Fairly Ranking the Most Brilliant Birds

After I finished the project of my last post I still had a giant collection of processed bird color data from BirdColorBase where every way I sliced it revealed new and mind-blowing birds. The minimal treatment I gave in that post just wasn’t enough. I had to dig in more. I set out to find the brightest, most brilliant, most colorful birds. I’m not going to make you wade through an essay’s worth of math and discussion before I show you my top 40. Here it is. Orange-breasted Bunting Andean Cock-of-the-rock Variable Kingfisher Asian Green Broadbill Black-hooded Oriole Plum-throated Cotinga Red-winged Parrot Emperor Fairywren Scarlet Macaw Blue-crowned Woodnymph Golden-tailed Sapphire Rose-bellied Bunting Guianan Cock-of-the-rock Paradise Tanager White-bellied Canary Green Thorntail Crimson Rosella Blue-and-yellow Macaw Crested Quetzal Scarlet Ibis Blue-headed Parrot Prothonotary Warbler Hunter’s Sunbird Yellow-crowned Amazon Violet-tailed Sunbird Doherty’s Bush-shrike Long-tailed Broadbill African Pygmy-kingfisher Beautiful Sunbird Scarlet Minivet Swallow-tailed Hummingbird Red-flanked Lorikeet Grant’s Bluebill Purple-breasted Cotinga Mayotte Sunbird Dusky-billed Parrotlet Papuan Lorikeet Opal-rumped Tanager Yellow Warbler Blue-chested Hummingbird This is ranked with math, but it is subjective math. If you disagree with the list, if you feel I have disrespected your favorites, or omitted some obvious contender, read on to see if I have treated your most beloved bird fairly, and what it even means in principle for a subjective ranking to be fair. I’m going to walk you through how I turned my subjective preferences into math. In the process we’ll touch on a number of common functions that are useful in information retrieval, but rather than ranking documents based on their relevance or quality, we’re ranking birds by their color. Chroma Where our story left off, we had found all the birds with plumage samples outside of sRGB and display-P3, the birds for whom at least some part of them is undisplayable on a conventional screen. This yielded pretty big lists of birds, so a natural next step to ask is, which are the brightest? What birds are behind all those dots at the edges? For that we need a notion of what it means to be “at the edge.” The simplest is the convex hull. There are a few problems here. The first is that there are too few birds. I want to see more birds than that. The second is that there are a lot of regions of the boundary very sparsely populated with birds. I want to see some magenta birds, but because the space of bird color curves inward there, no convex hull can show me. There is a clever generalization of the convex hull that can improve this, called the alpha shape. The gist of it is that rather than prohibiting any concavities in your hull, you allow concavities only up to a given size. If you can carve out a region with circles of radius more than 1/alpha without snaring any of the points in your circle, then you add the points on the boundary of the circle to your hull. This is much better. We have a lot of birds evenly spaced on the boundary now, and know now where to find all the most extreme colors that birds can show us. As a ranking of the most brilliant birds however, there are still two problems. First, this is showing the colors of individual plumage samples from birds, not full birds. If you are an entirely beige bird with one brilliant spot, you can easily earn a spot on this list. No offense to the fire fronted serin, its forehead spot is indeed very fiery, but one spot is just not enough. Or the cut throat finch, great Halloween costume and fantastic name, but we’re looking for a bit more than just a permanently bloody throat. Off with his head! The second problem is that this is scoring colors based on their saturation, but not their brightness. A bird that manages to achieve the same color of green but reflects twice as much light is meaningfully “more green,” and I want the ranking to reward that. This plot filters out feathers that were too dark to have a meaningful color, but to really rank birds, I’d like to incorporate brightness in a more continuous way rather than just a binary threshold that means “definitely not black.” The CIE colorspace has a transformation which extracts a component, chroma, that is very similar to what we call vibrance. It rewards both the saturation of the color, and the amount of light the color reflects, so this is a promising place to start. You might expect this to be some simple function of the quantities of light at different wavelengths, but it is quite complicated and abstruse. You first convert the spectrum to CIEXYZ, then CIELAB, then to CIElCh, using functions and parameters set empirically based on properties of the human eye without any intuitive interpretation. Before I found this I spent quite a lot of time experimenting with different ways to define the vibrance of a color using the more raw physical properties of the reflectance of the feathers. To quickly summarize all of the things I tried which didn’t work, let me leave you with an aphorism. If you find yourself reinventing color theory from scratch, stop. Let’s let CIE do all our work for us and just rank birds by their average chroma. Before we knock him off his perch, let’s spend a minute appreciating our current champion, the scarlet ibis. It is an avian stop sign, or perhaps a toddler eating a plate full of strawberries. It makes neotropical mangrove trees look like giant raspberry bushes. It is a very very red bird. The scarlet ibis looks exactly like an American white ibis that found a truck full of cherry fun dip and was very excited about it. Taxonomists aren’t entirely sure whether to classify it as a different species. The two ibises have overlapping regions, interbreed, and are identical in every anatomical detail except for their color, but what a color! “While showering dignity and color on the scarlet ibis, nature seems to have been reluctant in the bestowal of weapons. The bird’s beak was blunt, its toenails were unsharpened, and its eyes had a gentle, soft Bambi quality.” – Dr. Paul A. Zahl, Coro-Coro This is a promising demeanor for a bird we might crown as the prettiest, but it does suggest a bit of overconfidence, a bit of resting on one’s laurels, perhaps a nest made of such. If you simply want to maximize the amount of color that a bird shoves in your eyes regardless of any other aesthetic qualities, stop here. Aside from being bright solid red from crown to tail, it is also coincidentally quite a large bird, sure to fill your binoculars. For maximizing raw color per unit bird, we have a clear winner. Variability Let me explain why I’m not stopping here, and want to iterate more. Consider the first two birds. If you look closely at the chroma plots, the Andean cock-of-the-rock has nearly the same hues as the scarlet ibis, but every sample has higher chroma. It comes in second place only because it also has black wings. Similarly, compare the white bellied canary to the emperor fairywren, which it outranks by 18 positions. The emperor fairywren has higher chroma wherever it has color, but it has more black. Penalizing it for that doesn’t match how I think about the brilliance of a bird. Black among bright colors accentuates them by contrast, such as on an oriole or a goldfinch. If I’m given a choice between a bird with middling colors, and a bird with intense colors among patches of black, I would choose the half black bird to be more eye-catching every time. To express this in math, I changed from a simple mean of the chroma values to a weighted average, where the weight itself depends on the chroma, w=b+caw = b + c^a. Setting different values of bb changes how much we care whether a bird has blacks or whites, where 0 treats the bird as if it didn’t have them at all. Varying aa changes how much I care if a bird has beiges or browns. Setting bb too low yields lots of solid black birds with a few bright spots. Notice how almost all of them have red as their singular color. This will be important in a second. After some tweaking I finally opted for 0.25+c0.25 0.25+c^{0.25} as my weighting function, which gives the cock of the rock and the oriole a leg up among the top 5. My second and more significant quibble with the pure chroma ranking is that the scarlet ibis is just one color. It’s a lot of color, but just one. I wouldn’t want to say that it’s impossible for a monochromatic bird to be the brightest or prettiest of all birds, but right now our ranking doesn’t even reward a bird for being multiple colors at all, in fact it hurts it. Birds are very good at being red and yellow, due to the carotenoids that some birds transfer to their feathers from their diets. The CIE color space allows reds to have a high chroma even if they are very dim, and considers yellows to be very bright. This means that without any consideration for variety of color, the leaderboard will end up dominated by dim reds or tepid yellows. If a bird deviates from a pure red or pure yellow strategy, it is likely to do worse just because intense blues, greens, and magentas are harder for birds to make, and CIE expects more energy from them to raise the chroma. It is unfortunate that our ranking has this sensitivity rather than being based on simpler physical facts. One might be tempted to work with lower level primitives like the raw quantities of light, but as I mentioned before, when you are tempted to reinvent color theory, stop. Defining the vividness of birds to eyes necessarily requires opinions about eyes. Most of those opinions are best left to CIE. A natural concept to implement to reward birds with many colors over birds with just one is “diminishing returns.” Once a bird has a lot of red, you don’t want to keep giving it the same amount of additional credit for more red. You can instead say, “That’s enough red now, what else can you do?” This is naturally expressed by a power