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----
-title: "LED Characterization"
-date: 2018-05-02T11:18:38+02:00
----
-
-Preface
--------
-
-Recently, I have been working on a `small driver`_ for ambient lighting using 12V LED strips like you can get
-inexpensively from China. I wanted to be able to just throw one of these somewhere, stick down some LED tape, hook it up
-to a small transformer and be able to control it through Wifi. When I was writing the firmware, I noticed that when
-fading between different colors, the colors look *all wrong*! This observation led me down a rabbit hole of color
-perception and LED peculiarities.
-
-The idea of the LED driver was that it can be used either with up to eight single-color LED tapes or, much more
-interesting, with up to two RGB or RGBW (red-green-blue-white) LED tapes. For ambient lighting high color resolution was
-really important so you could dim it down a lot without flickering. I ended up using the same driver stage I used in the
-`multichannel LED driver`_ project for its great color resolution and low hardware requirements.
-
-.. raw:: html
-    
-    <figure>
-        <img src="images/rgb_cube.svg" alt="An illustration of the RGB color cube.">
-        <figcaption>An illustration of the RGB color cube.
-            <a href="https://commons.wikimedia.org/wiki/File:RGB_color_cube.svg">Picture</a> by
-            <a href="https://commons.wikimedia.org/wiki/User:Maklaan">Maklaan from Wikimedia Commons</a>,
-            <a href="https://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA 3.0</a>
-        </figcaption>
-    </figure>
-
-To make setting colors over Wifi more intuitive I implemented support for HSV colors. RGB is fine for communication
-between computers, but I think HSV is easier to work with when manually inputting colors from the command line. RGB  is
-close to how most monitors, cameras and the human visual apparatus work on a very low level but doesn't match
-higher-level human color perception very well. When we describe a color we tend to think in terms of "hue" or
-"brightness", and computing a measure of those from RGB values is not easy.
-
-Colors and Color Spaces
------------------------
-
-`Color spaces`_ are a mathematical abstraction of the concept of color. When we say "RGB", most of the time we actually
-mean `sRGB`_, a standardized notion of how to map three numbers labelled "red", "green" and "blue" onto a perceived
-color. `HSV`_ is an early attempt to more closely align these numbers with our perception. After HSV, a number of other
-*perceptual* color spaces such as `XYZ (CIE 1931)`_ and `CIE Lab/LCh`_ were born, further improving this alignment.  In
-this mathematical model, mapping a color from one color space into another color space is just a coordinate
-transformation.
-
-.. raw:: html
-    
-    <figure>
-        <img src="images/hsv_cylinder.png" alt="An illustration of the HSV color space as a cylinder.">
-        <figcaption>An illustration of the HSV color space as a cylinder.
-            <a href="https://commons.wikimedia.org/wiki/File:HSV_color_solid_cylinder.png">Picture</a> by
-            <a href="https://commons.wikimedia.org/wiki/User:SharkD">SharkD from Wikimedia Commons</a>,
-            <a href="https://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA 3.0</a>
-        </figcaption>
-    </figure>
-
-CIE 1931 XYZ is much larger than any other color space, which is why it is a good basis to express other color spaces
-in. In XYZ there are many coordinates that are outside of what the human eye can perceive. Below is an illustration of
-the sRGB space within XYZ. The wireframe cube is (0,0,0) to (1,1,1) in XYZ.  The colorful object in the middle is what
-of sRGB fits inside XYZ, and the lines extending out from it indicate the space that can be expressed in sRGB but not in
-XYZ. The fat white curve is a projection of the *monochromatic spectral locus*, that is the curve of points you get in
-XYZ for pure visible wavelengths.
-
-As you can see, sRGB is *much* smaller than XYZ or even the part within the monochromatic locus that we can perceive. In
-particular in the blues and greens we loose *a lot* of colors to sRGB.
-
-.. raw:: html
-
-    <figure>
-        <video controls loop>
-            <source src="video/sRGB.mkv" type="video/h264">
-            <source src="video/sRGB.webm" type="video/webm">
-            Your browser does not support the HTML5 video tag.
-        </video>
-        <figcaption>Illustration of the measured sRGB color space within XYZ. The thick, white line is the spectral
-            locus.
-            
-            <a href="video/sRGB.mkv">mkv/h264 download</a> /
-            <a href="video/sRGB.webm">webm download</a>
-        </figcaption>
-    </figure>
-
-The wrong colors I got when fading between colors were caused by this coordinate transformation being askew. Thinking
-over the problem, there are several sources for imperfections:
-
-* The LED driver may not be entirely linear. For most modulations such as PWM the brightness will be linear starting
-  from a certain value, but there is probably an offset caused by imperfect edges of the LED current. This offset can be
-  compensated with software calibration. I built a calibration setup for driver linearity in the `multichannel LED
-  driver`_ project. Below are pictures of ringing on the edges of an LED driver's waveform.
-
-* The red, green and blue channels of the LEDs used on the LED tape are not matched.  This skews the RGB color space.
-  In practice, the blue channel of my RGB tape to me *looks* much brighter than the red channel.
-
-* The precise colors of the red, green and blue channels of the LEDs are unknown. Though the red channel *looks* red, it
-  may be of a slightly different hue compared to the reference red used in `sRGB`_ which would also skew the RGB color
-  space.
-
-.. raw:: html
-
-    <figure>
-        <figure class="side-by-side">
-            <img src="images/driver_ringing_strong.jpg" alt="Strong ringing on the LED voltage waveform edge at about
-            100% overshoot during about 70% of the cycle time.">
-            <figcaption>The LED strip being at the end of a couple meters of wire caused extremely bad ringing at high
-            driving frequencies.</figcaption>
-        </figure><figure class="side-by-side">
-            <img src="images/driver_ringing_weak.jpg" alt="Weak ringing on the LED voltage waveform edge at about 30%
-            overshoot during about 20% of the cycle time.">
-            <figcaption>Adding a resistor in front of the MOSFET gate to slow the transition dampened the ringing
-            somewhat, but ultimately it cannot be eliminated entirely.</figcaption>
-        </figure>
-    </figure>
-
-These last two errors are tricky to compensate. What I needed for that was basically a model of the *perceived* colors
-of the LED tape's color channels. A way of doing his is to record the spectra of all color channels and then evaluate
-their respective XYZ coordinates. If all three channels are measured in one go with the same setup the relative
-magnitudes of the channels in XYZ will be accurate.
-
-To map any color to the LEDs, the color's XYZ coordinates simply have to be mapped onto the linear coordinate system
-produced by these three points within XYZ. LEDs are mostly linear in their luminous flux vs. current characteristic so
-this model will be adequate. The spectral integrals mapping the channels' measured responses to XYZ need only be
-calculated once and their results can be used as scaling factors thereafter.
-
-Measuring the spectrum
-----------------------
-
-In order to compensate for the cheap LED tape's non-ideal performance I had to measure the LED's red, green and blue
-channels' spectra. The obvious thing would be to go out and buy a `spectrograph`_, or ask someone to borrow theirs. The
-former is kind of expensive, and I did not want to wait two weeks for the thing to arrive. The latter I could probably
-not do every time I got new LED tape. Thus the only choice was to build my own.
-
-Luckily, building your own spectrometer is really easy. The first thing you need is something that splits incident light
-into its constituent wavelengths. In professional devices this is called the *`monochromator`_*, since it allows extraction
-of small color bands from the spectrum. The second thing is some sort of optics that project the incident light onto a
-screen behind the monochromator. In professional devices lenses or curved mirrors are used. In a simple homebrew job a
-pinhole as you would use in a `camera obscura`_ does a remarkably nice job.
-
-For the monochromator component several things could be used. A prism would work, but I did not have any. The
-alternative is a `diffraction grating`_. Professional gratings are quite specialized pieces of equipment and thus
-rather expensive. Luckily, there is a common household item that works almost as well: A regular CD or DVD. The
-microscopic grooves that are used to record data in a CD or DVD work the same as the grooves in a professional
-diffraction grating.
-
-Household spectra
------------------
-
-From this starting point, a few seconds on my favorite search engine yielded an `article by two researchers from the
-National Science Museum in Tokyo`_ providing a nice blueprint for a simple cardboard-and-DVD construction for use in
-classrooms. I replicated their device using a DVD and it worked beautifully. Daylight and several types of small LEDs I
-had around did show the expected spectra. Small red, yellow, green, and blue LEDs showed narrow spectra, daylight one
-continuous broad one, and white LEDs a continuous broad one with a distinct bright spot in the blue part. The
-single-color LED spectra are quite narrow since they are determined by the LED's semiconductor's band gap, which is
-specific to the semiconductor used and is quite precise. White LEDs are in fact a blue LED chip covered with a so-called
-*phosphor*. This phosphor is not elementary phosphorus but an anorganic compound that absorbs the LED chip's blue light
-and re-emits a broader spectrum of more yellow-ish wavelengths instead. The final LED spectrum is a superposition of
-both spectra, with some of the original blue light leaking through the phosphor mixing with the broadband yellow
-spectrum of the phosphor.
-
-.. raw:: html
-
-    <figure>
-        <figure class="side-by-side">
-            <img src="images/spectrograph_step1_parts.jpg">
-            <figcaption>The ingredients. The cup of coffee and Madoka Magica DVD set are essential to the eventual
-            function of the appartus.</figcaption>
-        </figure><figure class="side-by-side">
-            <img src="images/spectrograph_step2.jpg">
-            <figcaption>Step 1: Cut to size and mark down all holes as described in <a
-            href="http://www.candac.ca/candacweb/sites/default/files/BuildaSpectroscope.pdf">the manual</a></figcaption>
-        </figure>
-        <figure class="side-by-side">
-            <img src="images/spectrograph_step3.jpg">
-            <figcaption>Step 2: Cut out all holes</figcaption>
-        </figure><figure class="side-by-side">
-            <img src="images/spectrograph_step4_complete.jpg">
-            <figcaption>The finished result with the back side showing. The viewing window is on the bottom of the other
-            side.</figcaption>
-        </figure>
-    </figure>
-
-
-Now that I had a spectrograph, I needed a somewhat predictable way of measuring the spectrum it gave me.
-
-Measuring a spectrum
---------------------
-
-Pointing a camera at the spectrograph would be the obvious thing to do. This produces pretty images but has one critical
-flaw: I wanted to acquire quantitative measurements of brightness across the spectrum. Since I don't have a precise
-technical datasheet specifying the spectral response of any of my cameras I can't compare the absolute brightness of
-different colors on their pictures. Some other sensor was needed.
-
-.. raw:: html
-
-    <figure>
-        <img src="images/daylight_spectrum_dvd.jpg">
-        <figcaption>The daylight spectrum as seen using a DVD as a grating.
-            <a href="https://commons.wikimedia.org/wiki/File:SpectresSolaires-DVD.jpg">Picture</a> by
-            <a href="https://commons.wikimedia.org/wiki/User:Xofc">Xofc from Wikimedia Commons</a>,
-            <a href="https://creativecommons.org/licenses/by-sa/4.0/">CC-BY-SA 4.0</a>
-        </figcaption>
-    </figure>
-
-
-Measuring light intensity
-~~~~~~~~~~~~~~~~~~~~~~~~~
-
-Looking around my lab, I found a bag of `SFH2701`_ visible-light photodiodes. Their
-datasheet includes their spectral response so I can compensate for that, allowing precise-ish absolute intensity
-measurements. Just like LEDs, photodiodes are extremely linear across several orders of magnitude. The datasheet of the
-classic `BPW34`_ photodiode shows that this photodiode's light current is exactly proportional to illuminance over at
-least three orders of magnitude. The `SFH2701`_ datasheet does not include a similar graph but its performance will be
-similar. The `SFH2701`_ photodiodes I had at hand were perfect for the job compared to the vintage `BPW34`_ since their
-active sensing area is really small (0.6mm by 0.6mm) compared to the BPW34 (a whopping 3mm by 3mm). If I were to use a
-`BPW34`_ I would have to insert some small apterture in front of it so it does not catch too broad a part of the
-spectrum at once. The `SFH2701`_ is small enough that if I just point it at the projected spectrum directly I will
-already get only a small part of the spectrum inside its 0.6mm active area.
-
-To convert the photodiode's tiny photocurrent into a measurable voltage I built another copy of the `transimpedance
-amplifier`_ circuit I already used in the `multichannel LED driver`_. A `transimpedance amplifier`_ is an
-amplifiert that produces a large voltage from a small current. The weird name comes from the fact that it works kind of
-like an amplified resistor (which can be generalized as an *impedance* electrically). Apply a current to a resistor and
-you get a voltage. A transimpedance amplifiert does the same with the difference that its input always stays at 0V,
-making it look like an ideal current sink to the connected current source.
-
-Transimpedance amplifiers are common in optoelectronics to convert small photocurrents to voltages. In this instance I
-built a very simple circuit with a dampened transimpedance amplifier stage followed by a simple RC filter for noise
-rejection and a regular non-inverting amplifier using another op-amp from the same chip to further boost the filtered
-transimpedance amplifier output. I put all the passives setting amplifier response (the gain-setting resistors and the
-filter resistor and capacitors) on a small removable adapter so I could easily change them if necessary. I put a small
-trimpot on the virtual ground both amplifers use as a reference so I could trim that if necessary.
-
-.. raw:: html
-    
-    <figure>
-        <img src="images/preamp_schematic.jpg" alt="A drawing of the photodiode preamplifier's schematic">
-        <figcaption>The photodiode preamplifier schematic. Schematic drawn with an unlicensed copy of
-        DaveCAD.</figcaption>
-    </figure>
-
-Following are pictures of the preamplifier board. The connectors on the top-left side are two copies of the analog
-signal for the ADC and a small panel meter. The SMA connector is used as the photodiode input since coax cables are
-generally low-leakage and have built-in shielding. The circuit is powered via the micro-USB connector and the analog
-ground bias voltage can be adjusted using the trimpot.
-
-For easy replacement, all passives setting gain and frequency response are on a small, pluggable carrier PCB made from a
-SMD-to-DIP adapter.
-
-Flying-wire construction is just fine for this low-frequency circuit. In a high-speed photodiode preamp, the
-transimpedance amplifier circuit would be highly sensitive to stray capacitance, but we're not aiming at high speed
-here. 
-
-.. raw:: html
-
-    <figure>
-        <figure class="side-by-side">
-            <img src="images/preamp_front.jpg">
-            <figcaption>The front side of the preamplifier board.</figcaption>
-        </figure><figure class="side-by-side">
-            <img src="images/preamp_back.jpg">
-            <figcaption>The wiring of the photodiode preamp.</figcaption>
-        </figure>
-    </figure>
-
-Given a way to measure intensity what remains missing is a way to scan a single photodiode across the spectrum.
-
-Scanning the projection
-~~~~~~~~~~~~~~~~~~~~~~~
-
-A cheap linear stage can be found in any old CD or DVD drive. These drives use a small linear stage based on a
-stepper-driven screw to move the laser unit radially. Removing the laser unit and connecting a leftover stepper driver
-module I was left with a small linear stage with about 45 steps per cm without microstepping enabled. The driver I used
-was an `A4988`_ module that required at least 8V motor drive voltage. I used a small micro USB-input boost converter
-module to generate a stable 10V supply for the motor driver, with the USB's 5V rail used as a logic supply for the motor
-driver.
-
-The `SFH2701`_ can easily be mounted to the linear stage using a small SMD breakout board glued in place with thin wires
-connecting it to the transimpedance amplifier. The DVD drive linear stage is not very strong so it is important that
-this wire does not put too much strain on it.
-
-Above the photodiode, I mounted a small piece of paper on the linear stage to be used as a projection screen to align
-the linear stage in front of the spectrometer viewing window. A line on the screen paper points to the photodiode die in
-parallel to the linear stage allowing precise alignment.
-
-The whole unit with photodiode preamplifier, linear stage, photodiode and stepper motor driver finally looks like this:
-
-.. raw:: html
-
-    <figure>
-        <img src="images/electronics_whole.jpg" alt="The complete electronics setup of the spectrograph. In the back
-        there is the DVD drive stepper stage. In front of it, mounted on a piece of wood are a small USB-to-12V
-        switching-regulator module to power the stepper motor in the top left, below on the bottom left is the
-        photodiode preamp and on the right is a breadboard with the stepper driver module and lots of jumper wires
-        interconnecting everything. On the right of the breadboard, a buspirate is attached to interface everything to a
-        computer. On the bottom edge of the piece of wood, two LED panel meters are mounted for readout of the preamp
-        output and the stepper supply voltages.">
-        <figcaption>The complete electronics setup. The buspirate on the right interfaces to a computer and controls the
-        stepper driver and ADC'es the preamp output. The two panel meters show the preamp output and stepper voltage for
-        setup.</figcaption>
-    </figure>
-
-The projection of the spectrum can be adjusted by moving the light source relative to the entry slot and by moving
-around the grating DVD.
-
-The capture process
-~~~~~~~~~~~~~~~~~~~
-
-To capture a spectrum, first the light source has to be mounted near the spectrograph's entry slot. The LED tape I
-tested I just taped face-down directly into it. Next, the grating DVD has to be adjusted to make sure the spectrum
-covers a sensible part of the photodiode's path. Mostly, this boils down to adjusting the photodiode distance and height
-to match the vertical extent and wiggling the grating DVD to adjust the projection's horizontal position.
-
-After the optics are set-up, the photodiode preamplifier has to be adjusted. In my experiments, most LED tape at 5GΩ
-required a high-ish amplification. The goal in this step is to maximize the peak response of the preamp  to be just
-shy of its VCC rail to make best use of its dynamic range. To adjust the pre-amp, I took several very coarsely-spaced
-measurements to give me an estimate of the peak while I did not yet know its precise location.
-
-Since stray daylight totally swamped out the weak projection of the LED's spectrum I shielded the entire setup with a
-small box made of black cardboard and two black t-shirts on top. This shielding proved adequate for all my measurements
-but I had to be careful not to accidentially move the DVD that was stuck into the spectrograph with the shielding
-t-shirts.
-
-For capturing a single spectrum I wrote a small python script that will automatically move the stepper in adjustable
-intervals and take two measurements at each point, one with the LED tape off that can be used for offset calibration and
-one with the LED tape on. All measurements are stored in a sqlite database that can then be accesssed from other
-scripts.
-
-I built a small script that shows the progress of the current run and an jupyter notebook for data analysis. The jupyter
-notebook is capable of live-updating a graph with the in-progress spectrum's data. This was quite useful as a sanity
-check for when I made some mistake easy to spot in the resulting data.
-
-After one color channel is captured, the LED tape has to be manually set to the next color and the next measurement can
-begin.
-
-.. raw:: html
-    
-    <figure>
-        <img src="images/raw_plot_cheap_rgb.svg" alt="A plot with three wide peaks, two large peaks on both sides and
-        one smaller one in the middle. The middle one overlaps the two on the sides. The large ones are about 2.5V in
-        amplitude. Overall, the plot is about 300 stepper steps wide with each peak being around 130 steps wide.">
-        <figcaption>A plot of the raw preamp output voltage versus stepper position. From left to right, the three peaks
-        are blue, green and red. Step 0 corresponds to the bottommost stepper position and the shortest wavelength.
-        </figcaption>
-    </figure>
-
-
-Data analysis
-~~~~~~~~~~~~~
-
-Data analysis consists of three major steps: Offset- and stray light removal, wavelength and amplitude calibration and
-color space mapping.
-
-Offset removal
-**************
-The first task is to remove the offset caused by dark current as well as stray light of the LED's bright primary
-reflection on the DVD. The LED is very bright and only a small part of its light gets reflected by the grating towards
-the photodiode screen. The remaining part of the light is reflected onto the table in front of the DVD spectrograph.
-Though I covered all of this with black cardboard, some of that light ultimately gets reflected onto the photodiode.
-This causes a large offset, in particular in the blue part of the spectrum since in this part the photodiode is closest
-to the spectrograph's opening.
-
-The composite offset can be approximated with a second-order polynomial that is fitted to all the data outside of the
-main peak's area. Since at this point the wavelength of each data point is still unknown this is done with a rough first
-estimate of the three colors' peaks' locations and widths.
-
-Wavelength- and amplitude calibration
-*************************************
-The photodiode's response is strongly wavelength-dependent. In particular in the blue band, the photodiode's sensitivity
-gets very poor down to about 20% at the edge to ultraviolet. This effect is strong enough to move the apparent location
-of the blue peak towards red.
-
-.. raw:: html
-    
-    <figure>
-        <img src="images/photodiode_sensitivity.svg" alt="A plot of photodiode sensitivity against wavelength relative
-        to peak sensitivity at 820nm.  The sensitivity rises from 20% at 380nm approximately linearly to 80% at 620nm,
-        then the rise rolls off.">
-        <figcaption>A plot of the photodiode's relative sensitivity in the visible spectrum. The sensitivity is
-        normalized against its peak at 820nm.
-        </figcaption>
-    </figure>
-
-The problem is that in order to remove this non-linearity, we would already have to know the wavelength of the measured
-light. Since I don't, I settled for a two-step process. First, a coarse wavelength calibration is done relative to the
-red peak and the short-wavelength edge of the blue peak. The photodiode measurements are then sensitivity-corrected
-using this coarse measurement. Then all three channel peaks are measured in the resulting data and a fine wavelength
-estimate is produced by a least-squares fit of a linear function. This fine estimate is then used for a second
-sensitivity correction of all original measurements and the scale is changed from stepper motor step count to
-wavelength in nanometers.
-
-.. raw:: html
-    
-    <figure>
-        <img src="images/processed_plot_cheap_rgb.svg" alt="A plot with three wide peaks, all three of different
-        heights. The leftmost peak is highest at 6nA, the middle peak lowest at 1.6nA and the rightmost peak in between
-        at 4nA.  The middle one overlaps the two on the sides.  Overall, the plot spans about 300nm on its x axis with
-        each peak being around 100nm wide.">
-        <figcaption>A plot of the processed measurements. From left to right, the three peaks are blue, green and red.
-        </figcaption>
-    </figure>
-
-.. FIXME re-do these measurements, avoiding clipping
-.. FIXME re-do calibration using CCFL
-.. FIXME calibration for brightness imbalance due to wedge-shaped projection of spectrum
-
-Color space mapping
-*******************
-Finally, to achieve the objective of measuring the LED tape's channels' precise color coordinates the measured spetra
-have to be matched against the color spaces' *color matching functions*. The color matching functions describe how
-strong the color space's idealized *standard observer* would react to light at a particular wavelength. Going from a
-measured spectrum to color coordinates XYZ works by integrating over the product of the measurement and each color
-coordinate's color matching function.
-
-The result are three color coordinates X, Y and Z for each channel R, G and B yielding nine coordinates in total. When
-written as a matrix conversion between XYZ color space and LED-RGB color space is as simple as multiplying that matrix
-(or its inverse) and a vector from one of the color spaces.
-
-In XYZ space, the set of colors that can be produced with this LED tape is described by the `parallelepiped`_ spanned by
-the three channel's XYZ vectors. In the following figures, you can see a three-dimensional model of the RGB LED's color
-space (colorful) as well as sRGB (white) for comparison plotted within CIE 1931 XYZ. There is no natural map to scale
-both so for this illustration the LED color space has been scaled to fit. These figures were made with blender and a few
-lines of python.  The blender project file including all settings and the python script to generate the color space
-models can be found in the `project repo`_.
-
-.. raw:: html
-
-    <figure>
-        <video controls loop>
-            <source src="video/led_within_srgb_scale=1.0.mkv" type="video/h264">
-            <source src="video/led_within_srgb_scale=1.0.webm" type="video/webm">
-            Your browser does not support the HTML5 video tag.
-        </video>
-        <figcaption>Illustration of the measured LED color space scaled to fit within XYZ with sRGB (light gray) for
-            comparison. The thick, white line is the spectral locus.
-            
-            <a href="video/led_within_srgb_scale=1.0.mkv">mkv/h264 download</a> /
-            <a href="video/led_within_srgb_scale=1.0.webm">webm download</a>
-        </figcaption>
-    </figure>
-
-As you can see, the result is pretty disappointing. The LED's color space parallepiped is very narrow, which is because
-the blue channel is much brighter than the other two channels. An easy fix for this is to scale-up the RGB space and
-drop any values outside XYZ. The scaling factor is a trade-off between color space coverage and brightness. You can
-produce the most colors when you clip all channels to brightness of the weakest channel (green in this case), but that
-will make the result very dim. Scaling brightness like that stretches the RGB parallelepiped along its major axis. Up to
-a point the number of possible colors (the gamut) increases at expense of maximum brightness. When the parallelepiped is
-stretched far enought for all three channel vectors to be outside the 1,1,1 XYZ-cube, maximum brightness continues to
-decrease but the gamut stays constant. I don't know a simple scientific way to solve this problem, so I just played
-around with a couple of factors and settled on 2.5 as a reasonable compromise. Below is an illustration.
-
-.. raw:: html
-
-    <figure>
-        <video controls loop>
-            <source src="video/led_within_srgb_fancy_camera_path_scale=2.5.mkv" type="video/h264">
-            <source src="video/led_within_srgb_fancy_camera_path_scale=2.5.webm" type="video/webm">
-            Your browser does not support the HTML5 video tag.
-        </video>
-        <figcaption>Illustration of the measured LED color space at scale factor 2.5 within XYZ with sRGB (light gray)
-            for comparison. The thick, white line is the spectral locus.
-            
-            <a href="video/led_within_srgb_fancy_camera_path_scale=2.5.mkv">mkv/h264 download</a> /
-            <a href="video/led_within_srgb_fancy_camera_path_scale=2.5.webm">webm download</a>
-        </figcaption>
-    </figure>
-
-Firmware implementation
------------------------
-In the end, the above measurements yield two matrices: One for mapping XYZ to RGB, and one for mapping RGB to XYZ. Of
-the several versions of CIE XYZ I chose the CIE 1931 XYZ color space as a basis for the firmware because it is most
-popular. Mapping a color coordinate in one color space to the other is as simple as performing nine floating-point
-multiplications and six additions. Mapping Lab or Lch to RGB is done by first mapping Lab/Lch to XYZ, then XYZ to RGB.
-Lab to XYZ is somewhat complex since it requires a floating-point power for gamma correction, but any self-respecting
-libc will have one of those so this is still no problem. Lch also requires floating-point sine and cosine functions, but
-these should still be no problem on most hardware.
-
-My implementation of these conversions in the ESP8266 firmware of my `Wifi LED driver`_ can be found `on Github`_. You
-can view the Jupyter notebook most of the analysis above `here <http://nbviewer.jupyter.org/github/jaseg/led_drv/blob/master/doc/Spectrum%20Measurement.ipynb>`__.
-
-.. _`on Github`: https://github.com/jaseg/esp_led_drv/blob/master/user/led_controller.c
-.. _`project repo`: https://github.com/jaseg/led_drv
-.. _`Wifi LED driver`: {{<ref "posts/wifi-led-driver/index.rst">}}
-.. _`small driver`: {{<ref "posts/wifi-led-driver/index.rst">}}
-.. _`multichannel LED driver`: {{<ref "posts/multichannel-led-driver/index.rst">}}
-.. _`sRGB`: https://en.wikipedia.org/wiki/SRGB
-.. _`CC BY-SA 3.0`:  https://creativecommons.org/licenses/by-sa/3.0
-.. _`Color spaces`: https://en.wikipedia.org/wiki/Color_space
-.. _`HSV`: https://en.wikipedia.org/wiki/HSL_and_HSV
-.. _`CIE Lab/LCh`: https://en.wikipedia.org/wiki/Lab_color_space
-.. _`XYZ (CIE 1931)`: https://en.wikipedia.org/wiki/CIE_1931_color_space
-.. _`camera obscura`: https://en.wikipedia.org/wiki/Pinhole_camera
-.. _`article by two researchers from the National Science Museum in Tokyo`: http://www.candac.ca/candacweb/sites/default/files/BuildaSpectroscope.pdf
-.. _`spectrograph`: https://en.wikipedia.org/wiki/Ultraviolet%E2%80%93visible_spectroscopy
-.. _`monochromator`: https://en.wikipedia.org/wiki/Monochromator
-.. _`diffraction grating`: https://en.wikipedia.org/wiki/Diffraction_grating
-.. _`SFH2701`: https://dammedia.osram.info/media/resource/hires/osram-dam-2495903/SFH%202701.pdf
-.. _`BPW34`: http://www.vishay.com/docs/81521/bpw34.pdf
-.. _`transimpedance amplifier`: https://en.wikipedia.org/wiki/Transimpedance_amplifier
-.. _`A4988`: https://www.pololu.com/file/0J450/A4988.pdf
-.. _`parallelepiped`: https://en.wikipedia.org/wiki/Parallelepiped