# Vora Value

**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.

**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

**R**

_{s}(

**V**), where the subscript denotes the dimension of the vector space. For example,

**v**_{1},

**v**_{2},

**v**_{3}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

**R**

_{s}(

**V**), and the desired

*s*-stimulus vector is

**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

*for viewing illuminants D-50 and D-65. For another example,*

**f***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.

**g**is typically color corrected to obtain the best linear estimate of

*, \( {\boldsymbol{t}}_{\boldsymbol{est}}=\boldsymbol{A}\boldsymbol{g}, \) where color correction matrix*

**t***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*

**A***are assumed to be independent, identically distributed with variance*

**f***σ*

^{2}, it can be shown that

^{2}

*Trace*(

**V**

^{T}

**V**).

*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,

**R**

_{s}(

**V**). It measures how accurately

**t**can be estimated from

**g**after linear color correction and is defined as

*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

*consists of the inner products (dot products) of the vectors*

**t**

**v**_{1}

**v**_{2}…

**v**_{ s }with the vector

**f**[2]. Thus

*depends only on the*

**t***fundamental*of

*, which is defined as*

**f**

**P**_{ V }(

*), its orthogonal projection onto the target color space. On the other hand, any components of*

**f****f**that are orthogonal to all of

**v**_{1}

**v**_{2}…

**v**_{ s }will not contribute to

*. Thus, in particular, \( \mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) \) does not contribute to*

**t***.*

**t***s =*2. Vectors

**v**

_{1}and

**v**

_{2}define the target color space, the horizontal plane. The reflectance spectrum is represented by three-dimensional vector

**f**. The vector

*is completely determined by*

**t**

**P**_{ V }(

*), 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*

**f**

**v**_{1}or

**v**_{2}is zero and hence does not contribute to

*.*

**t***represent a single effective recording filter, obtained by combining the filter transmission function and the recording illuminant. Ignoring recording noise, the value*

**m**

**m**^{ T }

*or the inner product of vectors*

**f****m**and

**f**is the recorded value. The component of

*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*

**m***because it measures energy in \( \mathbf{f}-{\boldsymbol{P}}_{\boldsymbol{V}}\left(\boldsymbol{f}\right) \). This is illustrated in Fig. 1b.*

**t***,*

**m**

**P**_{ V }(

*) is useful for recording accuracy. The Colorimetric Quality Factor (*

**m***CQF*or q-factor, defined by Neugebauer in 1956) of

*measures the accuracy of*

**m****m**from this perspective, and is defined as the energy in the vector

**P**_{ V }(

*),as a fraction of the total energy in*

**m***.*

**m***q*(

*) is larger when*

**m**

**P**_{ V }(

*) 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.*

**m**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 }(

*) − that is, whether it measures the energy in*

**f**

**P**_{ V }(

*) in all directions of the target color space.*

**f***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

**m**

_{1}and

**m**

_{2}are both perfect, lie in

**R**

_{r}(

**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 }(

*) that is orthogonal to the filters. It is hence not capable of measuring all the energy in*

**f**

**P**_{ V }(

*).*

**f***s*=

*r*= 2 in Fig. 2a. The two recording filters

**m**

_{1}and

**m**

_{2}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 }(

*). The values that are recorded in practice depend on*

**f**

**P**_{ M }(

*).*

**f****R**

_{r}(

**M**) is contained in

**R**

_{s}(

**V**), or, equivalently, how much of the energy of orthogonal filters spanning

**R**

_{r}(

**M**) is contained in

**R**

_{r}(

**V**). It may be shown that

**R**

_{r}(

**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

*θ*

_{ i }is the

*i*

^{ th }principal angle [3] between the target color space,

**R**

_{s}(

**V**), and

**R**

_{r}(

**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.

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