Unifying optimization methods for color filter design

Through optimization we can solve for a filter that when the camera views the world through this filter, it is more colorimetric. Previous work solved for the filter that best satisfied the Luther condition: the camera spectral sensitivities after filtering were approximately a linear transform from the CIE XYZ color matching functions. A more recent method optimized for the filter that maximized the Vora-Value (a measure which relates to the closeness of the vector spaces spanned by the camera sensors and human vision sensors). The optimized Luther- and Vora-filters are different from one another. In this paper we begin by observing that the function defining the Vora-Value is equivalent to the Luther-condition optimization if we use the orthonormal basis of the XYZ color matching functions, i.e. we linearly transform the XYZ sensitivities to a set of orthonormal basis. In this formulation, the Luther-optimization algorithm is shown to almost optimize the Vora-Value. Moreover, experiments demonstrate that the modified orthonormal Luther-method finds the same color filter compared to the Vora-Value filter optimization. Significantly, our modified algorithm is simpler in formulation and also converges faster than the direct Vora-Value method

An improved optimization method for finding a color filter to make a camera more colorimetric

Recently, an iterative optimization method was proposed that determines the spectral transmittance of a color filter which, when placed in front of a camera, makes the camera more colorimetric [1]. However, the performance of this method depends strongly on the filter (guess) that initializes the optimization. In this paper, we develop a simple extension to the optimization where we systematically sample the set of possible initial filters and for each initialization solve for the best refinement. Experiments demonstrate that improving the initialization step can result in the effective ‘camera+filter’ imaging system being much more colorimetric. Moreover, the filters we design are smoother than previously reported (which makes them easier to manufacture)

Designing a Colour Filter for Making Cameras more Colorimetric

If a camera were to capture colour like a human observer, fundamentally, it should sense the light information as the way the human visual system does. It is necessary to either replicate the human visual sensitivity responses or reproduce the three-number colour representations - e.g. CIE XYZ tristimulus values - to obtain an accurate colour measurement. In practice, however, the camera sensors generally deviate from the ideal sensitivities of the human visual system. Consequently, the colour triplets a camera records are device-dependent, which generally differ from the standard observer tristimulus values. The colorimetric performance can be improved by either correcting camera responses to the reference ground-truth values using sophisticated mathematical transformations or using more imaging sensors/filters to capture more information about the incident light. These methods have their disadvantages: the former increases the computational complexity and the latter increases the system complexity and the overall cost. In this thesis, we aim to make the digital camera capture colours more like the human visual perception by placing a colour filter in front of the camera so as to alter its spectral sensitivity functions as desired. The central contribution of this study is to carefully design a colour filter for a given camera so that the ‘filter+camera’ setting having the new sensitivities becomes almost colorimetric, i.e. recording the colour triplets that can be linearly transformed to the ground-truth XYZ tristimulus values. The starting point for this thesis is to design the filter that makes the filtered camera best achieve the Luther condition, i.e. the new effective camera sensitivity functions after filtering are a linear combination of the colour matching function of the human visual system. Under this condition, the camera can capture any incoming colour signal accurately in the sense that the captured RGBs are almost a linear transform from the XYZ tristimuli. Next, we reformulate the problem formulation for finding the optimal filter that targets the more generalised Vora-Value goodness measure. The Vora-Value, by definition, measures the similarity between the vector spaces spanned by the spectral sensitivities of a camera and the XYZ colour matching functions underpinning the human visual system. The Vora-Value has the advantage that the best filter is related to the target human visual space and not fixed coordinates (e.g. the XYZ and RGB colour matching functions have different coordinate values but are in the same vector space). As well as developing a method that finding a filter maximises the Vora-Value (makes the vector spaces most similar), we examine the relationship between the Vora-Value and Luther condition optimisations. We show that the Luther-condition optimisation also maximises the Vora-Value if we find the filter that makes a linear combination of the camera sensitivities most similar to a linear transform of XYZ (that is orthonormal). This is an important result as the Luther optimisation is much simpler to implement and faster to execute. So we can use the simpler Luther-condition formulation to maximise the Vora-Value measure using a more straightforward algorithm. A strength and weakness of the Luther and Vora-Vora optimisations is that they assume - as an explicit part of their formulations - that all spectra are equally likely. But, this is not the case in real imaging applications. So we extend our filter design algorithms in a data-driven manner that it optimises for the best colorimetric estimates given a collection of illuminants and surface reflectance data. Our extended method uses quadratic programming that allows us to add linear inequality constraints into the problem formulation. We show how to find filters that have smooth distribution and bounded transmittance (e.g. transmit at least 50% of the light) across the spectrum. Constraints like these make the filters more useful and feasible could make the filters easier to manufacture. We show that we can find smooth and highly transmissive colour filters that when placed in front of a digital camera can make the camera significantly more colorimetric and hence can be used for colour measurement applications with high demand in colour accuracy

Designing a Color Filter via Optimization of Vora-Value for Making a Camera more Colorimetric

Previous work has proposed to solve for a filter which, when placed in front of a camera, improves the colorimetric property by best satisfying the Luther condition. That is, the filtered spectral sensitivities of a camera - after a linear transform - are as close to the color matching functions of the human visual system as possible. By construction, the prior art solves for a filter for a given set of human visual sensitivities, e.g.the XYZ color matching functions or the cone response functions. However, depending on the target spectral sensitivity set, a different optimal filter is found. In this paper, we set out a method to solve for a filter that works equally well for all possible target sensitivity sets of the human visual system. We observe that the cone fundamentals, the CIE XYZ color matching functions or any linear combination thereof, span the same vector space. Thus, we solve for a filter that makes the vector space spanned by the filtered camera sensitivities as similar as possible to the space spanned by human vision sensors. We argue that the Vora-Value is a suitable way to measure subspace similarity and we develop an optimization method for finding a filter that maximizes the Vora-Value measure. Experiments demonstrate that our new optimization leads to the filtered camera sensitivities which have a significantly higher Vora-Value and improved colorimetric performance compared with antecedent methods

Designing color filters that make cameras more colorimetric

When we place a colored filter in front of a camera the effective camera response functions are equal to the given camera spectral sensitivities multiplied by the filter spectral transmittance. In this article, we solve for the filter which returns the modified sensitivities as close to being a linear transformation from the color matching functions of the human visual system as possible. When this linearity condition - sometimes called the Luther condition- is approximately met, the 'camera+filter' system can be used for accurate color measurement. Then, we reformulate our filter design optimisation for making the sensor responses as close to the CIEXYZ tristimulus values as possible given the knowledge of real measured surfaces and illuminants spectra data. This data-driven method in turn is extended to incorporate constraints on the filter (smoothness and bounded transmission). Also, because how the optimisation is initialised is shown to impact on the performance of the solved-for filters, a multi-initialisation optimisation is developed. Experiments demonstrate that, by taking pictures through our optimised color filters, we can make cameras significantly more colorimetric

A mathematical investigation into the design of prefilters that make cameras more colorimetric

By placing a color filter in front of a camera we make new spectral sensitivities. The Luther-condition optimization solves for a color filter so that the camera’s filtered sensitivities are as close to being linearly related to the XYZ color matching functions (CMFs) as possible, that is, a filter is found that makes the camera more colorimetric. Arguably, the more general Vora-Value approach solves for the filter that best matches all possible target spectral sensitivity sets (e.g., any linear combination of the XYZ CMFs). A concern that we investigate here is that the filters found by the Luther and Vora-Value optimizations are different from one another. In this paper, we unify the Luther and Vora-Value approaches to prefilter design. We prove that if the target of the Luther-condition optimization is an orthonormal basis—a special linear combination of the XYZ CMFs which are orthogonal and are in unit length—the discovered Luther-filter is also the filter that maximizes the Vora-Value. A key advantage of using the Luther-condition formulation to maximize the Vora-Value is that it is both simpler to implement and converges to its optimal answer more quickly. Experiments validate our method

Robust estimation of bacterial cell count from optical density

Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data