect the 122 123 resulting image to be slightly translated. This expectation is confirmed 123 when alternatively viewing an error diffused image and the corresponding DBS 124 halftone.124 visually when rapidly switching between an error diffused image and the 125 corresponding DBS halftone. 125 126 126 127 \begin{figure} … … 134 135 This small translation is visually innocuous but we found that it means a lot 135 136 in terms of error computation. A common way to compute the error between an 136 image $h_{i,j}$ and the corresponding halftone $b_{i,j}$ is to compute the137 mean square error between modified versions of the images, in the form:137 image $h_{i,j}$ and the corresponding binary halftone $b_{i,j}$ is to compute 138 the mean square error between modified versions of the images, in the form: 138 139 139 140 \begin{equation} … … 144 145 convolution and $v$ is a model for the human visual system. 145 146 146 To compensate for the slight translation experienced in the halftone, we147 use thefollowing error metric instead:147 To compensate for the slight translation observed in the halftone, we use the 148 following error metric instead: 148 149 149 150 \begin{equation} … … 152 153 153 154 \noindent where $t_{dx,dy}$ is an operator which translates the image along the 154 $(dx,dy)$ vector. 155 $(dx,dy)$ vector. By design, $E_{0,0} = E$. 155 156 156 157 A simple example can be given using a Gaussian HVS model: … … 185 186 \end{figure} 186 187 187 For instance, a FloydSteinberg dither of \textit{Lena} ,with $\sigma = 1.2$188 yields a perpixel mean square error of $ 8.51\times10^{4}$. However, when189 taking the displacement into account, the error becomes $ 7.77\times10^{4}$190 for $(dx,dy) = (0.167709,0.299347)$. The new, corrected error is significantly 191 smaller,with the exact same input and output images.188 For instance, a FloydSteinberg dither of \textit{Lena} with $\sigma = 1.2$ 189 yields a perpixel mean square error of $3.67\times10^{4}$. However, when 190 taking the displacement into account, the error becomes $3.06\times10^{4}$ for 191 $(dx,dy) = (0.165,0.293)$. The new, corrected error is significantly smaller, 192 with the exact same input and output images. 192 193 193 194 Experiments show that the corrected error is always noticeably smaller except … … 196 197 vision sets and from the image board \textit{4chan}, providing a representative 197 198 sampling of the photographs, digital art and business graphics widely exchanged 198 on the Internet .199 200 In addition to the classical FloydSteinberg and Jarvis , Judice and Ninke201 kernels, we tested two serpentine error diffusion algorithms: Ostromoukhov's 202 simple error diffusion \cite{ostromoukhov}, which uses a variable coefficient 203 kernel,and Wong and Allebach's optimum error diffusion kernel \cite{wong}.199 on the Internet nowadays \cite{4chan}. 200 201 In addition to the classical FloydSteinberg and JarvisJudiceNinke kernels, 202 we tested two serpentine error diffusion algorithms: Ostromoukhov's simple 203 error diffusion \cite{ostromoukhov}, which uses a variable coefficient kernel, 204 and Wong and Allebach's optimum error diffusion kernel \cite{wong}. 204 205 205 206 \begin{center} 206 \begin{tabular}{lll} 207 \hline 208 & $E$ & $E_{min}$ \\ \hline 209 raster FloydSteinberg & 0.00089705 & 0.000346514 \\ \hline 210 raster JaJuNi & 0.0020309 & 0.000692003 \\ \hline 211 Ostromoukhov & 0.00189721 & 0.00186343 \\ \hline 212 % raster optimum kernel & 0.00442951 & 0.00135092 \\ \hline 213 optimum kernel & 0.00146338 & 0.00136522 \\ 207 \begin{tabular}{lcc} 208 \hline 209 &~ $E\times10^4$ ~&~ $E_{min}\times10^4$ ~\\ \hline 210 ~raster FloydSteinberg ~&~ 3.7902 ~&~ 3.1914 ~\\ \hline 211 ~raster JaJuNi ~&~ 9.7013 ~&~ 6.6349 ~\\ \hline 212 ~Ostromoukhov ~&~ 4.6892 ~&~ 4.4783 ~\\ \hline 213 ~optimum kernel ~&~ 7.5209 ~&~ 6.5772 ~\\ 214 214 \hline 215 215 \end{tabular} … … 227 227 visual error measurement than $E(h,b)$. However, its major drawback is that it 228 228 is highly computationally expensive: for each image, the new $(dx,dy)$ values 229 need to be calculated to minimise the e nergyvalue.229 need to be calculated to minimise the error value. 230 230 231 231 Fortunately, we found that for a given raster or serpentine scan … … 271 271 272 272 \begin{center} 273 \begin{tabular}{l llll}274 \hline 275 & $E$ & $dx$ & $dy$ & $E_{fast}$\\ \hline276 raster FloydSteinberg & 0.00089705 & 0.16 & 0.28 & 0.00083502\\ \hline277 raster JaJuNi & 0.0020309 & 0.26 & 0.76 & 0.00192991\\ \hline278 Ostromoukhov & 0.00189721 & 0.00 & 0.19 & 0.00186839\\ \hline279 optimum kernel & 0.00146338 & 0.00 & 0.34 & 0.00138165\\273 \begin{tabular}{lcccc} 274 \hline 275 &~ $E\times10^4$ ~&~ $dx$ ~&~ $dy$ ~&~ $E_{fast}\times10^4$ ~\\ \hline 276 ~raster FloydSteinberg ~&~ 3.7902 ~&~ 0.16 ~&~ 0.28 ~&~ 3.3447 ~\\ \hline 277 ~raster JaJuNi ~&~ 9.7013 ~&~ 0.26 ~&~ 0.76 ~&~ 7.5891 ~\\ \hline 278 ~Ostromoukhov ~&~ 4.6892 ~&~ 0.00 ~&~ 0.19 ~&~ 4.6117 ~\\ \hline 279 ~optimum kernel ~&~ 7.5209 ~&~ 0.00 ~&~ 0.34 ~&~ 6.8233 ~\\ 280 280 \hline 281 281 \end{tabular} … … 291 291 diffusion kernels. According to the original authors, the coefficients were 292 292 found "mostly by trial and error" \cite{fstein}. With our improved metric, we 293 now have the tools to confirm or infirm Floyd and Steinberg's initial proposal.293 now have the tools to confirm or infirm Floyd and Steinberg's initial choice. 294 294 295 295 We chose to do an exhaustive study of every $\frac{1}{16}\{a,b,c,d\}$ integer 296 combination. We deliberately chose positive integers whose sum is 16. Error296 combination. We deliberately chose positive integers whose sum was 16: error 297 297 diffusion coefficients smaller than zero or adding up to more than 1 are known 298 to be unstable \cite{stability}, and diffusing less than 100\% of the error is 299 known to cause important error in shadow and highlight areas of the image. 300 301 First we studied all possible coefficients on a pool of 250 images with an 302 error metric $E$ based on a standard Gaussian HVS model. Since we are studying 303 algorithms on different images but error values are only meaningful for a given 304 image, we chose a Condorcet voting scheme to determine winners. $E_{min}$ is 305 only given here as an indication and had no role in the computation: 298 to be unstable \cite{stability}, and diffusing less than 100\% of the error 299 causes important loss of detail in the shadow and highlight areas of the image. 300 301 We studied all possible coefficients on a pool of 3,000 images with an error 302 metric $E$ based on a standard Gaussian HVS model. $E_{min}$ is only given here 303 as an indication and only $E$ was used to elect the best coefficients: 306 304 307 305 \begin{center} 308 306 \begin{tabular}{cccc} 309 307 \hline 310 rank & coefficients & $E$ & $E_{min}$\\ \hline311 1 & 8 3 5 0 & 0.00129563 & 0.000309993\\ \hline312 2 & 7 3 6 0 & 0.00131781 & 0.000313941\\ \hline313 3 & 9 3 4 0 & 0.00131115 & 0.000310815\\ \hline314 4 & 9 2 5 0 & 0.00132785 & 0.000322754\\ \hline315 5 & 8 4 4 0 & 0.0013117 & 0.00031749\\ \hline316 \dots & \dots & \dots & \dots\\308 ~ rank ~&~ coefficients ~&~ $E$ ~&~ $E_{min}$ ~\\ \hline 309 ~ 1 ~&~ 8 3 5 0 ~&~ 0.00129563 ~&~ 0.000309993 ~\\ \hline 310 ~ 2 ~&~ 7 3 6 0 ~&~ 0.00131781 ~&~ 0.000313941 ~\\ \hline 311 ~ 3 ~&~ 9 3 4 0 ~&~ 0.00131115 ~&~ 0.000310815 ~\\ \hline 312 ~ 4 ~&~ 9 2 5 0 ~&~ 0.00132785 ~&~ 0.000322754 ~\\ \hline 313 ~ 5 ~&~ 8 4 4 0 ~&~ 0.00131170 ~&~ 0.000317490 ~\\ \hline 314 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ 317 315 \hline 318 316 \end{tabular} … … 320 318 321 319 The exact same operation using $E_{min}$ as the decision variable yields very 322 different results. Again, $E$ is only given here as an indication:320 different results. Similarly, $E$ is only given here as an indication: 323 321 324 322 \begin{center} 325 323 \begin{tabular}{cccc} 326 324 \hline 327 rank & coefficients & $E_{min}$ & $E$\\ \hline328 1 & 7 3 5 1 & 0.000306251 & 0.00141414\\ \hline329 2 & 6 3 6 1 & 0.000325488 & 0.00145197\\ \hline330 3 & 8 3 4 1 & 0.000313537 & 0.00141632\\ \hline331 4 & 7 3 4 2 & 0.000336239 & 0.00156376\\ \hline332 5 & 6 4 5 1 & 0.000333702 & 0.00147671\\ \hline333 \dots & \dots & \dots & \dots\\325 ~ rank ~&~ coefficients ~&~ $E_{min}$ ~&~ $E$ ~\\ \hline 326 ~ 1 ~&~ 7 3 5 1 ~&~ 0.000306251 ~&~ 0.00141414 ~\\ \hline 327 ~ 2 ~&~ 6 3 6 1 ~&~ 0.000325488 ~&~ 0.00145197 ~\\ \hline 328 ~ 3 ~&~ 8 3 4 1 ~&~ 0.000313537 ~&~ 0.00141632 ~\\ \hline 329 ~ 4 ~&~ 7 3 4 2 ~&~ 0.000336239 ~&~ 0.00156376 ~\\ \hline 330 ~ 5 ~&~ 6 4 5 1 ~&~ 0.000333702 ~&~ 0.00147671 ~\\ \hline 331 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ 334 332 \hline 335 333 \end{tabular} 336 334 \end{center} 337 335 338 Our improved metric was able to confirm that the original FloydSteinberg 339 coefficients were indeed the best possible for raster scan. 336 Our improved metric allowed us to confirm that the original FloydSteinberg 337 coefficients were indeed amongst the best possible for raster scan. 338 More importantly, using $E$ as the decision variable may have elected 339 $\frac{1}{16}\{8,4,4,0\}$, which is in fact a poor choice. 340 340 341 341 For serpentine scan, however, our experiment suggests that … … 350 350 the optimum coefficients $\frac{1}{16}\{7,4,5,0\}$ that improve 351 351 on the standard FloydSteinberg coefficients in terms of visual 352 quality for a given HVS model}352 quality for the HVS model studied in section 3.} 353 353 \label{fig:lena7450} 354 354 \end{center} … … 365 365 work is only the beginning: future work may cover more complex HVS models, 366 366 for instance by taking into account the angular dependance of the human eye 367 \cite{sullivan}. And now that we have a proper metric, we plan to improve all 368 error diffusion methods that may require finetuning of their propagation 369 coefficients. 367 \cite{sullivan}. We plan to use our new metric to improve all error diffusion 368 methods that may require finetuning of their propagation coefficients. 370 369 371 370 % … … 401 400 SPIE 3648, 485494 (1999) 402 401 403 \bibitem[6]{ quality}402 \bibitem[6]{kite} 404 403 T. D. Kite, 405 404 \textit{Design and Quality Assessment of Forward and Inverse ErrorDiffusion … … 410 409 \bibitem[7]{halftoning} 411 410 R. Ulichney, 412 \textit{Digital Halftoning} 411 \textit{Digital Halftoning}. 413 412 MIT Press, 1987 414 413 415 414 \bibitem[8]{spacefilling} 416 415 L. Velho and J. Gomes, 417 \textit{Digital halftoning with spacefilling curves} 416 \textit{Digital halftoning with spacefilling curves}. 418 417 Computer Graphics (Proceedings of SIGGRAPH 91), 25(4):8190, 1991 419 418 420 419 \bibitem[9]{peano} 421 420 I.~H. Witten and R.~M. Neal, 422 \textit{Using peano curves for bilevel display of continuoustone images} 421 \textit{Using peano curves for bilevel display of continuoustone images}. 423 422 IEEE Computer Graphics \& Appl., 2:4752, 1982 424 423 425 424 \bibitem[10]{nasanen} 426 425 R. Nasanen, 427 \textit{Visibility of halftone dot textures} 426 \textit{Visibility of halftone dot textures}. 428 427 IEEE Trans. Syst. Man. Cyb., vol. 14, no. 6, pp. 920924, 1984 429 428 430 429 \bibitem[11]{allebach} 431 430 M. Analoui and J.~P. Allebach, 432 \textit{Modelbased halftoning using direct binary search} 431 \textit{Modelbased halftoning using direct binary search}. 433 432 Proc. of SPIE/IS\&T Symp. on Electronic Imaging Science and Tech., 434 433 February 1992, San Jose, CA, pp. 96108 … … 436 435 \bibitem[12]{mcnamara} 437 436 Ann McNamara, 438 \textit{Visual Perception in Realistic Image Synthesis} 437 \textit{Visual Perception in Realistic Image Synthesis}. 439 438 Computer Graphics Forum, vol. 20, no. 4, pp. 211224, 2001 440 439 441 440 \bibitem[13]{bhatt} 442 441 Bhatt \textit{et al.}, 443 \textit{Direct Binary Search with Adaptive Search and Swap} 442 \textit{Direct Binary Search with Adaptive Search and Swap}. 444 443 \url{http://www.ima.umn.edu/20042005/MM8.110.05/activities/WuChai/halftone.pdf} 445 444 … … 450 449 \bibitem[15]{wong} 451 450 P.~W. Wong and J.~P. Allebach, 452 \textit{Optimum errordiffusion kernel design} 451 \textit{Optimum errordiffusion kernel design}. 453 452 Proc. SPIE Vol. 3018, p. 236242, 1997 454 453 455 454 \bibitem[16]{ostromoukhov} 456 455 Victor Ostromoukhov, 457 \textit{A Simple and Efficient ErrorDiffusion Algorithm} 456 \textit{A Simple and Efficient ErrorDiffusion Algorithm}. 458 457 in Proceedings of SIGGRAPH 2001, in ACM Computer Graphics, Annual Conference 459 458 Series, pp. 567572, 2001 … … 461 460 \bibitem[17]{lsmb} 462 461 T.~N. Pappas and D.~L. Neuhoff, 463 \textit{Leastsquares modelba ed halftoning}462 \textit{Leastsquares modelbased halftoning}. 464 463 in Proc. SPIE, Human Vision, Visual Proc., and Digital Display III, San Jose, 465 464 CA, Feb. 1992, vol. 1666, pp. 165176 … … 467 466 \bibitem[18]{stability} 468 467 R. Eschbach, Z. Fan, K.~T. Knox and G. Marcu, 469 \textit{Threshold Modulation and Stability in Error Diffusion} 468 \textit{Threshold Modulation and Stability in Error Diffusion}. 470 469 in Signal Processing Magazine, IEEE, July 2003, vol. 20, issue 4, pp. 3950 471 470 472 \bibitem[19]{kite} 473 T.~D. Kite, 474 \textit{Design and quality assessment of forward and inverse error diffusion 475 halftoning algorithms} 476 Ph.~D. in Electrical Engineering (Image Processing), August 1998 477 478 \bibitem[20]{sullivan} 471 \bibitem[19]{sullivan} 479 472 J. Sullivan, R. Miller and G. Pios, 480 \textit{Image halftoning using a visual model in error diffusion} 473 \textit{Image halftoning using a visual model in error diffusion}. 481 474 J. Opt. Soc. Am. A, vol. 10, pp. 17141724, Aug. 1993 482 475
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