**Abstract**:

**Abstract**
**In almost all of the recent vision experiments, stimuli are controlled via computers and presented on display devices such as cathode ray tubes (CRTs). Display characterization is a necessary procedure for such computer-aided vision experiments. The standard display characterization called “gamma correction” and the following linear color transformation procedure are established for CRT displays and widely used in the current vision science field. However, the standard two-step procedure is based on the internal model of CRT display devices, and there is no guarantee as to whether the method is applicable to the other types of display devices such as liquid crystal display and digital light processing. We therefore tested the applicability of the standard method to these kinds of new devices and found that the standard method was not valid for these new devices. To overcome this problem, we provide several novel approaches for vision experiments to characterize display devices, based on linear, nonlinear, and hybrid search algorithms. These approaches never assume any internal models of display devices and will therefore be applicable to any display type. The evaluations and comparisons of chromaticity estimation accuracies based on these new methods with those of the standard procedure proved that our proposed methods largely improved the calibration efficiencies for non-CRT devices. Our proposed methods, together with the standard one, have been implemented in a MATLAB-based integrated graphical user interface software named Mcalibrator2. This software can enhance the accuracy of vision experiments and enable more efficient display characterization procedures. The software is now available publicly for free.**

- These new methods have a much broader application because they do not presume the internal model of the display device and can handle nonlinearity of the device. Therefore, our methods are suitable to calibrate non-CRT devices such as LCD, DLP, and even future display devices such as the organic-EL (Electro-Luminescence) and Laser displays.
- These methods are most useful for experiments with visual stimuli that contain only a relatively small number of luminance and chromaticity.
- These methods are considerably robust against noise in the measured data since they estimate chromaticities based on only a small limited color spaces.
- These methods can achieve fast and accurate calibrations of target chromaticities within one to five repetitions of measurements (1-2 min for estimating a single chromaticity value).
- Even without an assumption of monotonicity in the display gamma functions, these methods can theoretically estimate valid video input values to display required chromaticities. This is because our methods adopt goal-seeking algorithms within a small limited color space.

- These methods cannot produce chromaticity values successively in real time, whereas standard procedures can estimate the required chromaticities very quickly based on simple linear transformations, once a global color transformation matrix has been acquired.
- These methods cannot model RGB-phosphor cross-talks, though this is also true for the standard calibration procedure.
- These methods are unable to model quantization effects due to bit-depth, though this is also true for the standard calibration procedure.
- Though our methods are flexible, they may not be able to model DLP projectors when they use a RGBW, not RGB, color filter wheel.

*L*is the modeled luminance,

*gain*and

*offset*are constant variables to scale the magnitude and the section of the fitting curve,

*x*is video input value (0 ≦ × ≦ 1, digital),

*x*

_{0}is the starting point from which luminance is above zero, and

*γ*describes a nonlinear form of the typical gamma function. We then calculate the inverse function

*x*=

*f*

^{−1}(

*y*) of the fit for each of the RGB phosphors and get the adjusted RGB video input values as color lookup tables (CLUT) so that luminance increments follow a linear function against the adjusted video inputs (Figure 2c-e).

*L*is maximum luminance value of the target phosphor.

_{max}*XYZ*) for maximum RGB video inputs and create an array consisting of these values, such as, Here,

*rXYZ*,

*gXYZ*, and

*bXYZ*are row vectors of

*XYZ*values of each of the RGB phosphors, and

*pXYZ*is a 3 × 3 matrix. Here, we call this

*pXYZ*transformation matrix a “global” transformation matrix to differentiate our “local” linear transformation procedures described later. Then, the tristimulus values

*XYZ*for the

*rgb*video inputs are calculated as a linear sum of these values by

*rgb*video input values to produce the desired chromaticity () can be acquired by

*xyY_to_XYZ( )*is a function to convert CIE1931 xyY values to tristimulus values,

*XYZ*, and defined as where,

*x*,

*y*, and

*Y*correspond to CIE1931 xyY values and

*T*is the transposition of the matrix. We subtract zero-level

*XYZ*values (flare) in advance of calculation if the flare cannot be ignored (Figure 3; Equation 2).

*i*is an iterator,

*N*is the maximum number of iterations, and the estimations will stop when

*i*reaches the maximum number of iterations.

*T*, as

_{1}*T*=

_{1}*pXYZ*. Here,

^{−1}*pXYZ*is a global transformation matrix calculated by Equation 3.

^{−1}*rgb,*required to produce the target chromaticity

*wXYZ*, using

*T*and Equation 5. Then, measure CIE1931 xyY for

_{i}*rgb*and calculate the error matrix,

*errXYZ*, by subtracting

*wYXZ*from the actual measurement.

*S*, measure CIE1931 xyY values for randomly generated 18 RGB video inputs,

*sRGB*(3[RGB] × 18 matrix). Convert them to

*XYZ*values,

*sXYZ*(3 × 18 matrix). Then, using

*sXYZ*and

*sRGB*, estimate the next local color transformation matrix,

*T*, based on the least-squares estimation,

_{i+1}*rgb*, required to produce the target chromaticity,

*wXYZ*, using

*T*and Equation 5. Then, measure CIE1931 xyY for

_{i+1}*rgb*.

*RMSE*) between the target chromaticity and the actual measurement. Then add 1 to

*i*. If the error is smaller than the terminating condition that was originally set, or if

*i*reaches the number of maximum iterations, terminate the estimation and go to the next step. Otherwise, go to Step 2 and repeat the estimations in smaller space.

*rgb*with the best estimation from all the iterations and finish.

*i*is an iterator.

*i*≦ 3. Obtain the

*rgb*with the best estimation as an initial starting point of nonlinear estimation.

*rgb*; evaluate the error; and terminate the estimation.

*i*≦ 3. Obtain the

*rgb*values with the best estimation to use for the initial starting point of the line search.

*rgb*; evaluate the error; and terminate the estimation.

Abbreviation | Display device | Photometer | Results |

CRT1 | CRT display GLM-20E21 (Silicon Graphics, Fremont, CA) | Konica-Minolta, CS-100A | Figure 3, Figure 6, Figure 8, Figure 9, Figure 10, Figure 11, Table 2, Table 3, Table 4, Table 5 |

LCD1 | Thinkpad T61 laptop computer, LCD panel (Lenovo, Beijing, China) | ||

LCD2 | MultiSync PA241W LCD display (NEC, Tokyo, Japan) | ||

DLP | DLP projector U2-1130 (PLUS Vision, Tokyo, Japan) | ||

CRT2 | ViewSonic P225f (ViewSonic Corporation, Walnut, CA) | Admesy Brontes-LL | Figure 12, Figure 13, Table 6, Table 7 |

LCD3 | SyncMaster 913N (Samsung, Seoul, Korea) |

*RMSE*s for each of RGB phosphors separately (Figure 8). The CLUTs were generated based on inverse functions of the best GOG fits and used for later chromaticity estimations. Furthermore, the linearity of the luminance outputs against the video input values after gamma correction was tested separately for each of the RGB phosphors. The tests were performed by remeasuring luminance values for 18 video input values, which were not used in initial fits (equally spaced from 0.05 to 0.95), and fitting linear functions considering the flare at zero-level video input (Figure 9). These results were evaluated by

*RMSE*s (Tables 2 and 3).

Method | Phosphor | CRT1 | LCD1 | LCD2 | DLP |

GOG | Red | 0.0781 | 0.1069 | 0.0178 | 0.4064 |

Green | 0.0900 | 0.0747 | 0.0218 | 0.3606 | |

Blue | 0.0867 | 0.0576 | 0.0466 | 0.3925 | |

Mean | 0.0849 | 0.0797 | 0.0287 | 0.3865 | |

Cubic spline | Red | 0.0072 | 0.0002 | 0.0001 | 0.0067 |

Green | 0.0034 | 0.0001 | 0.0002 | 0.0001 | |

Blue | 0.0244 | 0.0001 | 0.0029 | 0.0086 | |

Mean | 0.0117 | 0.0001 | 0.0011 | 0.0051 | |

(RMSE) |

Method | Phosphor | CRT1 | LCD1 | LCD2 | DLP |

GOG | Red | 0.0386 | 0.1289 | 0.0168 | 0.1867 |

Green | 0.0428 | 0.0936 | 0.0096 | 0.1544 | |

Blue | 0.0232 | 0.0473 | 0.0130 | 0.1587 | |

Mean | 0.0349 | 0.0899 | 0.0131 | 0.1666 | |

Cubic spline | Red | 0.0237 | 0.0657 | 0.0243 | 0.0525 |

Green | 0.0270 | 0.0369 | 0.0195 | 0.0467 | |

Blue | 0.0149 | 0.0116 | 0.0117 | 0.0161 | |

Mean | 0.0219 | 0.0381 | 0.0185 | 0.0384 | |

(RMSE) |

*RMSE*s (Figures 8 and 9; Tables 2 and 3). The linearity after gamma correction was tested and evaluated using the same procedures as the standard GOG-based gamma-correction method. For all the later color estimation procedures, CLUTs generated by GOG or this cubic spline–based method were used in correcting video input values to ensure the linearity of the display input/output relationship.

*global*color transformation matrix) were done following Equations 3 to 6 for 50 chromaticities. These 50 chromaticities were randomly generated in CIE1931 xyY space once for each display device, with a restriction that they were within a triangle enclosed by x and y values of RGB phosphors (see Figure 10). This restriction means that the corresponding video input values for the target chromaticity fall from 0.0 to 1.0 and that the target chromaticity can theoretically be reproduced based on Equations 3 to 6. The actual CIE1931 xyY values for the estimated RGB video inputs based on Equation 5 were measured. The estimation accuracies were evaluated by

*RMSE*s and

*delta *E Lab*errors. Here,

*RMSE*s were calculated by the formula below after converting the measured error in the CIE1931 xyY space to percentage residuals because the scale of Y is relatively larger than x and y in the raw data.

*RMSE*s) to 1.0× errors during the estimation iterations. We set two termination conditions: when the

*RMSE*s between the estimation and the target came to less than 1 or the number of repetitions reached the maximum (five iterations).

*RMSE*s were calculated based on Equations 9 and 10. Thus, when a percentage error in each of the xyY values is 1.0 (%), the

*RMSE*is 1.73. The goal of the present study is to provide accuracy to within 1 of

*RMSE*. The criterion is relatively demanding, but our methods actually achieved this for non-CRT devices, as shown in the Results section.

*RMSE*s.

*RMSE*s for RGB phosphors were 1.59 times smaller for the CRT, 2.36 times for the LCD1, and 4.34 times for the DLP, whereas the standard GOG was 1.4 times better for the LCD2 (Table 3). One reason that we may have observed a reduction in linearization for the LCD2, despite better initial fittings, may be due to an overfitting of the cubic spline curves. However, as the absolute values of

*RMSE*s for the LCD2 were smaller than those for the other devices, we can conclude that both methods were successful for the LCD2.

- Method 1: GOG model-based gamma correction + linear color transformation
- Method 2: cubic spline–based gamma correction + linear color transformation
- Method 3: cubic spline–based gamma correction + recursive linear color estimation
- Method 4: cubic spline–based gamma correction + linear/nonlinear hybrid color estimation
- Method 5: cubic spline–based gamma correction + line search color estimation

*RMSE*s (

*RMSE*≤ 1.0 was the preset termination criterion) and

*delta *E Lab*errors of chromaticity estimations for 50 randomly generated CIE1931 xyY values. A mixed-design analysis of variance (ANOVA) on

*RMSE*s for 50 reproduced chromaticities found significant differences between estimation methods (

*F*(4, 196) = 1134.862

*, p*< 0.00001). A significant interaction was also observed for Display Device × Method

*F*(12, 784) = 133.364,

*p*< 0.00001), but the interactions were only for Display × The Standard Global Estimation Procedures (

*F*(3, 980) = 285.911,

*p*< 0.00001, for Method 1 and

*F*(3, 980) = 127.101,

*p*< 0.00001, for Method 2). Further multiple comparisons (corrected with Ryan's method,

*p*< 0.05) showed that all three of our methods (Methods 3, 4, and 5) significantly improved the estimation accuracies for all devices when compared with the standard method (Methods 1 and 2).

Method | RMSE | Delta *E Lab | ||||||

CRT1 | LCD1 | LCD2 | DLP | CRT1 | LCD1 | LCD2 | DLP | |

Method 1 | 2.7443 | 3.2804 | 9.6667 | 11.1526 | 4.7555 | 10.3887 | 2.6495 | 12.0693 |

Method 2 | 0.7411 | 1.4808 | 4.4507 | 4.9502 | 2.0338 | 10.7232 | 1.6921 | 1.9483 |

Method 3 | 0.3764 | 0.5988 | 0.9600 | 1.2907 | 0.9993 | 0.8304 | 0.6645 | 0.7955 |

Method 4 | 0.8005 | 1.2398 | 1.0467 | 1.9792 | 1.7840 | 0.8878 | 1.6584 | 0.8217 |

Method 5 | 0.5788 | 0.8360 | 0.7880 | 1.4404 | 1.3705 | 0.9237 | 1.1532 | 0.5180 |

*SD*s) and minimum/maximum errors across methods should be also important as well as averaged errors. As shown in Table 5, the smallest error for each device was obtained using a different method. Notably, our proposed methods could give better estimations with smaller errors, especially for non-CRT displays compared with the standard procedure. The different estimation accuracies observed for different displays likely derive from differences in the profiles of the display devices. Some of our methods such as recursive linear transformations are based only on linear transformation and are not suitable for displays with a nonlinear profile. In contrast, although the nonlinear or direct search methods can handle nonlinearity, they are not necessarily suitable for some linear display devices because they may overfit to the local values. We therefore need to select different methods for different display devices. Our software Mcalibrator2 can overcome this linearity/nonlinearity problem by preparing both linear and nonlinear direct search calibration algorithms.

Method | Delta *E Lab | |||||||||||

CRT1 | LCD1 | LCD2 | DLP | |||||||||

SD | Min | Max | SD | Min | Max | SD | Min | Max | SD | Min | Max | |

Method 1 | 2.0948 | 2.0371 | 14.6091 | 1.9862 | 5.8364 | 14.8696 | 1.9971 | 1.1320 | 15.5537 | 3.9332 | 6.7672 | 21.6162 |

Method 2 | 3.0835 | 0.5183 | 20.9798 | 2.4852 | 5.5208 | 15.2245 | 2.0760 | 0.5057 | 15.5822 | 0.8778 | 0.5560 | 3.8892 |

Method 3 | 2.8602 | 0.0591 | 19.2288 | 2.0356 | 0.0868 | 11.9246 | 1.8185 | 0.1027 | 13.1948 | 2.0172 | 0.0810 | 10.6824 |

Method 4 | 3.3935 | 0.1217 | 22.2754 | 2.5548 | 0.0629 | 17.2483 | 2.1303 | 0.0636 | 13.6405 | 1.7606 | 0.0827 | 10.0818 |

Method 5 | 3.2580 | 0.0511 | 22.0912 | 2.7172 | 0.0764 | 17.9090 | 2.1463 | 0.0818 | 14.4913 | 0.3629 | 0.0570 | 1.8337 |

*RMSE*s against the number of iterations of Method 3 and Method 4 for LCD1 and DLP devices. For LCD1, although all the chromaticities estimated by a global transformation method were above our predefined criteria (RMSE > 1.0, see Equations 9 and 10), 43 of 50 chromaticities converged to the termination criterion within three iterations. For DLP, after two iterations, 47 of 50 chromaticities reached the termination. When linear (Method 3, red line in Figure 11) and nonlinear (Method 4, blue line in Figure 11) search methods are compared, the linear method converged faster than the nonlinear procedures. Notably, the nonlinear method was more unstable; it could give better estimations for some chromaticities, as shown in Figure 11, whereas it also gave worse estimations for some chromaticities (for DLP, only 42 of 50 colors reached the termination conditions when the nonlinear method was applied). Thus, the best strategy would be that we should try both procedures and take the better one, if we can spend enough time for daily calibrations.

*eX*,

*eY*, and

*eZ*are residuals between the target CIE1931 xyY values and the actual measurements in the corresponding estimation step, and

*ss*is a coefficient that gradually decreases against the repetitions of estimations (from 2.0 [the first estimation] to 1.0 [fifth estimation] in the 0.2 step). Also note that luminance values of 50 CIE1931 xyY chromaticities estimated in this test were limited within 5 to 15 cd/m

^{2}. This is because we already found, through several previous tests, that chromaticity estimations with higher luminance values were considerably stable and good enough, and we could not see any clear differences between methods. We thus had to test efficiency in noisy conditions.

*RMSE*errors against the iterations of recursive linear transformations for 50 CIE1931 xyY values. Table 6 shows average

*RMSE*errors obtained from the random and grid-sampling procedures, together with Method 2 as a comparison. We found that both the random and grid-sampling procedures improved estimation accuracy compared with the standard global linear transformation method. Furthermore, although we performed the comparisons in noisy conditions, we did not find any differences between the random (red lines) and grid (blue lines) sampling procedures. A mixed-design ANOVA (Device [CRT2 and LCD3] × Method (random vs. grid) × Iteration [one to five steps]) with 50

*RMSE*samples showed significant differences between devices (

*F*(1, 98) = 9.338,

*p*< 0.029) and iteration (

*F*(4, 392) = 94.568,

*p*< 0.00001) and interactions of Device × Iteration (

*F*(4, 392) = 58.271,

*p*< 0.00001) but never found differences between sampling methods (

*F*(1, 98) = 0.045,

*p*= 0.83). The reason that both random and grid samplings worked effectively may be that 18 chromaticity values are large enough to estimate a 3 × 3 local color transformation matrix, even when it is selected randomly. It may be also possible that random and grid selections may never bias pooled data when they are performed in a fairly local and small color space. We can therefore conclude that the random sampling taken in our methods is efficient enough for estimating a local color transformation matrix.

Method | Delta *E Lab | |||||||||

CRT2 | LCD3 | |||||||||

RMSE | delta *E | SD | Min | Max | RMSE | delta *E | SD | Min | Max | |

Method 2 | 18.8312 | 4.7055 | 1.3082 | 2.4358 | 7.7162 | 8.3665 | 8.0636 | 3.2751 | 2.8745 | 18.0133 |

Method 3 (random sampling) | 2.0479 | 1.2180 | 0.9187 | 0.2226 | 4.2213 | 1.4851 | 1.2772 | 2.5188 | 0.0864 | 16.2654 |

Method 3 (grid sampling) | 2.0072 | 1.1629 | 0.7961 | 0.2075 | 3.6344 | 1.4731 | 1.2813 | 2.5754 | 0.0960 | 16.8452 |

*M*that maps

*xyY*to

*RGB*video input values. We have a current set of

*RGB*values (

*RGB*), and they produce a current measured output

_{i}*xyY*. Our goal is to obtain target

_{i}*xyY*values. Then, we can compute The linear model says that to correct for this error, we can add Unfortunately, as the present study has shown, the linear model is not exactly right, especially for non-CRT displays. However, this sort of linear residual adjustment may be good enough to head things in the right direction, especially if it is combined with a rate parameter

*C*to decrease and adjust

*RGB*. This sort of update might converge very quickly on good RGB values even based on the global transformation matrix,

_{delta}*M*. We thus investigated how accurately chromaticity values were reproduced by adjusting errors, following Equations 13 and 14 together with multiplying a coefficient

*C*(−3.0 <

*C*< 3.0) on the

*RGB*. Namely, Equation 14 was modified as Here,

_{delta}*C*was adjusted by a standard linear optimization procedure implemented in MATLAB. We also compared color reproduction accuracies when RGB phosphor errors were adjusted simultaneously (

*C*was a scalar) and when they were modified independently (

*C*was a 1 × 3 vector).

*t*

_{196}= 1.017,

*p*> 0.31, after Ryan's correction for multiple comparisons). The estimations became worse compared with the global transformation when RGB errors were adjusted independently (

*t*

_{196}= 14.5430,

*p*< 0.00001, with Ryan's correction).

Method | Delta *E Lab | |||||||||

CRT2 | LCD3 | |||||||||

RMSE | delta *E | SD | Min | Max | RMSE | delta *E | SD | Min | Max | |

Error adjustment, RGB simultaneously | 7.3554 | 3.6439 | 1.9828 | 0.7156 | 9.8965 | 6.9713 | 6.9742 | 3.6069 | 0.2723 | 17.6009 |

Error adjustment, RGB separately | 18.8312 | 4.7055 | 1.3082 | 2.4358 | 7.7162 | 8.3665 | 8.0636 | 3.2751 | 2.8745 | 18.0133 |

*RMSE*< 1.0; see Figure 11). This difference in the required number of iterations may be due to the difference in termination conditions between the two methods; the termination condition in the Olds et al. study was based on the residuals of multiple regressions, whereas our termination was set based on

*RMSEs*(see Equations 9 and 10). In addition, the most likely reason for the difference would be the differences between the numbers of samples used in estimation step. Their method first obtained seven local chromaticity values around the target CIE1931 xyY for each of the RGB phosphors (thus, 7 × 7 × 7 = 343 chromaticity values in total). Then, the whole of these values were input to multiple regression procedures. Thus, although their method gave fairly good estimations even with a single iteration, their method required establishing relatively large samples in an initial step. In contrast, our method measured only 18 chromaticity samples for each step. We showed that even with this small population of samples, our method gave considerably good results after three iterations of the estimations. Therefore, it is not valid simply to compare the number of iterations. Finally, although the procedure is slightly different, the present study clearly extended an idea of piecewise linearity of Olds et al. (1999) by applying the idea to non-CRT displays and exploring its efficiency in details, together with introducing novel linear/nonlinear hybrid search algorithms.

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