"IT'S A LITTLE PIECE of garbage math," said Brian, the tall, clean-cut engineer with the British accent at Integrated Silicon Technology Corporation, or IST as it is known in the business. "You don’t need to understand it; it just works."
But since the piece of "garbage math" was the core of IST’s product, or at least what made it work, I pressed Brian for some details. What’s more, I like to understand what I’m writing about no matter whether it’s chicken feed, garden sprinklers, or computer chips. He looked annoyed but began to explain what he called the discrete cosine transform or DCT.
We were sitting in one of those fancy glassed-in conference rooms in IST corporate headquarters. You probably know the scene: plush carpet, long polished oak table, big grease-pen blackboard at one end, and fancy speaker phone for conference calls--all a far cry from the dinner-nook that doubles for an office in my trailer with the varnished plywood, pull-down table (up you can sleep there). I sipped cautiously from a cup bearing the IST logo and glanced at a photo blow-up on the wall of an IST chip (where was the Minotaur among all those lines?), as Brian began to explain:
"The DCT is used to convert pixel values to frequency space. That allows us to do the compression." (Was Brian the Minotaur, and had he escaped?)
By compression he meant that less data could be used to represent a picture on a computer screen. That saved computer memory and processing, which was essential if certain types of computer graphics applications were going to be developed by companies--for instance "teleconferencing," in which you not only hear but see the person you are talking to on the phone. Okay, so who wants to see the person they are talking to? Well, not me. That’s the great thing about phones; you don’t have to stare, or be stared at, unnecessarily. But marketing people see things differently; they claim a lot of people like to stare, and that the wants, needs, or whatnots of those people are being thwarted by technological "bottlenecks" related to data compression. Whatever the case may be--you be the judge!--the conversion process allowed the programs, or the computer hardware, to eliminate data that the eye was relatively insensitive to. In layman’s terms the idea was this: if the eye can’t see it, why bother with it? (Would Daedalus be charmed? Probably.)
But I am getting away from the main point, which had to do with the little piece of "garbage math" that made it all possible.
Brian grabbed a blue marker and drew a large block on the grey-white blackboard. Then he divided the block then into quadrants. Next he grabbed a red marker, and inside the quadrants he nervously and rapidly produced row after row of dots representing the many pixels of data on a computer screen. The board now looked like it had a bad case of the German measles. It looked terminal as he drove home his point:
"Say you have 8 input values of y mapped to 8 output valued of . . . let us say z." He drew an equation on the board that reminded me of the Fourier transform, or at least part of it.
He began to talk about average intensity, DC and AC values, and scanning along a zig-zag path, which he attempted to depict by drawing a black wiggly line across the matrix of red pock marks--until my mind simply shut down, as though to protect itself from infection.
The other information, which we had covered earlier, such as "motion compensation" and "entropy encoding," was all understandable; the DCT, however, was not. Sensing some confusion on my part, he tried to explain it further, but my mind was securely boarded up; I listened, I took notes, but nothing got through. Not wanting, however, to appear at a total loss on the subject, I nodded intelligently, deciding, since I had lots of notes and was recording the meeting on tape, I would go back to the woods and clear it all up there. Things always take on a new perspective in the woods, you know--under pines, redwoods, oaks, and madrone; under a full moon; under a billion blazing stars at night truth stands out more brilliantly, and the missing link, like the wanderer Ulysses arriving home at last, falls into place. The missing link might be charmed somewhere by beauty, or it might be high on drugs, but eventually it would hear the call of duty and sail for home. Yes, back in the woods I would comprehend. Never fails!
Only this time I didn’t, not at all. I felt like Daedalus defeated by German measles. And I had an entire book devoted to the Fourier transform. In fact I had the definitive book on the Fourier transform by Stanford University professor Ron Bracewell, and some years ago as a student at that university I had studied the transform thoroughly.
Back in the woods I successfully excavated Professor Bracewell’s book from a storage compartment under the right cushion of the dinner nook--things get buried in a fifteen-foot trailer--fixed myself a cup of coffee strong enough to grow owl feathers on your wrists, and stared at the book. I recollected what a wonderful thing the Fourier transform had seemed when I studied it back at the university. Not only was it a beautiful and elegant piece of abstract mathematics, but in the physical world it evoked such pictorial wonders as the convolution of graphically precise physical functions. To me it had seemed one of the great beauties of mathematics that had fallout in many different fields as well--not just in "graphics quantization" where engineers were trying to save memory chip costs by fooling the eye--the human one, anyway.
But don’t take my word for it. Here is what Professor Bracewell had to say about the Fourier transform in his highly respected book:
"Fourier transforms play an important part in the theory of many branches of science. While they may be regarded as purely mathematical functions, as is customary in the treatment of other transforms, they also assume in many fields just as definite a physical meaning as the functions from which they stem."
As a graduate student at the university, my own interest had been in the application of the Fourier transform to music; I was interested in generating musical timbres that had never been heard before. And I was a little like Icarus, Daedalus’ son; I wanted to fly right into the sun. That was back in the Utopian 70s when there seemed no end to time and money for doing anything new while watching your family fall apart like a playing-card pagoda. Need I say it? Professor Bracewell’s book brought back memories.
Sipping coffee from a logoless cup with a chipped handle, I traced down the "little piece of garbage": It was the cosine portion of the Fourier transform calculated at discrete points, rather than as a smooth and continuous function of the variables. If you don’t understand that, don’t worry. Look at it this way: It was a kind of double degradation of the elegant transform named for the famous French mathematician. Or look at it like this: It was a Berlioz symphony played on the bagpipes to the accompaniment of a kettle drum.
Okay, I will back off a little. This type of thing is done in the applied sciences. There is even a kind of cleverness about it. But "garbage math"? No! Garbage products would be much more accurate.
But before I forget it, let me note one other thing about IST’s "application": There was a twist that Brian had not made clear at all--and it began to make me wonder whether he really understood the Fourier transform. The discrete cosine transform, as IST was using it, was applied in two dimensions: in both a horizontal (x) dimension and a vertical one (y).
I checked back in my notes and found that Brian had in fact written the equation down incorrectly; he had written it for a one-dimensional transform only!
But enough of this garbage. You can smell the problem, can’t you? Great scientific thought and knowledge dragged from the university and forced into servitude in the manufacturing whorehouse of consumer products--products, by the way, that no consumer ever expressed the least desire for.
But I will shut up. I am beginning to sound self-righteous, a worse affliction among mortal minds than honest whoring for a buck. And to cut this episode short, I will just say that the article was written and accepted by the magazine; but due to an executive far up the corporate ladder who felt that IST was taking a risk paying for the article before it actually appeared in print--a couple of months off--I was not paid soon enough to attend my kid’s graduation. Not what I had in mind.