Rúbricas de evaluación
Actividades 1 y 2: Lectura del cuento “El monstruo de los colores” y el emocionómetro.
For constructing ICC proles for this research, primarily 3-D to 3-D spline tting was the es- sential hurdle to overcome for building the CLUTs. Additionally, 1-D to 1-D tting (Fig.7.3 left) is useful for building the pre-linearisation tables per input channel, as cameras usually produce gamma corrected output. Therefore, we have tested the suitability for these given cases. For ease of analysing multi-dimensional cases, we have also conducted experiments with 2-D to 2-D tting experiments (Fig.7.3right), as it is much easier to perform a visual evaluation for these.
In the 1-D case on the left of Fig. 7.3, a linear spline curve is tted to a set of ve data points (red diamonds in the middle graph). The data points have been computed from a quadratic function with added random noise. The tting is performed in a multi grid fashion, starting out with a ve node mesh ranging from 0.0 to 255.0. After ve iterations, the spline reached its convergence criteria of a maximum iteration displacement for a node of 0.5 units per iteration step. The convergence criteria is plotted in the top graph, with the solid line representing the maximum node displacement of the iteration, and the dashed line representing the average displacement. The mesh was rened to a ner resolution (from 5 to 9 nodes) in the same range, by splitting each interval in half. This ner resolution converged after four iterations (9thtotal iteration). Each tted resolution spline is displayed
in the centre plot. Further renements were conducted through 17 to 33 nodes for the nal destination spline shape in the bottom plot. The convergence plot shows a typical shape in the plot with a logarithmic ordinate scale, as it can also be observed for higher dimensional problems.
The 2-D sample on the right hand side of Fig. 7.3 was created in a similar way. 200 randomly distributed data points were computed, and random noise was added. The data
0 5 10 15 20 10-2 10-1 100 101 102 103 c o n v e rg e n c e 0 50 100 150 200 250 300 50 0 50 100 150 200 250 300 fi tt e d r e s o lu ti o n s 0 50 100 150 200 250 300 50 0 50 100 150 200 250 300 fi n a l fi tt e d
}
}
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5 nodes 9 nodes 17 nodes 33 nodes
iterations model coordinates model coordinates wo rld co or d in ates wo rld co or d in ates un its o f di spl ace m en t
Figure 7.3: Spline interpolations in 1-D and 2-D. (left top) Maximum and average node displacement per iteration; (left middle) Converged tted 1-D spline curves for dierent resolutions; (left bottom) Converged 1-D spline curve at nal highest resolution. (right) Converged 2-D linear spline approximating a test function with common dicult features: rotated and shifted rectangular and uniform source data space; outer shape contains concave and convex curvatures; non-uniform node spacing.
points are rendered as little cubes, whereas the mesh nodes are rendered as little spheres. The function for computing the data points was chosen to test certain features: The rectangular and uniform source data space is rotated and shifted; the overall outer shape contains concave as well as convex curvature; and it features a non-uniform node spacing. These are all features that can potentially be found in ICC prole CLUTs. One can clearly see the reconstructed topography of the original function transformation. The major deviations from ideal ts are clearly due to the added (synthetic) noise of the data points, as well as to areas more sparsely sampled with data points. Balancing the relationship between the coecients of the data springs and the second dierence springs leads to a trade-o between a close t to the (noisy) data points and a smooth shape of the spline.
Fig.7.4illustrates the 3-D to 3-D spline tting for a CLUT. The data points are actual measurements from a captured test target, they are rendered in the less orderly distributed colourful cubes in the twoL∗a∗b∗diagrams. The two diagrams represent dierent stages in the multi grid tting sequence. To better show the distribution of the data points and the
7.4. RESULTS 95
Figure 7.4: 3-D rendering of the interpolation volumes for an ICC prole. The 285 coloured blocks represent the locations of the test chart's (Christophe Métairie Photographie DigitaL TargeT 003) colour patches inL∗a∗b∗ space, the coloured spheres are the LUT node posi- tions. (left) With93 interpolation nodes and (right) at a later rened mesh with173nodes
at a dierent angle.
shape of the distorted interpolation cubes, the two diagrams have been rendered at dierent angles inL∗a∗b∗ space. As it can be seen, the test target manufacturers use a distribution of colours to cover the neutral/grey axis as well as chains of colours of similar hue but with diering intensities and saturations from dark to light. The intention is to sample a volume in colour space, that is as big as possible, using only a relatively small number of colour patches.
In principle, we have discovered that the LUTs resulting from the spline tting algorithm, as discussed in this chapter, are suciently good. Actually, the algorithm can even be simplied by cutting some corners without a degradation in quality. The rst dierence spring coecients for example can be reduced to zero (0.0), eectively disabling them, and saving computational time. Furthermore, the problem can be satisfactorily solved as an LSF by discarding further iterations in the stages 2 and 3 for the NSF. Finally, the coupling was strong enough using the homogeneous second dierence spring terms only (Pxx, Pyy
andPzz), and disregarding the diagonal second dierence terms (Pxy,Pyz andPxz), again
speeding up calculations.
However, we could see that the tendency for divergence of the solver, as well as the convergence rate were improved by retaining weak rst dierence springs (kx,kyandkz), in
comparison to the data springs and second dierence springs (which were both approximately four times as high).
Figure 7.5: Rendering of standard deviation of data point measurements for a test chart in RGB colour space. Size of the spheroids proportional to measured standard deviations.
As information propagates much more quickly downstream with the iteration than upstream, experiments with dierent sweeping sequences have been conducted. Uni- directional sweeps, random node sweeps as well as alternating direction (forward/backward) sweeps were tried. As already suggested by Bone [62], the alternating direction showed to be most ecient in the cases present.