Numerical Stability of LUT-Based Color Transformations

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


The ideal black & white printer

L*
paper

toner counts
0 31 63 95 127 159 191 213 255


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


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

Determining the dither cell

L*
paper

ℓ1

toner counts
0 31 63 95 127 159 191 213 255
(increments of 8)


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


Piecewise linear interpolation

L*
paper measure
measure

measure
measure
toner measure counts
0 31 63 95 127 159 191 213 255


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

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


Tolerance

L*
paper

toner counts
0 31 63 95 127 159 191 213 255
Note the tolerances are not constant!


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


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!


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


Possible experiment 1

Determine the tolerance for an actual monochrome printer and
calculate the recommended maximal number of measurements for
strict monotonicity



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



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


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?


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


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


Numerical Stability of LUT-Based Color Transformations

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