Description

The eye's ability to separate two distinct colors depends of the hue, saturation and brightness of the observed color.
In this model, we want to design a procedures that take the desired image as input and the image needed to be project on the bacteria as output.

Approach

RGB is a convenient color model for computer graphics because the human visual system works in a way that is similar — though not quite identical — to an RGB color space.
First, we find the image coordinates in RGB color space and extract three matrices for primary color red, green and blue. secondly, simply assume the RGB value is linearly related to fluorescent protein quantity. Gene expression output is closely related to light intensity input.Then, Hill function was used to describe this relationship. finally, three transfered matrices were compound to one picuture as output.

Assumptions

• Emission peaks of fluorescent proteins are close to primary wavelengths which form the three axes for RGB color space respectively.
• In the CcaS-CcaR system, red light competitively inhibits activation by green light. So growing under red light reduce the sensitivity of CcaS-CcaR to green light while preserving the full output range.
• The Cph8-OmpR and YF1-Fixj system are both light–intensity sensors.
• Steady-state response with respect to light intensity of these light sensors can be described by Hill function. Fluorescent protein quantity is perceptually uniform thus can be obtained from the RGB coordinates through linear transformation. H=b+(aI^n)/(I^n+k^n )

RGB components

Here we works with The CIE 1931 color space standard which consisted of monochromatic primaries red, green and blue.
The tristimulus value extracted from sample image can be associated with a RGB color space where axes range for 0 to 255.
We use Matlab to perform this color-protein-color transformation.
By importing the sample image into Matlab, we obtain its RGB matrix from which we extract each primaries separately so that we have three two-dimensional matrices for red, green and blue. Labeled as r,g,b.

Protein expression output

As mentioned in assumption, we assume that two colors that are equally distant in the color space are equally different in the amount of fluorescent protein, which means relationship between the RGB values and required protein quantity can be characterized by a linear function.

We believe the Hill function can describe the ligand-binding situation in our light-sensing system by determining the degree of cooperativeness of the ligand binding to receptors. This has also been demonstrated in previous papers.
P=b+(aI^n)/(I^n+k^n )
I-light intensity.
P-protein quantity produced by sensing light.
For red and blue sensor, required light intensity is thus
Ig=k(a/(P-b)-1)^((-1)/n)
Ib=k(a/(P-b)-1)^((-1)/n)

However the green sensor needs to grow under red light to perform a similar response as the red one.
For green sensor
Ir=k(a/(P-b)-1)^((-1)/n)+r0
a, b, k- Hill coefficients. From data provided in paper,
bg=19.9;ag=65.3;ng=2.49;kg=0.138;
br=10.6;ar=97.9;nr=1.42;kr=0.0239;
bb=16.0;ab=75.3;nb=1.58;kb=0.03;
We select a proper area for linear transformation.
Light intensity 0-0.4
Protein quantity 20-80
Thus RGB(0-255) will be transfer to Protein(20-80) follow the equation:
protein=0.2353R/G/B +20
I=k(a/(0.2353r+20-b)-1)^((-1)/n)*255/0.4(+r0)
Constant r0=150

matrix R plot in 3D space
matrix G plot in 3D space
below is the image we finally get through this whole procedures.