Encyclopedia of Color Science and Technology

2016 Edition
| Editors: Ming Ronnier Luo

Vora Value

Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-8071-7_48

Synonyms

Definition

The Vora Value, ν, measures the recording accuracy of a set of color filters [1]. The definition of ν takes into account the viewing and recording illuminants and the target color space. The original definition is independent of the data set to be recorded, does not take into account recording noise, and is based on the Euclidean distance measure in the target color space.

Let \( \mathbf{V}=\left[{\mathbf{v}}_1{\mathbf{v}}_2 \dots {\mathbf{v}}_{\mathrm{s}}\right] \) be an \( N\times s \) matrix whose columns are the vectors (or functions) that define the target color space for a specified viewing illuminant. If we assume that the vectors of V are independent, then they define a basis for the span of V, \( Span\Big({\left\{{\mathbf{v}}_i\right\}}_{i=1}^s \). To simplify notation, we denote the span of V by Rs(V), where the subscript denotes the dimension of the vector space. For example, v1, v2, v3 could represent the three CIE XYZ matching functions for viewing illuminant D65. Let the column vector f of length N represent the reflectance spectrum of the sample to be measured. Then the target color space is Rs(V), and the desired s-stimulus vector is
$$ \mathbf{t}={\mathbf{V}}^T\mathbf{f} $$
In our example, t would represent the CIE XYZ values of f for viewing illuminant D-65. The value of s need not be restricted to s = 3. For example, one may use s = 6 to measure the CIE XYZ values of f for viewing illuminants D-50 and D-65. For another example, s = 4 may be used to aid in color correction. For hyperspectral cameras and other multiband image recording systems, s may take on other, larger, values.
Let \( \boldsymbol{M} = \left[{\boldsymbol{m}}_1\;{\boldsymbol{m}}_2 \dots {\boldsymbol{m}}_r\right] \) be an \( N\times r \) matrix whose columns represent the effective recording filters (combination of the scanning filters and scanning illuminant). If recording noise can be ignored, the vector of recorded values is
$$ \mathbf{g}={\mathbf{M}}^T\mathbf{f} $$
It is typically physically impossible − or, at least, impractical − to construct vectors \( {\boldsymbol{m}}_i={\boldsymbol{v}}_i \) that match the ideal color matching functions exactly. Further, recording and viewing illuminants generally differ. In general, \( r\ne s \), \( \boldsymbol{M}\ne \boldsymbol{V} \), and \( \boldsymbol{t}\ne \boldsymbol{g} \). The vector g is typically color corrected to obtain the best linear estimate of t, \( {\boldsymbol{t}}_{\boldsymbol{est}}=\boldsymbol{A}\boldsymbol{g}, \) where color correction matrix A is chosen so as to minimize the mean square estimation error, \( E\left\{{\left\Vert \mathbf{t}-{\mathbf{t}}_{est}\right\Vert}^2\right\}, \) where \( E\left\{\cdot \right\} \) is the expected value operator. If the components of f are assumed to be independent, identically distributed with variance σ2, it can be shown that
$$ \mathbf{A}={\mathbf{V}}^{\mathrm{T}}\mathbf{M}{\left({\mathbf{M}}^{\mathrm{T}}\mathbf{M}\right)}^{-1} $$
and
$$ \begin{array}{l}E\left\{{\left\Vert \mathbf{t}-{\mathbf{t}}_{est}\right\Vert}^2\right\}=\hfill \\ {}{\upsigma}^2 Trace\left({\mathbf{V}}^{\mathrm{T}}\mathbf{V}-{\mathbf{V}}^{\mathrm{T}}\mathbf{M}{\left({\mathbf{M}}^{\mathrm{T}}\mathbf{M}\right)}^{-1}{\mathbf{M}}^{\mathrm{T}}\mathbf{V}\right)\hfill \end{array} $$
The maximum value of \( E\left\{{\left\Vert \mathbf{t}-{\mathbf{t}}_{est}\right\Vert}^2\right\} \) is σ2Trace(VTV).
The Vora Value, ν, is a measure of the effectiveness of the set of recording filters \( {{\left\{{\boldsymbol{m}}_i\right\}}_{i=1}}^r \), in recording the color of a sample in the target color space, Rs(V). It measures how accurately t can be estimated from g after linear color correction and is defined as
$$ \nu =\left({\sigma}^2 Trace\left({\mathbf{V}}^T\mathbf{V}\right)-E\left\{{\left\Vert \mathbf{t}-{\mathbf{t}}_{est}\right\Vert}^2\right\}\right)/{\sigma}^2 Trace\left({\boldsymbol{V}}^T\boldsymbol{V}\right) $$
Note that ν increases linearly with a decrease in mean square estimation error and ranges in value from zero to one, where the maximum value of one corresponds to perfect recording.

The definition of ν does not take into account recording noise, real data sets, or perceptual error measures, each of which would influence the error expression. For measures that take these into account, please see the section on Other Measures below. The original form of ν is simple and provides an effective rough estimate of the accuracy of the filter set.

Motivation: The Measure ν and the Q-Factor

To understand the properties of a good filter set, note that t consists of the inner products (dot products) of the vectors v1v2v s with the vector f [2]. Thus t depends only on the fundamental of f, which is defined as P V (f), its orthogonal projection onto the target color space. On the other hand, any components of f that are orthogonal to all of v1v2v s will not contribute to t. Thus, in particular, \( \mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) \) does not contribute to t.
$$ \begin{array}{l}\boldsymbol{t}={\boldsymbol{V}}^T\boldsymbol{f}={\boldsymbol{V}}^T{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) + {\boldsymbol{V}}^T\left(\mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right)\right) = {\boldsymbol{V}}^T{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right)\ \\ {}+\boldsymbol{0}\hfill \end{array} $$
Figure 1a illustrates this for s = 2. Vectors v1 and v2 define the target color space, the horizontal plane. The reflectance spectrum is represented by three-dimensional vector f. The vector t is completely determined by P V (f), the orthogonal projection of f onto the horizontal plane. The inner product of \( \mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) \) with either of v1 or v2 is zero and hence does not contribute to t.
Vora Value, Fig. 1

(a) f-P V (f) does not contribute to t. (b) m-P V (m) measures parts of f-P V (f)

Let m represent a single effective recording filter, obtained by combining the filter transmission function and the recording illuminant. Ignoring recording noise, the value m T f or the inner product of vectors m and f is the recorded value. The component of m that is orthogonal to the target color space, \( \boldsymbol{m}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{m}\right) \), contributes to inaccuracy in the recording of t because it measures energy in \( \mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) \). This is illustrated in Fig. 1b.
$$ \begin{array}{l}\boldsymbol{g}={\boldsymbol{m}}^T\boldsymbol{f} = \left({\boldsymbol{P}}_{\boldsymbol{V}}{\left(\boldsymbol{m}\right)}^T + {\left(\boldsymbol{m}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{m}\right)\right)}^T\right)\hfill \\ {}\left({\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) + \left(\mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right)\right)\right)\hfill \\ {}={\boldsymbol{P}}_{\boldsymbol{V}}{\left(\boldsymbol{m}\right)}^T{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right)\hfill \\ {} + {\left(\boldsymbol{m}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{m}\right)\right)}^T\left(\mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right)\right)\hfill \end{array} $$
Thus we have seen that only the fundamental of m, P V (m) is useful for recording accuracy. The Colorimetric Quality Factor (CQF or q-factor, defined by Neugebauer in 1956) of m measures the accuracy of m from this perspective, and is defined as the energy in the vector P V (m),as a fraction of the total energy in m.
$$ q\left(\boldsymbol{m}\right) = {\left\Vert {\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{m}\right)\right\Vert}^2/{\left\Vert \boldsymbol{m}\right\Vert}^2 $$
The value of q(m) is larger when P V (m) is a larger fraction of m. If \( \boldsymbol{m}\in {\mathbf{R}}_{\mathrm{s}}\left(\mathbf{V}\right) \), \( {\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{m}\right)=\boldsymbol{m} \), and m has a maximum q-factor of one.

While the CQF captures an important requirement of color scanning filters, recording filters are not used by themselves and it is necessary to evaluate filter sets as opposed to single filters. As in the case of a single filter, it is important for the filter set to measure the energy in P V (f). Thus the approach underlying the q-factor is useful. However, while a single filter can record energy only in one “direction” of the target color space (e.g., a single filter can measure one of the CIE tristimulus values) the purpose of a filter set is to measure energy in all the directions of the target color space. Thus, for example, if we wish to record CIEXYZ tristimulus values, the filter set should accurately measure X, Y, and Z values. An evaluation criterion for a set of recording filters should hence evaluate whether it measures all the energy in P V (f) − that is, whether it measures the energy in P V (f) in all directions of the target color space.

The generalization of the q-factor to the evaluation of a filter set is not trivial. To see this, consider, for example, a set of three perfect red filters, each with a q-factor of one. While these are clearly inadequate to measure tristimulus values, any function of individual q-factors will indicate that the set is perfect. This limitation is illustrated in Fig. 2a for s = 2. The two filters m1 and m2 are both perfect, lie in Rr(M) and have q-factors of 1. However, one is a multiple of the other. Hence the filter set is not capable of measuring that part of P V (f) that is orthogonal to the filters. It is hence not capable of measuring all the energy in P V (f).
Vora Value, Fig. 2

(a) Left: Two or more scanning filters should not record energy in a single direction. (b) Right: Differences between Rr(M) and Rs(V) lead to inaccurate recording

The general case is illustrated for s = r = 2 in Fig. 2a. The two recording filters m1 and m2 do not lie in the target color space, which is the horizontal plane, but in the shaded plane. The desired recorded values depend on P V (f). The values that are recorded in practice depend on P M (f).
$$ \begin{array}{ll}\boldsymbol{g}& ={\boldsymbol{M}}^T\boldsymbol{f}={\boldsymbol{M}}^T{\boldsymbol{P}}_M\left(\boldsymbol{f}\right) + {\boldsymbol{M}}^T\left(\boldsymbol{f}-{\boldsymbol{P}}_M\left(\boldsymbol{f}\right)\right)\hfill \\ {}& ={\boldsymbol{M}}^T{\boldsymbol{P}}_M\left(\boldsymbol{f}\right) + \boldsymbol{0}\hfill \end{array} $$
Hence the set of filters is perfect when \( {\mathbf{R}}_{\mathrm{r}}\left(\mathbf{M}\right)={\mathbf{R}}_{\mathrm{s}}\left(\mathbf{V}\right) \), and the error is not dependent on individual filters but on the spaces. Thus \( E\left\{{\left\Vert \mathbf{t}-{\mathbf{t}}_{est}\right\Vert}^2\right\} \) depends on how much of the energy of Rr(M) is contained in Rs(V), or, equivalently, how much of the energy of orthogonal filters spanning Rr(M) is contained in Rr(V). It may be shown that
$$ v\left(\mathbf{V},\mathbf{M}\right)={\displaystyle \sum_{i=1}^r\frac{q\left({\mathbf{o}}_i\right)}{s}}, $$
where \( {\left\{{\mathbf{o}}_i\right\}}_{i=1}^r \) is an orthogonal basis for Rr(M) and q(o i ) is the q-factor of o i . The use of orthogonal filters prevents correlation among filters from artificially increasing the value of the measure (as in Fig. 2a).

Additional insight into the power of the Vora Value is obtained by considering the case where the target space is the horizontal plane in 3-space. A set of three independent vectors, none of which lie in the plane, form a basis of the 3-space. None of the vectors will have a q-factor of one. It is clear that the span of the three vectors includes the target space. For this case, ν = 1, as it should. Thus, we can create a filter set that produces perfect tristimulus values from imperfect filters. The cost of this is that it requires more filters than the dimension of the target color space, i.e., r > s.

The Measure ν and Principal Angles

It may be shown that
$$ v\left(\mathbf{V},\mathbf{M}\right)={\displaystyle \sum_{i=1}^s\frac{{ \cos}^2\left({\theta}_i\right)}{s}} $$
where θ i is the i th principal angle [3] between the target color space, Rs(V), and Rr(M).

Other Measures

The measure ν is based on a mean-square error in the target color space assuming independent, identically distributed data. It does not take into account real data sets, perceptual error, or recording noise.

Finlayson and Drew [4, 5] address the issue of real data sets. They propose a similar measure based on the White Point Preserving Vora Error With Maximum Ignorance With Positivity (WPPVEMIP), which assumes that one is maximally ignorant of the distribution of f except for knowing that all values in the vector f are positive and that the color correction procedure preserves white points. They demonstrate that the WPPVEMIP is a more accurate estimate of the real error, and hence that the corresponding measure is more accurate than ν.

Wolski et al. [6] address the issue of perceptual error by using a linear approximation of CIE Lab error in the neighborhood of the points in the (known) data set. The filter sets designed by this method minimize an error measure that is a better estimate of perceptual error than the mean square error used to define ν.

Sharma and Trussell [7] address the issue of recording noise and propose a Figure of Merit (FOM) based on the approach of Wolski et al. The FOM is derived from an error that takes into consideration recording noise.

References

  1. 1.
    Vora, P.L., Joel Trussell, H.: Measure of goodness of a set of color scanning filters. J. Opt. Soc. Am. A 10(7), 1499–1508 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    Neugebauer, H.E.J.: Quality factor for filters whose spectral transmittances are different from color mixture curves, and its application to color photography. J. Opt. Soc. Am. 46(10), 821–884 (1956)ADSCrossRefGoogle Scholar
  3. 3.
    Golub, Gene H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996).Google Scholar
  4. 4.
    Finlayson, G.D, Drew, M.S.: The maximum ignorance assumption with positivity. 4th color imaging conference: color, science, systems and applications, IS&T/SID, pp. 202–205 (1996)Google Scholar
  5. 5.
    Finlayson, G.D, Drew, M.S.: White-point preservation enforces positivity. 6th color imaging conference: color, science, systems and applications, pp. 47–52 (1998)Google Scholar
  6. 6.
    Mark, W., Bouman, C.A., Allebach, J.P., Eric, W.: Optimization of sensor response functions for colorimetry of reflective and emissive objects. IEEE Trans. Image Proc. 5(3), 507–517 (1996)CrossRefGoogle Scholar
  7. 7.
    Sharma, G., Trussell, H.J.: Figures of merit for color scanners. IEEE Trans. Image Proc. 6(7), 990–1001 (1997)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceThe George Washington UniversityWashingtonUSA
  2. 2.Department of Electrical and Computer EngineeringNorth Carolina State UniversityRaleighUSA