What would a conference built around the computer graphics industry be like? Pretty boring, right? There’d be talks on eking out another million triangles per second or whatever they do these days, with boring engineers talking about slightly larger and faster registers and pipelines so kids can play slightly more elaborate computer games or whatever. Dead wrong. That conference is SIGGRAPH (Special Interest Group on GRAPHics and Interactive Techniques) and it’s batshit insanely cool.

The most shocking thing about innovations in computer graphics (understood pretty widely, mind) is that they’re so simple and obvious once you hear about them. You really notice that as a field of study it’s actually pretty young. Example: content-aware image resizing. It was presented at SIGGRAPH 2007 and it appears to be a totally novel and simple idea about how to naturally destroy or create information in images.

It works something like this:
1) Detect edges in the image (borders between areas of colours). There are numerous algorithms to do this.
2) For each pair of pixels opposite each other at the top and bottom of the image, find a path from one to the other that crosses the least edges in the image. This is called the least-energy path and it contains “the least information”. (Everything works the same way if you want to destroy/create horizontal lines, just use pixels at the sides).
3) If you want to reduce the image horizontally, remove paths (the previously computed least-energy ones) from the image starting from the one with the least energy. To enlarge the image, take the path with the most energy, compute the average of that and its highest-energy neighbour and insert it between them.

Pretty simple, right? And does it work well? Like magic. The video has numerous interesting details I omitted here.

Now, this is more cute than really interesting, but still: what do you think this inequality describes?

(the square brackets without tops represent the floor function)

Well, the outermost modulo operation is of the form mod( f(x,y), 2 ) so perhaps it codes information from x and y into bits somehow. Almost, but not quite. Let n be an integer and graph the points for which the inequality is true in the plane-slice 0 < x < 106 and n < y < n + 17. This will draw information contained in n - in fact an arbitrary image that fits into a 105x16 grid. For n = 9609393799189588849716729621278527547150 0433966012930665150551927170280239526642 4689642842174350718121267153782770623355 9932372808741443078913259639413377234878 5773574982392662971551717371699516523289 0538221612403238855866184013235585136048 8286933379024914542292886670810961844960 9170518345406782773155170540538162738096 7602565625016981482083418783163849115590 2256100036523513703438744618483787372381 9822484986346503315941005497470059313833 9226497249461751545728366702369745461014 6559979337985374831437868418065934222278 98388722980000748404719, the inequality describes a little picture of the formula of the inequality itself:

As I say, cute. This and more is detailed in a SIGGRAPH paper here.

This is a total meta-post, so just skip it unless you’re bored.

My previous post made me think of two things:

Where did that thing in the subject come from? Least pessimum? I’ve just remembered it’s an adaptation from The Story of Mel, an excellent and rather long programming anecdote. Excerpt:

Mel never wrote time-delay loops, either, even when the balky Flexowriter required a delay between output characters to work right. He just located instructions on the drum so each successive one was just *past* the read head when it was needed; the drum had to execute another complete revolution to find the next instruction. He coined an unforgettable term for this procedure. Although “optimum” is an absolute term, like “unique”, it became common verbal practice to make it relative: “not quite optimum” or “less optimum” or “not very optimum”. Mel called the maximum time-delay locations the “most pessimum”.

The other thing was that optimism bias made me think of sexual strategies, which made me think of the best spam email I ever received. It advertised, among other things, the best combination of sex drugs ever: NYMPHOMAX and SUREGASM. I mean, that could be straight out of The Simpsons.

Also: the subject “line” of this post is the subject of a spam message I received this morning.

Half Sigma has been writing about optimism bias recently. I found a post about the tendency to overestimate one’s intelligence to be especially striking. A chart I stole from it:

Frequency Distribution
Cells contain: -Column percent -N of cases		Distribution
V174	1: FAR BLOW:(1)	1.1 150
	2: BELOW AV:(2)	1.5 216
	3: SL BELOW:(3)	4.3 597
	4: AVERAGE:(4)	31.4 4,389
	5: SL ABOVE:(5)	24.9 3,476
	6: ABOVE AV:(6)	28.6 3,993
	7: FAR ABOV:(7)	8.3 1,158
	COL TOTAL	100.0 13,979

The chart tells you what amount of subjects self-assessed themselves to be in which “intelligence group”. The study was on US high school seniors who I guess might be more optimistic than Finnish kids, but still. Under seven percent of subjects considered themselves to be below average in intelligence and over sixty percent considered themselves to be above average. Almost any way you interpret this, if someone tells you they’re of above average intelligence, the best assumption you can make on that information alone is that they’re not.

An interesting question about optimism bias (like this) is its cause. My first idea was an evolutionary explanation: that optimistic males will try out more things and be more adventurous – in particular they’ll go after more females even after failure. For well-known biological reasons it would then make sense for females to have a smaller optimism bias than males, but this is not supported by evidence (in humans). Maybe it happens with other animals?

Anyway, it seems possible that the majority of optimism bias in humans is caused instead by humanness, ie. culture, society and so on. But how can this be? I’ve always assumed that it would be extra painful to have a high opinion of oneself and to be proven wrong all the time. Perhaps it’s even more painful to self-admit one’s mediocrity/suckiness.

As for the intelligence thing: I’ve always kind of assumed that I scrape into “above average”, but considering this I’m not so sure anymore. The only real way to know is to get tested, and I don’t want to. Too scary.

One of the more curious things I’ve discovered in HAKMEM:

Although not new (cf Coxeter, Introduction to Geometry, 1st ed. p393), the following coloring number (chromatic number) may be useful to have around:

N = [[(7 + sqrt(48 H + 1))/2]]

where N is the number of colors required to color any map on an object which has H holes (note: proof not valid for H = 0)

(Beeler, M., Gosper, R.W., and Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb. 29, 1972)

[Note: “color” here means “colour so that no two regions that share a length of border have the same colour”. “Map” means map in the intuitive sense; a partition into sensible sorts of bits of the surface. Think countries on a map of the world. The outermost square brackets mean the floor function (the greatest integer less than the function’s argument).]

Isn’t that cool? HAKMEM is obviously not all that mathemathically sophisticated; it doesn’t tell you what kind of object. I guess in this case it’s just bounded and connected and stuff, in R^3. There’s probably also some nuance to do with the concept of “hole”. But to think that this can be found in a book called Introduction to Geometry! Further evidence that all math books have “introduction”, “elementary”, “approach” or something similar in the title just to make everyone feel small.

If you’re having difficulties picturing this, think about a torus. The hole is one extra way for each region to “come around” to meet each other (in contrast to regions in the plane), but one is “stuck in the middle” so the chromatic number is 7.

Another way of thinking about the torus: start by drawing a band of colour around the entire outer rim of the torus. The rest is now split into two “colourable planes” that can conflict through the hole, so you need 4 + 4 – 1 colours (the outer rim gets counted twice, once for each plane, so we subtract it once).

If this kind of thing is totally new to you, get some paper and try to find out how many colours you need (at most) for a map in two dimensions, ie. a plane.

Question: how would this work in R^n?

Question: do people find it easier to be around friends than non-friends? Shurely.

If you know me, you probably know that it’s difficult for me to be comfortable around other people. There are many reasons for this and it’s a pretty common characteristic. But this is odd: it’s not particularly better around friends. I even feel a kind of special pressure to make the situation function, to keep everyone from getting embarrassed or bored (by me). This is a pretty stupid way to think about one’s friends, but I do it anyway.

Does this mean that I don’t really think of my friends as friends, but as social obligations? I don’t know, maybe. Probably not. It’s not as if I feel obligated to get into many more social situations than I want to (if you know me you probably also know that I seldom go anywhere or do anything).

Probably the reason I can sometimes be oblivious to social pain around non-friends is that I care about them less, and (these days) have sufficient self-confidence to think that maybe they’re bored because they’re boring.

Bottom line: it sucks to care about people.

What I meant to say about Freaks and Geeks was that it is good in the way normal people who love what they’re doing come up with. It’s crafted but often flawed, opaque, meaty, mostly obvious and simple. Warm. It makes you feel good and, in this particular case, happy. If you like the people behind it you end up liking the thing, and as previously mentioned, I like Freaks and Geeks. But I can accept that plenty of people aren’t particularly impressed (it got cancelled 2/3 through the first season due to poor ratings).

Then there is a kind of goodness that is more scary than “nice”. Bach doesn’t sound good because he loved what he was doing, he sounds good because he’s on a mission from God. He was prolific to the point of nonchalance, he doesn’t seem to have made a lot of fuss about himself and he probably wasn’t much fun to be around. But there can’t be a composer, musician or serious music listener who isn’t impressed by his work. It just sounds perfect.

Of course, there’s no tv equivalent of Bach (as far as I know). A television show is too complicated a thing to make with a “perfect” touch. Maybe something like Angels in America represents a serious attempt. Sopranos is also obviously distinct from what I described earlier (I’ll call it “Tarantino-good”) – it’s too huge, organised, expensive and Shakespearian to allow a brief description of motivation or nature.

My reassuringly expensive Freaks and Geeks box set arrived. The plastic case is pretty badly broken; worse luck. I’m trying to decide whether I want to live with the temporary hassle of sending it back and getting a new one or with the lifelong frustration of owning such a cruel combination of perfection and imperfection.

But what I really wanted to do here was to quote Judd Apatow (executive producer) from the enclosed booklet:

Q: Is this the end of Freaks and Geeks?
A: No. I shall use the warm feelings that this country showers on this show to prolong my career well past the point where I have stopped making quality work. I will use the credit to secure a job on a mediocre sitcom, where I will not pitch any jokes in the writers’ room. Instead I will bore everyone with stories about the making of Freaks and Geeks and make sure to mention at least twice a night that that show was much better than this show. When the show is cancelled, I will be unable to secure another job and will be forced to teach at USC, where I will make the kids listen to the commentary tracks on this DVD over and over and over and over, never listening to their screams for mercy.

Quentin Tarantino said it and I always believed it (and shall now paraphrase it): if you really love what you’re doing, you can’t help being good.

edit: I didn’t really mean “perfection”, I meant something else. I’ll try to come up with another word.