+
+
+
+
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.
+
+ 
+ An illustration of the RGB color cube.
+ Picture by
+ Maklaan from Wikimedia Commons,
+ CC-BY-SA 3.0
+
+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.
+
+ 
+ An illustration of the HSV color space as a cylinder.
+ Picture by
+ SharkD from Wikimedia Commons,
+ CC-BY-SA 3.0
+
+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.
+
+
+ Illustration of the measured sRGB color space within XYZ. The thick, white line is the spectral
+ locus.
+
+ mkv/h264 download /
+ webm download
+
+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.
+
+
+
+ 
+ The shift register logic output of the multichannel LED driver directly driving a small mosfet's
+ gate through an inch or so of PCB trace caused extremely bad ringing at high driving
+ frequencies.
+
+ 
+ Adding a resistor dampened the ringing somewhat, but ultimately it cannot be eliminated
+ entirely.
+
+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.
+
+
+ 
+ The ingredients. The cup of coffee and Madoka Magica DVD set are essential to the eventual
+ function of the appartus.
+
+ 
+ Step 1: Cut to size and mark down all holes as described in the manual
+
+
+ 
+ Step 2: Cut out all holes
+
+ 
+ The finished result with the back side showing. The viewing window is on the bottom of the other
+ side.
+
+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.
+
+ 
+ The daylight spectrum as seen using a DVD as a grating.
+ Picture by
+ Xofc from Wikimedia Commons,
+ CC-BY-SA 4.0
+
+
+
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.
+
+ 
+ The photodiode preamplifier schematic. Schematic drawn with an unlicensed copy of
+ DaveCAD.
+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.
+
+
+ 
+ The front side of the preamplifier board.
+
+ 
+ The wiring of the photodiode preamp.
+
+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:
+
+ 
+ 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.
+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.
+
+ 
+ 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.
+
+
+
+
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.
+
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.
+
+ 
+ A plot of the processed measurements. From left to right, the three peaks are blue, green and red.
+
+
+
+
+
+
+
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.
+
+
+ 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.
+
+ mkv/h264 download /
+ webm download
+
+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.
+
+
+ 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.
+
+ mkv/h264 download /
+ webm download
+
+
+
+
+
+
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.
+
+