Effect of image path bit depth on image quality Edgar Bernal*, Robert P. Loce Xerox Corporation, 800 Phillips Road, Webster, NY, USA 14580 ABSTRACT Digital Tone Reproduction Curves (TRCs) are applied to digital images for a variety of purposes including compensation for temporal engine drift, engine-to-engine color balancing, user preference, spatial nonuniformity, and gray balance. The introduction of one or more compensating TRCs can give rise to different types of image quality defects: Tonal errors occur when the printed value differs from the intended value; contours occur when the output step size is larger than the intended step size; pauses occur when two adjacent gray levels map to the same output level. Multiple-stage TRCs are implemented when compensation operations are performed independently, such as independent adjustment for temporal variation and user preference. Multiple TRCs are often implemented as independent operations to avoid complexity within an image path. The effect of each TRC cascades as an image passes through the image path. While the original image possesses given and assumed desirable quantization properties, the image passed through cascaded TRCs can possess tonal errors and gray level step sizes associated with a much lower bit-depth system. In the present study, we quantify errors (tonal errors and changes in gray-level step size) incurred by image paths with cascaded TRCs. We evaluate image paths at various bit depths. We consider real-life scenarios in which the local gray-level slope of cascaded compensating TRCs can implement an increase by as much as 200% and decrease by as much as 66%. Keywords: Bit depth, image quality, tone reproduction curve, tonal errors, contours, pauses, gradient
1. INTRODUCTION Gray-level mapping of digital pixel values is required in many print engines to account for temporal engine drift, engineto-engine color balancing, user preference, spatial nonuniformity, and gray balance. Gray-level mapping of pixel values is typically performed through the use of Tone Reproduction Curves (TRCs) [1], implemented as look-up tables. In the present study, we quantify errors (tonal errors and changes in step size) incurred by image paths that utilize single and multiple cascaded TRCs, and we examine those errors for image paths operating at different bit depths. Multiple-stage TRCs, rather than one composite TRC, may be implemented to ease the complexity of system design. To understand the system design issues, consider an image path with two types of TRCs. A user-preference TRC may be selected and applied within a digital front end (DFE) at the upstream portion of an image path. After application of that TRC, the image may be sent to a printer or archived and later sent to a printer. Downstream, a print engine receives the digital image and may apply one or more TRCs that are intended to compensate for print engine behavior, such as temporal drift or spatial nonuniformity. The upstream and downstream TRCs are applied at locations in the image path that may not have the ability to communicate each other’s actions. Also, the TRCs have very different temporal behavior, where the user-preference TRC can be constant for the duration of a print job, and print engine compensation TRCs can vary over long or short time intervals. In such a scenario, independent application of the TRCs is much simpler to implement than implementing one composite TRC. The bit-depth of an original image possesses given and assumed desirable quantization properties. Those properties change as the image is passed through cascaded TRCs. The processed image can possess tonal errors and gray level step sizes associated with a much lower bit-depth system. Thus, for an 8-bit printer that produces a largest ∆E from paper Dmax, the Dmax/255 gray level interval can be significantly and detrimentally increased in a printer with an image path utilizing multiple cascaded TRCs. In the present study, we quantify errors (tonal and changes in gray-level step size) incurred by image paths with cascaded TRCs. We evaluate image paths at various bit depths. We consider real-life
*
[email protected]; phone 1 585 422-0102; fax 1 585 422-6117; www.xerox.com
Image Quality and System Performance VI, edited by Susan P. Farnand, Frans Gaykema, Proc. of SPIE-IS&T Electronic Imaging, SPIE Vol. 7242, 72420J · © 2009 SPIE-IS&T CCC code: 0277-786X/09/$18 · doi: 10.1117/12.812881 SPIE-IS&T/ Vol. 7242 72420J-1
scenarios in which the local gray-level slope of cascaded compensating TRCs can implement an increase by as much as 200% and decrease by as much as 66%. The structure of this paper is as follows. In Sec. 2 we introduce the different types of response and reproduction curves used throughout the present study. Section 3 examines the effect of the engine drift and compensating TRCs on the print engine output. The different types of errors are quantified as a function of bit-depth of the image path. Finally, we present a discussion of the results and the conclusions.
2. BACKGROUND ON RESPONSE CURVES The halftone screen geometry and spot function determine the tone response of a particular engine. Specifically, the halftone fill order and the print engine response to it, define the raw Engine Response Curve (ERC). The response can be quantified with a metric such as CIELab ∆E, ∆L, or other related metric. In the present study we employ ∆E, and the ERC is obtained by measuring the ∆E-from-paper value corresponding to halftone level of the halftone fill order for the given halftone screen. Since the number of halftone levels in the fill order depends on the halftone screen, in an ERC it is convenient to plot ∆E-from-paper as a function of fractional digital area coverage, as illustrated in Fig.1.
ERC 100 90
∆E from Paper
80 70 60 50 40 30 20 10 0
0
20
40
60
80
100
Digital Area Coverage (%)
Fig. 1. Example of an Engine Response Curve (ERC).
Typically, the number of halftone fill order levels is different than the number of gray levels that the image path can utilize, which is dependent upon on the bit depth of the print engine: an n-bit print engine has 2n digital gray values. For instance, a “super cell” halftone containing many halftone subcells can have many thousands of unique digital area coverage patterns, while an 8-bit image path can utilize only 256 of those patterns. The Halftone Tone Response Curve (HTRC) is obtained by assigning a particular halftone fill, or digital area coverage pattern, to each of the digital levels of the image path. This assignment is typically made in a way that results in an approximately linear HTRC. Linearity is achieved by sampling the ∆E-from-paper axis in the ERC at 2n points on uniform intervals and finding the corresponding digital area coverage values for the selected ∆E-from-paper values as illustrated by Fig. 2.
Linearized HTRC 100
90
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∆ E from Paper
∆E from Paper
ERC 100
70 60 50 40 30
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Digital Area Coverage (%)
100
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Fig. 2. Building an 8-bit Halftone Tone Response Curve (HTRC) from the ERC of Fig. 1.
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The digital level assignment is performed at a point in time for a particular machine state. The ERC of a print engine is not perfectly stable over time. It can drift due to environmental conditions, the state of the print materials, and wear of printer components. The drift in ERC results in a drift of the HTRC. While the original HTRC had an intended linear shape and targeted ∆E values, the drifted HTRC can lose its linearity and its ∆E values will differ from the intended and assumed values. Figure 3 illustrates a drifted ERC and its effect on an HTRC.
Actual HTRC 100
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∆E from Paper
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Fig. 3. Effects of engine drift on HTRC: as the engine drifts from its initial state (solid ERC curve) to a different state (dotted ERC curve), the HTRC is no longer linear and the originally targeted ∆E values are not achieved. Engine drift can occur within a short time frame and building a new HTRC can be a laborious task. To avoid the cost and time of rebuilding an HTRC, print engine drift is often compensated by introducing a digital look-up table, or Tone Response Curve (TRC), whose function is to map the halftone response to a curve as linear (or as close to a given desired response) as possible. This is achieved by mapping the available 2n input digital levels to a set of 2n output digital levels that produce a response as close as possible to the desired response, as illustrated in Fig. 4.
Nonlinear HTRC
TRC
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∆ E from Paper
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Fig. 4. Tone Reproduction Curves (TRCs) are digital tone-to-tone mappings used to compensate for variations in the HTRC. In addition to compensating for print engine drift, TRCs may also be used for several other purposes. As mentioned above, TRCs may be used for engine-to-engine color balancing, user preference, spatial nonuniformity, and gray balance. Also mentioned above, to simplify systems design, TRCs may be implemented in multiple stages rather than as a single digital mapping that is a composite of multiple mapping functions. In a multi-stage implementation, the effect of the TRCs cascade as an image is passed through the mappings. A multi-stage implementation is illustrated in Fig. 5.
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Non-linear response
TRC2
TRC1
User preference
Linearization
HTRC ERC
Fig. 5. TRCs applied at different stages along an image path.
3. IMAGE QUALITY DEFECTS INTRODUCED BY THE USE OF TRCS As described in Sec. 2, TRCs are used to map the existing 2n digital levels to new values in order to best approximate a desired printed output response. The mapping only approximates the desired result because the choice of output digital levels is limited, and, as a consequence, the choice of output ∆E-from-paper levels is limited. The difference between the intended response curve and the one actually achieved can be understood and quantified according to several error metrics. To understand the errors, consider the TRC mapping of Fig. 6 and resulting output gray-level sweep of Fig. 7. Figure 6 illustrates the use of a TRC intended to compensate an HTRC drift in the highlight portion of the digital scale from the initial intended response (depicted with crosses) to a drifted response (depicted with circles). A TRC is introduced to approximate the intended behavior of the crosses with the available circle output values. A comparison between the desired and the obtained output sequence (vertical scale in between plots) illustrates the types of errors incurred by the use of a TRC.
HTRC (highlight portion)
Output digital levels (scaled by drifted ∆E steps) Compensating
TRC (highlight portion)
0
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Input digital levels (scaled by ideal ∆E steps)
Ideel HTRC Drifted HTRC
C
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Fig. 6. A digital TRC (right) is introduced to compensate for the drift of the HTRC (left). The vertical scale (middle) shows the difference between the intended output (crosses) and the obtained output (circles). A pictorial illustration of the types of errors incurred by the use of the TRC is shown in Fig. 7. There are three types of errors. Tonal errors occur when the output for a given input digital value is different than the intended or desired output for that level. In Fig. 6, tonal errors are indicated by non-horizontal arrows: the larger the skew of the arrow, the larger the tonal error. A tonal error is illustrated in Fig. 7 as an output gray level being significantly different than the ideal gray level. Pauses occur when two adjacent digital values are mapped to the same output value and may appear as tone reversals due to an effect related to Mach bands. There is also an associated information loss due to two input levels becoming indistinguishable. In Fig. 6, pauses occur when two arrows point at the same gray level. A pause is shown in two regions of Fig. 7. When such a gray-level sweep is printed at a suitable scale, the human visual system can cause the right side of the pause to be perceptibly lighter than the left side. Contours are caused by an increase in the output step
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size. In Fig. 6, the desired step size is given by the distance between the arrow tails while the obtained step size is given by the distance between the arrow tips.
Ideal HTRC
Drifted HTRC with slope gain Output of TRC + drifted HTRC
Tonal error
Output step size larger than original step size
Pause may appear as tone reversal
Fig. 7. Illustration of the types of errors incurred by the use of compensating TRCs. 3.1 Error bounds for print engine with drifted HTRC Tonal errors incurred by a printing system are often expressed by a value near the maximum value, such as the 95th percentile, 2σ, or 3σ errors. These near-maximum values are used to communicate tone reproduction performance because large errors can produce noticeable defects that can be more significant than the mean error of a printing system. The extreme maximum values are typically not used because tone reproduction studies are usually performed using measured print data, and print noise and measurement noise would have more of an effect on the extreme maximum rather than on near-maximum values. Since the present study is not encumbered by noise, we will typically quantify performance using largest ∆E step size values that could occur at some point in the tone scale. We will express this performance as a mathematical bound and as numeric values determined via simulation. The error bounds depend on the bit depth of the image path and the drift in slope of the print engine response. For a fixed image path bit-depth, larger drift of engine response slope increases the error bound. For a fixed drift in engine response slope, larger bit-depth reduces the error bounds. In this paper, we quantify engine drift by the magnitude of change in the local slope of a response curve and we use values that have been found in practice. We also employ a reference gray-level step size based on typical printer image paths. While these numbers employed here are not exact, they do simplify the discussion, and specific values for a given printer can easily be substituted into the analysis. For simplicity, and without loss of generality, we assume that a linearized engine produces 0∆E from paper at 0% area coverage, 100∆E from paper at 100% area coverage, and that its response is linear in between the two extreme points. This means that the reference step size for a nominal 8-bit engine is s0 = 100∆E / 255 ≈ 0.4∆E. We use that reference value as an indicator for the potential for contouring. For very low noise systems, a gray level step size a little larger than 0.4∆E may be perceptible as a contour, while printing systems with some noise could tolerate a larger step without producing a contour. As the bit depth of the print engine increases linearly, the nominal step size of the engine decreases exponentially, as illustrated in Fig. 8.
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Linearized HTRC 100
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Step Size vs Bit Depth for Linearized Engine
41
Linearized HTRC
Step Size Larger Slope Step Size Nominal Slope
40.2 40
Step Size Smaller Slope
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Bit Depth
Fig. 8. Nominal step size as a function of the bit depth of the print engine. To simplify the discussion we will perform the analysis for a drifted HTRC, and generally refer to the slope as the slope of drifted HTRC, which requires an equivalent but inverse TRC. These slopes can actually be in one or more of the types of TRCs indicated above. By examining a variety of applied TRCs, we have found that a typical range of local slope is from 33% to 300% of its nominal value. For such a range of drifted HTRC, the output step size s of an 8-bit engine without a compensating TRC is located in the range s0/3 = 0.13∆E ≤ s ≤ 1.20∆E = 3s0. As the bit depth of the print engine increases linearly, the range of the step size decreases exponentially. The exponential relationship can be understood by the difference in the linear trends in Fig. 9. Since the potential for contouring increases as the output step size increases, print engines with larger bit-depth image paths are more robust to contouring artifacts as drift occurs. Specifically, for a print engine with an n-bit image path without any compensating TRCs, the output step size s is the HTRC step size sH, which is bounded by
3s0 s0 / 3 n -8 ≤ s = s H ≤ n -8 2 2
(1)
Bound for Output Step Size [∆E]
Bound for Output Step Size vs. Slope and Bit Depth 1.4
8 bits 10 bits 12 bits Detection threshold for step size
1.2 1
Potential for contours
0.8 0.6 0.4
Low potential for contours
0.2 0
0
0.2
0.4
0.6
0.8
Slope of Drifted HTRC
1
1.2
nominal slope
Fig. 9. Output ∆E step size as a function of HTRC slope and bit depth. 3.2 Error bounds for engine with drifted HTRC and single compensating TRC When a digital TRC is introduced to compensate for the effects of engine drift –modeled here as a deviation in the local slope of the HTRC from its nominal value– there are two cases to consider: the case in which the HTRC step size sH is larger than the nominal step size s0 and the case in which sH is smaller than that amount. In the first case, as illustrated
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by Fig. 6, the output step size s is either sH or 0 (when a pause occurs). In the second case, as illustrated by Fig. 10, no pauses occur but the largest output step size is an integer multiple of sH.
HTRC (highlight
Input digital levels Output digital levels (scaled by ideal (scaled by drifted ∆E steps) portion) ∆E steps)
Compensating TRC (highlight portion)
0
IdeI HTRC
0
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Fig. 10. A digital TRC (right) is introduced to compensate for the drift of the HTRC (left) which results in a decrease in the local slope of the HTRC from its nominal slope.
0 '0
Figure 11(a) shows the upper bounds for the output step size s when sH ≥ s0 as a function of drift in HTRC slope sH and bit depth n. In that case, as illustrated by Fig. 6, the upper bound for the output step size is equal to the largest step size of the drifted HTRC, that is
s≤
sH 2n -8
(2)
e
Fig. 11(b) shows the upper bounds for the output step size s when sH < s0 as a function of drift in the slope sH and bitdepth n. In that case, as illustrated by Fig. 10, the upper bound for the output step size is a multiple of the largest step size of the drifted HTRC. More specifically, if the local HTRC slope sH is in the range
[
s0 s0 j +1 j
,
), where j = 1, 2, 3, …,
the output step size is upper bounded by
s≤
( j + 1) sH 2 n -8
Bound for Step Size vs. Slope and Bit Depth
Bound for Step Size vs. Slope and Bit Depth 8 bits 10 bits 12 bits Step size @ 8 bits without drift
0.9
1 0.8 0.6 0.4 0.2 0 0.2
(a)
1 8 bits 10 bits 12 bits Step size @ 8 bits without drift
Bound for Step Size [∆E]
Bound for Step Size [∆E]
1.4 1.2
(3)
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.4
0.6
0.8
1
Slope of Drifted HTRC
nominal slope
1.2
0 0.1
1.4
0.15
(b)
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0.25
0.3
0.35
Slope of Drifted HTRC
0.4
0.45
nominal slope
Fig. 11. Output (TRC + HTRC) step size bounds for the cases when (a) HTRC step size is larger than the nominal step size, and (b) HTRC step size is smaller than the nominal step size.
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As with any quantization process, increases in step size imply that the magnitude of the quantization error (in this case tonal error e) is upper bounded by one half of the step size [1]. More specifically, the bound for the largest difference in ∆E units between a printed color and an intended color is for a given HTRC step size sH and print engine bit-depth n is:
e≤
sH 2n -7
(4)
Equation 2 and the plots in Fig. 11 are bounds that specify how large the errors can be. However, these extreme values are not always attained. In order to determine actual maximum values, we performed a numerical analysis in which all possible levels from 0 to 255 were input into a TRC/HTRC combination for different values of HTRC slope and the resulting output step values and tonal errors were determined. Figure 12(a) shows the output step size as a function of input gray level for a system with a TRC that compensates for a 30% increase in the local slope of the HTRC. Figure 12(b) shows the shadow portion of Fig. 12(a) in detail to highlight the occurrence of pauses and the increase in the output step size with respect to the step size of the ideal HTRC.
Output Step Size
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Step Size TRC + Drifted HTRC Step Size Ideal HTRC Step Size Bound
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Fig. 12. Output step size for a TRC + HTRC tandem as a function of input gray level. Figure 13 shows the tonal error as a function of input gray level for a system with a TRC that compensates for a 30% increase in the local slope of the HTRC as well as the tonal error bound from Eq. 4. It can be seen that the tonal error bound is not attained.
Tonal Error Tonal Error TRC + Drifted HTRC Tonal Error Bound
Tonal Error [∆E]
0.3
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Fig. 13. Tonal error for a TRC + HTRC tandem as a function of input gray level. Figure 14 shows the comparison between the bounds illustrated in Fig. 11 and in Eq. 2 and the maximum values obtained through simulation for different values of drifted HTRC slope on an 8-bit print engine.
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1.4
Bounds and Actual Values (8 bits)
Bounds and Actual Values (8 bits) 0.8
Largest Tonal Error Tonal Error Bound Largest Step Size Step Size Bound
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Largest Tonal Error Tonal Error Bound Largest Step Size Step Size Bound
0.7 0.6
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∆E
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0 0.1
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Fig. 14. Comparison between theoretical bounds and numerical simulation values of tonal errors and step sizes for an 8-bit system: (a) HTRC step size is larger than nominal step size; (b) HTRC step size is smaller than nominal step size. 3.3 Error bounds for engine with drifted HTRC and multiple cascaded compensating TRCs Consider a tone compensation setting where two different TRCs are applied for two compensation purposes; examples of such compensations include: correction of temporal engine drift, engine-to-engine color balancing, user preference, and gray balance. For a variety of reasons, described above, it may be simpler to implement each compensation separately at different stages of the image path, as shown in Fig. 5. When TRCs are applied for different purposes at different stages of the image path, a given one of the TRC may not bring the TRC + HTRC response close to the overall desired system response. Consider for example the case depicted in Fig. 15 in which the HTRC (solid line) and the TRC1 + HTRC tandem behave as depicted in Fig. 15(a) (dash-dotted line). The compensating TRC2 from Fig. 15(b) is introduced to map the response of TRC1 + HTRC (dash-dotted line) to the desired linear response of Fig. 15(a) (dotted line).
Response at Each Stage
Compensating TRC 2
100 250
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Fig. 15. (a) HTRC and TRC1 + HTRC responses, which can differ significantly from the overall system desired response due to a specific goal of TRC1. (b) A compensating TRC2 used to map the TRC1 + HTRC response to the desired overall system response. Intuitively, as compensating TRCs are cascaded, two possible scenarios can occur regarding tonal or quantization errors: in the first scenario, the tonal error introduced by TRC1 has the opposite sign with respect to the tonal error introduced by TRC2. In that case, the overall tonal error magnitude is smaller than the largest tonal error magnitude. A less fortuitous scenario occurs when both tonal errors have the same sign and the resulting tonal error has a larger magnitude than either of the errors. The conditions in the latter scenario imply that as TRCs are cascaded, tonal error (and step size) bounds increase.
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Like in previous sub-sections, engine response drifts are modeled by changes in the local slope of the curves. In this case, however, there are two slopes to consider: the local HTRC slope sH and the local TRC1 + HTRC tandem slope, which we will denote by s1. As before, it is assumed that the local HTRC slope sH varies from 33% to 300% of its nominal value. With respect to s1, it is assumed that it varies from sH−s0 to sH+s0, where s0 is the nominal slope defined in Sec. 3.1. Simulations show that tonal errors may be as large as the largest step size of the local HTRC when two digital TRCs are cascaded under these assumptions. This corresponds to a 100% increase in the tonal error bound with respect to the single TRC case, as illustrated in Fig. 16. As compensating TRCs are cascaded, the bounds for the output step size also increase. Simulations have shown that step size bounds for two cascaded TRCs are twice as large with respect to the bounds for the single TRC case as illustrated in Fig. 17.
Tonal Error
1.8
Tonal Error TRC2 + TRC1 + Drifted HTRC
1.6
Tonal Error Bound for Single TRC Tonal Error Bound for Double TRC
Tonel Error [∆E]
1.4 1.2 1 0.8 0.6 0.4 0.2 0
0
50
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200
Digital Value
250
Fig. 16. Tonal error a TRC2 + TRC1 + HTRC tandem as a function of input gray level.
Output Step Size [ ∆E] Step Size TRC2 + TRC1 + Drifted HTRC Step Size Ideal HTRC Step Size Bound for Single TRC Step Size Bound for Double TRC
Output Step Size
3
2.5
2
1.5
1
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0
0
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250
Fig. 17. Output step size for a TRC2 + TRC1 + HTRC tandem as a function of input gray level. Figure 18(a) shows the tonal error bounds as a function of the bit-depth of the print engine, and as s1 varies from sH−s0 to sH+s0 with sH=3s0. Figure 18(b) shows the bounds for output step size as a function of the bit-depth of the print engine, and as s1 varies from sH−s0 to sH+s0 with sH=3s0. The dotted line indicates the threshold above which there is potential for contouring (horizontal dashed line). It can be seen that when two TRCs are cascaded, a bit depth of 12 bits is required to remain below the contouring under the tested set of operating conditions.
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Bounds for Tonal Error vs. TRC 1 + HTRC Slope and Bit Depth 1.4 1.2
8 bits 10 bits 12 bits
3.5
8 bits 10 bits 12 bits
3
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Bounds for Step Size vs. TRC 1 + HTRC Slope and Bit Depth B o u n d s fo r S te p S iz e [ ∆ E ]
B o u n d s fo r T o n a l E rro r [ ∆ E ]
1.6
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1.6
1
Fig. 18. (a) Tonal error bounds and (b) output step size bounds for a two-TRC system (TRC2 + TRC1 + HTRC) as a function of the bit-depth of the print engine, and as the TRC1 + HTRC tandem slope varies from sH−s0 to sH+s0 with sH=3s0.
4. CONCLUSIONS In the present study, we have quantified the bounds for tonal errors and changes in step size as a function of the bit depth of the image path. The use of compensating TRCs introduces tonal errors, pauses and increase in the output step size. We considered real-life scenarios in which the local slope of the engine response can increase by as much as 200% and decrease by as much as 66%. For a single TRC system, 10-bit image paths yield acceptable performance as the error bounds remain well below the perceptibility thresholds for contouring across the full set of operating conditions considered. As multiple TRCs are cascaded, errors accumulate and larger bit depths are necessary to maintain an acceptable performance. This paper only considers the color error perceptibility aspect of the use of TRCs. Additional aspects such as perceived visual noise and other spatial considerations are the subject of separate studies.
REFERENCES [1] [2]
Sharma, G., [Digital Color Imaging Handbook], CRC Press, Florida, 750-753 (2003). Proakis, J. and Manolakis, D., [Digital Signal Processing], Prentice Hall, New Jersey, 272-273 (1996).
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