Changeset 2293 for research/2008-displacement/paper/paper.tex
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research/2008-displacement/paper/paper.tex
r2292 r2293 102 102 (LSMB) \cite{lsmb}. 103 103 104 HVS models are usually low-pass filters. Nasanen \cite{nasanen}, Analoui 105 and Allebach \cite{allebach} found that using Gaussian models gave visually 106 pleasing results, an observation confirmed by independent visual perception 107 studies \cite{mcnamara}. 104 HVS models are usually low-pass filters. Nasanen \cite{nasanen}, Analoui and 105 Allebach found that using Gaussian models gave visually pleasing results, an 106 observation confirmed by independent visual perception studies \cite{mcnamara}. 108 107 109 108 DBS yields halftones of impressive quality. However, despite efforts to make … … 116 115 Boustrophedonic (serpentine) scanning has been shown to cause fewer visual 117 116 artifacts \cite{halftoning}, but other, more complex processing paths such as 118 Hilbert curves \cite{spacefilling} , \cite{peano} are seldom used as they do not119 im prove the image quality significantly.117 Hilbert curves \cite{spacefilling} are seldom used as they do not improve the 118 image quality significantly. 120 119 121 120 Intuitively, as the error is always propagated to the bottom-left or … … 169 168 Experiments show that for a given image and a given corresponding halftone, 170 169 $E_{dx,dy}$ has a local minimum almost always away from $(dx,dy) = (0,0)$ (Fig. 171 \ref{fig:lena- min}). Let $E$ be an error metric where this remains true. We170 \ref{fig:lena-values}). Let $E$ be an error metric where this remains true. We 172 171 call the local minimum $E_{min}$: 173 172 … … 177 176 178 177 \begin{figure} 179 \begin{center} 180 \input{lena-min} 181 \caption{Mean square error for the \textit{Lena} image. $v$ is a simple 182 $11\times11$ Gaussian convolution kernel with $\sigma = 1.2$ and 183 $(dx,dy)$ vary in $[-1,1]\times[-1,1]$.} 184 \label{fig:lena-min} 178 \begin{minipage}[c]{0.8\textwidth} 179 \input{lena-values} 180 \end{minipage} 181 \begin{center} 182 \caption{Mean square error for the \textit{Lena} image ($\times10^4$). $v$ 183 is a simple $11\times11$ Gaussian convolution kernel with $\sigma 184 = 1.2$ and $(dx,dy)$ vary in $[-1,1]\times[-1,1]$.} 185 \label{fig:lena-values} 185 186 \end{center} 186 187 \end{figure} … … 202 203 we tested two serpentine error diffusion algorithms: Ostromoukhov's simple 203 204 error diffusion \cite{ostromoukhov}, which uses a variable coefficient kernel, 204 and Wong and Allebach's optimum error diffusion kernel \cite{wong} .205 and Wong and Allebach's optimum error diffusion kernel \cite{wong}: 205 206 206 207 \begin{center} … … 268 269 the error computed at $(dx,dy)$. As $E_{fast}$ does not depend on the image, it 269 270 is a lot faster to compute than $E_{min}$, and as it is statistically closer to 270 $E_{min}$, we can expect it to be a better error estimation than $E$ .271 $E_{min}$, we can expect it to be a better error estimation than $E$: 271 272 272 273 \begin{center} 273 \begin{tabular}{|l|c|c|c|c| }274 \hline 275 &~ $E\times10^4$ ~&~ $ dx$ ~&~ $dy$ ~&~ $E_{fast}\times10^4$ ~\\ \hline276 ~raster Floyd-Steinberg ~&~ 3.7902 ~&~ 0.16 ~&~ 0.28 ~&~ 3.3447 ~\\ \hline277 ~raster Ja-Ju-Ni ~&~ 9.7013 ~&~ 0.26 ~&~ 0.76 ~&~ 7.5891 ~\\ \hline278 ~Ostromoukhov ~&~ 4.6892 ~&~ 0.00 ~&~ 0.19 ~&~ 4.6117 ~\\ \hline279 ~optimum kernel ~&~ 7.5209 ~&~ 0.00 ~&~ 0.34 ~&~ 6.8233 ~\\274 \begin{tabular}{|l|c|c|c|c|c|} 275 \hline 276 &~ $E\times10^4$ ~&~ $E_{min}\times10^4$ ~&~ $dx$ ~&~ $dy$ ~&~ $E_{fast}\times10^4$ ~\\ \hline 277 ~raster Floyd-Steinberg ~&~ 3.7902 ~&~ 3.1914 ~&~ 0.16 ~&~ 0.28 ~&~ 3.3447 ~\\ \hline 278 ~raster Ja-Ju-Ni ~&~ 9.7013 ~&~ 6.6349 ~&~ 0.26 ~&~ 0.76 ~&~ 7.5891 ~\\ \hline 279 ~Ostromoukhov ~&~ 4.6892 ~&~ 4.4783 ~&~ 0.00 ~&~ 0.19 ~&~ 4.6117 ~\\ \hline 280 ~optimum kernel ~&~ 7.5209 ~&~ 6.5772 ~&~ 0.00 ~&~ 0.34 ~&~ 6.8233 ~\\ 280 281 \hline 281 282 \end{tabular} … … 309 310 ~ 1 ~&~ 7 3 6 0 ~&~ 4.65512 ~&~ 3.94217 ~\\ \hline 310 311 ~ 2 ~&~ 8 3 5 0 ~&~ 4.65834 ~&~ 4.03699 ~\\ \hline 311 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\\hline312 \hline 312 313 ~ 5 ~&~ 7 3 5 1 ~&~ 4.68588 ~&~ 3.79556 ~\\ \hline 313 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\\hline314 \hline 314 315 ~ 18 ~&~ 6 3 5 2 ~&~ 4.91020 ~&~ 3.70465 ~\\ \hline 315 316 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ … … 327 328 ~ 1 ~&~ 6 3 5 2 ~&~ 3.70465 ~&~ 4.91020 ~\\ \hline 328 329 ~ 2 ~&~ 7 3 5 1 ~&~ 3.79556 ~&~ 4.68588 ~\\ \hline 329 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\\hline330 \hline 330 331 ~ 15 ~&~ 7 3 6 0 ~&~ 3.94217 ~&~ 4.65512 ~\\ \hline 331 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\\hline332 \hline 332 333 ~ 22 ~&~ 8 3 5 0 ~&~ 4.03699 ~&~ 4.65834 ~\\ \hline 333 334 ~ \dots ~&~ \dots ~&~ \dots ~&~ \dots ~\\ … … 339 340 coefficients were indeed amongst the best possible for raster scan. 340 341 More importantly, using $E$ as the decision variable may have elected 341 $\frac{1}{16}\{8,4,4,0\}$, which is in fact a poor choice. 342 $\frac{1}{16}\{7,3,6,0\}$ or $\frac{1}{16}\{8,3,5,0\}$, which are in fact poor 343 choices. 342 344 343 345 For serpentine scan, however, our experiment suggests that … … 348 350 \begin{figure} 349 351 \begin{center} 350 \includegraphics[width=0.8\textwidth]{lena.eps} 351 \caption{halftone of \textit{Lena} using serpentine error diffusion and 352 the optimum coefficients $\frac{1}{16}\{7,4,5,0\}$ that improve 353 on the standard Floyd-Steinberg coefficients in terms of visual 354 quality for the HVS model studied in section 3.} 352 \includegraphics[width=0.4\textwidth]{output-7-3-5-1-serp.eps} 353 ~ 354 \includegraphics[width=0.4\textwidth]{output-7-4-5-0-serp.eps} 355 \end{center} 356 \begin{center} 357 \includegraphics[width=0.4\textwidth]{crop-7-3-5-1-serp.eps} 358 ~ 359 \includegraphics[width=0.4\textwidth]{crop-7-4-5-0-serp.eps} 360 \caption{halftone of \textit{Lena} using serpentine error diffusion 361 (\textit{left}) and the optimum coefficients 362 $\frac{1}{16}\{7,4,5,0\}$ (\textit{right}) that improve on the 363 standard Floyd-Steinberg coefficients in terms of visual quality 364 for the HVS model used in section 3. The detailed area 365 (\textit{bottom}) shows fewer structure artifacts in the regions 366 with low contrast.} 355 367 \label{fig:lena7450} 356 368 \end{center} … … 419 431 Computer Graphics (Proceedings of SIGGRAPH 91), 25(4):81--90, 1991 420 432 421 \bibitem[9]{peano} 422 I.~H. Witten and R.~M. Neal, 423 \textit{Using peano curves for bilevel display of continuous-tone images}. 424 IEEE Computer Graphics \& Appl., 2:47--52, 1982 425 426 \bibitem[10]{nasanen} 433 \bibitem[9]{nasanen} 427 434 R. Nasanen, 428 435 \textit{Visibility of halftone dot textures}. 429 436 IEEE Trans. Syst. Man. Cyb., vol. 14, no. 6, pp. 920--924, 1984 430 437 431 \bibitem[1 1]{allebach}438 \bibitem[10]{allebach} 432 439 M. Analoui and J.~P. Allebach, 433 440 \textit{Model-based halftoning using direct binary search}. … … 435 442 February 1992, San Jose, CA, pp. 96--108 436 443 437 \bibitem[1 2]{mcnamara}444 \bibitem[11]{mcnamara} 438 445 Ann McNamara, 439 446 \textit{Visual Perception in Realistic Image Synthesis}. 440 447 Computer Graphics Forum, vol. 20, no. 4, pp. 211--224, 2001 441 448 442 \bibitem[1 3]{bhatt}449 \bibitem[12]{bhatt} 443 450 Bhatt \textit{et al.}, 444 451 \textit{Direct Binary Search with Adaptive Search and Swap}. 445 452 \url{http://www.ima.umn.edu/2004-2005/MM8.1-10.05/activities/Wu-Chai/halftone.pdf} 446 453 447 \bibitem[1 4]{4chan}454 \bibitem[13]{4chan} 448 455 moot, 449 456 \url{http://www.4chan.org/} 450 457 451 \bibitem[1 5]{wong}458 \bibitem[14]{wong} 452 459 P.~W. Wong and J.~P. Allebach, 453 460 \textit{Optimum error-diffusion kernel design}. 454 461 Proc. SPIE Vol. 3018, p. 236--242, 1997 455 462 456 \bibitem[1 6]{ostromoukhov}463 \bibitem[15]{ostromoukhov} 457 464 Victor Ostromoukhov, 458 465 \textit{A Simple and Efficient Error-Diffusion Algorithm}. … … 460 467 Series, pp. 567--572, 2001 461 468 462 \bibitem[1 7]{lsmb}469 \bibitem[16]{lsmb} 463 470 T.~N. Pappas and D.~L. Neuhoff, 464 471 \textit{Least-squares model-based halftoning}. … … 466 473 CA, Feb. 1992, vol. 1666, pp. 165--176 467 474 468 \bibitem[1 8]{stability}475 \bibitem[17]{stability} 469 476 R. Eschbach, Z. Fan, K.~T. Knox and G. Marcu, 470 477 \textit{Threshold Modulation and Stability in Error Diffusion}. 471 478 in Signal Processing Magazine, IEEE, July 2003, vol. 20, issue 4, pp. 39--50 472 479 473 \bibitem[1 9]{sullivan}480 \bibitem[18]{sullivan} 474 481 J. Sullivan, R. Miller and G. Pios, 475 482 \textit{Image halftoning using a visual model in error diffusion}. … … 479 486 480 487 \end{document} 481
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