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The numerical stability of LUT-based color  transformations
 

The numerical stability of LUT-based color transformations

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    The numerical stability of LUT-based color  transformations The numerical stability of LUT-based color transformations Presentation Transcript

    • The numerical stability of LUT-based color transformations Giordano B. Beretta Print Production Automation Lab Hewlett-Packard Laboratories Palo Alto, California 1 April 2010 G. Beretta (HP Labs) speculative 1 April 2010 1 / 20
    • Disclaimer For the following I have neither data nor an authoritative reference This presentation is purely speculative G. Beretta (HP Labs) speculative 1 April 2010 2 / 20
    • The ideal black & white printer L* paper toner counts 0 31 63 95 127 159 191 213 255 G. Beretta (HP Labs) speculative 1 April 2010 3 / 20
    • The real black & white printer L* paper measure a te r pol inte toner measure counts 0 31 63 95 127 159 191 213 255 G. Beretta (HP Labs) speculative 1 April 2010 4 / 20
    • Halftoning Gray tones are interpolated through halftoning Simplest case: spatial (printer) or temporal (display) dithering Example: 8 × 8 cell for 32 gray values (Bayer) 1 17 5 21 2 18 6 22 25 9 29 13 26 10 30 14 7 23 3 19 8 24 4 20 31 15 27 11 32 16 28 12 2 18 6 22 1 17 5 21 26 10 30 14 25 9 29 13 8 24 4 20 7 23 3 19 32 16 28 12 31 15 27 11 To achieve a given gray level L = 1 , use the interpolation line to find the inverse of the number of pixels in the required dither matrix G. Beretta (HP Labs) speculative 1 April 2010 5 / 20
    • Determining the dither cell L* paper ℓ1 toner counts 0 31 63 95 127 159 191 213 255 (increments of 8) G. Beretta (HP Labs) speculative 1 April 2010 6 / 20
    • Printing is not linear Printers are not linear Dot gain, tribo-electric effects, etc. There is actually no simple model Solution: printer characterization: print swatches for the various counts measure each swatch invert the lookup table Question: How many measurements do we need? After printer linearization, simple linear interpolation is sufficient G. Beretta (HP Labs) speculative 1 April 2010 7 / 20
    • Piecewise linear interpolation L* paper measure measure measure measure toner measure counts 0 31 63 95 127 159 191 213 255 G. Beretta (HP Labs) speculative 1 April 2010 8 / 20
    • Error sources intra-instrument 8 7 inter-instrument 6 ICC maker 5 4 3 2 1 proof vs. press press drift color transform proofer drift G. Beretta (HP Labs) speculative 1 April 2010 9 / 20
    • Do not forget your error bars Printer characterization is a physics experiment Many sources of error (source: Ing. Rainer Wagner): short term measurement repeatability: ∆E ∈ [0.02, 0.34] long term measurement repeatability: ∆E ∈ [0.07, 0.62] 2 hour drift after calibration: ∆E ∈ [0.15, 2.04] difference between instruments: ∆E ∈ [1.38, 4.90] difference between sheets in an offset run: ∆E ∈ [0.26, 1.51] difference between proofs over days: ∆E ∈ [0.62, 1.85] maximal difference in 70% of an offset run: ∆E ∈ [3.20, 3.60] error introduced by separation software: ∆E ∈ [2.37, 4.82] difference between proof and print if technologies are different, ICC workflow: ∆E ∈ [2.52, 5.33] difference between proof and print if technologies and ICC producers are different: ∆E ∈ [3.08, 6.64] It is important to keep the error bars in mind The interpolated values will be in an interval G. Beretta (HP Labs) speculative 1 April 2010 10 / 20
    • Tolerance L* paper toner counts 0 31 63 95 127 159 191 213 255 Note the tolerances are not constant! G. Beretta (HP Labs) speculative 1 April 2010 11 / 20
    • How many measurements do we need? To avoid artifacts, tone reproduction must be strictly monotonic Naïve thought: because the printer is far from linear, the more measurements we perform, the better the LUT However: if the error bars overlap, monotonicity is no longer guaranteed increasing measurements can backfire to improve print quality, increase printer accuracy first, then more measurements are useful G. Beretta (HP Labs) speculative 1 April 2010 12 / 20
    • Loss of monotonicity L* paper non-monotonic non-monotonic toner counts 0 31 63 95 127 159 191 213 255 On the coarser grid, this function is unchanged! G. Beretta (HP Labs) speculative 1 April 2010 13 / 20
    • What are the consequences? Are too many measurements damaging? In practice, the artifacts from loss of monotonicity are small relative to the printer instability, so they do not make things much worse There is no change on the coarser grid There is no benefit having more measurements when monotonicity is lost G. Beretta (HP Labs) speculative 1 April 2010 14 / 20
    • Possible experiment 1 Determine the tolerance for an actual monochrome printer and calculate the recommended maximal number of measurements for strict monotonicity G. Beretta (HP Labs) speculative 1 April 2010 15 / 20
    • Possible experiment 2 Separations introduce additional variance because the mapping is non-injective (different ink combinations have the same CIELAB value) Example: add a gray ink Typically, in the mid-tones there are multiple options for the separations (Marc Mahy patent) Determine the maximum number of meaningful measurements when a gray separation is added G. Beretta (HP Labs) speculative 1 April 2010 16 / 20
    • Possible experiment 3 Consider a gray image printed on a color printer Example: add cyan, magenta, and yellow inks Printer characterization is iterative 1 perform gray balancing 2 determine 3-dimensional lookup table for color correction 3 rebalance the grays 4 determine a new color correction table 5 usually print quality does not improve after 2 iterations Determine the maximum number of measurements for which the tone scale is monotonic when a color printer is used for grayscale printing G. Beretta (HP Labs) speculative 1 April 2010 17 / 20
    • From grayscale to full color Tessellation of the printer’s color space Failure of monotonicity becomes incorrect tessellation in one or more dimensions a vertex has a non-monotonic coordinate value and the tetrahedron is “is folded over” there can be holes tetrahedra can intersect How is the argument scaled from grayscale to full color? G. Beretta (HP Labs) speculative 1 April 2010 18 / 20
    • Scaling to n inks There is order only in a 1-dimensional space There is no order in a higher dimensional space Heuristic method: print thousands of color scales and examine them for transposed colors More formally: consider a sequence or family F = (xi )i∈I where the xi are byte counts in n dimensions (separations) consider the set {Fj } of all families that are monotonic in all dimensions let T be the color transformation then the condition is that all T (Fj ) must be monotonic G. Beretta (HP Labs) speculative 1 April 2010 19 / 20
    • My ignorance I do not remember the math of all this An intelligent paper should use a theorem of the underlying math to declare a corollary that proves something non-obvious that cannot be gleaned from studying the tessellation problem by itself An opportunity for a small investigation G. Beretta (HP Labs) speculative 1 April 2010 20 / 20