The Realm of Finite Possibilities

Pixels, Pixels, and more Pixels

We begin with a 2x2 grid of squares. Imagine that the squares in the grid are “pixels” in a binary image. We start with 4 pixels, forming a 2x2 grid.

2x2 grid of pixels.

What I am going to be talking about for now is what is called a “bitonal image”. Really it’s just a black and white image, where pixel values can be either black OR white, with no other possible values they can take on.

The square you see above would be 4 white pixels. What I want to do is give you an idea of how many possible “permutations” there are of just these 4 pixels, given that there are only 2 possible “values” per pixel, black or white.

Simply put, with a 2x2 matrix, we have 16 possible permutations for a bitonal image, 2^4 (“two-to-the-power-of-four”) which = 16. Let us see what those possible permutations look like, since there are only 16.

16 possible “permutations” of a 2x2 pixel matrix with 2 possible values (black and white, “bitonal image”).

What I really want to get to is the idea of a bitonal image with black and white pixels that is 256 pixels by 256 pixels square. How many possible “configurations” or “arrangements” of black and white pixels do you think such an image could possibly take on?

The answer is 2^65,536 (“two-to-the-power-of-sixty-five-thousand-five-hundred-and-thirty-six”). Just to give you an idea of the size of this number, it happens to be a number with 19,729 decimal digits. Also to give you an idea, 2^64 — “two-to-the-power-of-sixty-four” — is equal to 18,446,744,073,709,551,616, a number with 20 decimal digits. That would be the number of possible “arrangements” of pixels for an image that was 8 pixels wide by 8 pixels high, given as before that there are only 2 possible pixel values, 0 or 1, black or white.

Imagine for a second that we were dealing with a 256 pixel by 256 pixel image in full 24-bit color, meaning that each pixel could have a value from not 2 possibilities, but 16,777,216 different possibilities. That means that we would be dealing with a number of possible arrangements that is represented by a very, very huge number 16,777,216^65536, a number coincidentally with 473,480 decimal digits. A very, very big number indeed.

Now let’s go back to our 2x2 pixel matrix of black and white pixels, only 2 possible values for each of 4 pixels, making 16 possible “configurations”. For a 2x2 grid, we have 16 possible permutations. For a 4x4 grid, we have 65,536 possibilities, and for an 8x8 grid, 18,446,744,073,709,551,616 possibilities.

As you can see, when we double the size of the image, from 4 pixels to 16 pixels, and then to 64 pixels, i.e. from a 2x2 grid to a 4x4 grid, to an 8x8 one, the number of possible permutations grows exponentially.

Poor attempt at plotting powers of 2 to show exponential quality of the curve.

What does all this mean, and what does it have to do with PAINTING?

Short answer: “Picture Space”, a.k.a. “Finite Picture Space”.

256 pixel by 256 pixel image with random black & white pixels, a.k.a. “pure Gaussian white noise”.

There seems to be a common misconception about visual artists, about painters, that there are “infinite possibilities” for what a painting could be. Often one hears something like this about the artist, about her imagination and creativity and about how “anything is possible”, i.e. given basic technique and lots of imagination and creativity “anything is possible”.

The truth of the matter is that painting is very limited as a tradition, limited BY its own tradition, and that’s a good thing. If we think of a composer like Ludwig van Beethoven, his “genius” if you will was not that he did some fantastic, wonderful thing with music in and of itself, but that he did what he did WITHIN THE TRADITION in which he was “thrown”, historically.

While it is true that for a given size canvas, say a canvas of dimensions 16 inches by 20 inches, it is theoretically possible to do a vast number of things with it. One is not limited by “number of pixels” and “pixel values” as in a bitonal image, as described in the examples above. One can use any number of colors and textures, collage elements, and any number of brushstrokes and other techniques. I would still argue that there is an upper limit to what is visually “possible” in the sense of a painting being some form of image that is “discernible” from another image. At some point, one will end up making copies of the same thing. In any case, I believe that for any given subject and genre, there is an upper limit of some kind, though massively huge.

Even if there was no limit to how many different images one could make, a painter is still constrained by countless other things. In that regard, if one thinks of each painter as having a unique “Picture-Space”, then in their lifetime there is a limited number of paintings they can feasibly make. There are other feasibility constraints, constraints on number and size of paintings produced in any given period, constraints related to inventory control problems, to storage space, to the artist’s ability to “ship” paintings onto the market once produced. There are constraints of tradition itself, too.

It’s not true, and never was true, that a professional painter just sits in front of her canvas and just “let’s herself loose”, suddenly possessed by the power “to do anything”. Not everything is possible. One is also limited by one’s previous oeuvre or corpus. If I only ever made paintings of the mountainside, and my mountainside paintings have a certain value on the market, I have to take this into consideration if I decide that I want to start painting automobiles or cows. I am constrained by my previous work, by the work of current working artists, and by the works produced in the entire history of the arts, going back tens of thousands of years. Work isn’t automatically “relevant”. There is a method to the madness. One is actually much more constrained than “free”.

What Does This Have To Do With Machine Intelligence?

Believe it or not, this has a great deal to do with current developments in machine learning, pattern recognition, image classifiers, even Deep Learning initiatives, or machine intelligence in general. For example, often systems using supervised learning start with image datasets, the training data. These are notoriously costly to make or acquire, and yet, for our purposes, they are made up of IMAGES. They are FINITE in number, that is to say that the dataset is made up of a finite number of images. But what makes a good “image” for machine learning tasks, for an image dataset?

The other perhaps even more important question has to do with recent developments attempting to “marry” machine intelligence and artistic creation. That is to say, various ventures involved in research around “computational creativity” and “computational aesthetics”, from generative arts to algorithmic composition, etc., have EVERYTHING to do with the same “Picture Space” that I previously mentioned. If a machine can “generate” an image from scratch that is “aesthetically pleasing”, then this image is a subpopulation of the “population of images” that makes up the complete, total, “image space” of all possible images, of those dimensions and other parameters, characteristics.

The thing is, for digital images, bitonal images and so forth, it really all comes down to various “combinations of pixel values”. And the impact can be of CRITICAL IMPORTANCE. For instance, machine learning systems are vulnerable to adversarial examples. This means that it’s possible to “fool” or else “exploit” or even “break” classifiers in machine learning systems. This can be of extreme, critical importance, for autonomous systems even as simple as autonomous vehicles, cars. No one wants autonomous cars to accidentally crash themselves because their machine learning systems were “vulerable to adversarial examples”. This has EVERYTHING to do with the “Picture-Space” of which I have been speaking, which I will make abundantly clear in future articles in this same Medium Collection.

It’s not because I am a “painter in the classical tradition” that my work, my lifetime of research and experiments, is not pertinent to present-day innovations and related concerns in machine intelligence and other like things. I am all too aware of current technological developments, and I am working on solutions 24/7. It starts with imagining the “space of all possible pixel configurations”. It starts with “noise functions”, what I call “seed noise”, which I use to start most of my experiments in image processing. The problem has to do with AMBIGUITY. That is the main problem, in my book, with these kinds of problems with “adversarial examples”, because they precisely have to do with the “decision boundary” of the classifier, which has EVERYTHING to do with AMBIGUITY BETWEEN CATEGORIES.

That is the thing. I worked for years in content analysis where classification schemes were of utmost, critical importance to the work we did. I understand classification problems INTIMATELY. My addition, or innovation, to the discourse, was the creation of the mathematical expression “nearly-indistinguishable-from”, as in the expression, which I commonly used, of something being “nearly indistinguishable from noise”. (For instance, a pure noise and another functional output being “nearly indistinguishable from noise” means that formally, mathematically speaking, they are co-equal, it is impossible for any computational or algorithmic system to “tell the difference”. That is the meaning of “nearly” here, it is “nearly indistinguishable”, that is formally, mathematically indistinguishable).

Are There Any Precedents For Any Of This? Where Is This Coming From?

I’d like to start by answering the second self-generated, self-imposed question. It stems from a lifetime of observation, a lifetime of study and experiments in the arts, in the laboratory of my Atelier, or art studio (interdisciplinary). It comes from doing mountains of research, testing it on the blackboard, trying to see if I missed anything, doing more studies, making more observations, etc. Of course it’s analogous to a kind of cyclical process in that it is the same method being used over and over again for almost everything. But I would say it is an iterative and evolutionary, ambient, and experimental design workflow methodology. It just happens to be applicable to painting also, which I discovered with the invention of the Project-based Serial Method.

So that is what led me to the creation of various different “sciences” which, for all intents and purposes, in the market for scientific research, or in the academies, is either pure make-believe or else “pseudo-science”, or worse. I would argue that they are proto-sciences, tiny science blocks. Modules, if you will, of a kind of “systematic abstract science”, to make a kind of analogy with the work done by Stephen Wolfram in his book, “A New Kind of Science” (Wolfram Media, 2002, 1197 pp). Great, but is there any precedent for this BEFORE any of what I just said?

Well, yes, and yes. It depends on how one defines and understands precedence and how it is being used, what it is being used for, and a host of other constraints. Countless efforts were made since the dawn of time at the codification of rules of art. One such example and possibly one that is most pertinent to me as visual artist, is the written work of Leon Battista Alberti (1404-1472), such as De pictura. De re aedificatoria, another book by Alberti, is also very pertinent to me even though I’m not an architect or even building anything. If one takes the time to work through these kinds of big, heavy, often highly abstract and theoretical works, despite their often being difficult, maybe even frustrating reads, and one “tests” it out in practise, I for one have found the results of experimenting empirically in the studio with such sound, time-tested art theories — let us say — rather illuminating.

Can I be more specific? Yes. Yes, and a much more recent example of a kind of precedent related precisely to my concept of “Picture-Space”, though approached from another angle entirely. It is Wassily Kandinsky’s “Point-Ligne-Plan: Contribution à l’analyse des éléments picturaux” (original in German, 1926). Here we have something similar to the codification of rules of art of others before him, such as the aforementioned Italian polymath, Leon Battista Alberti. It essentially comes down to the same thing. (And it is gruelling work. Taking a break, coming back to this in a bit. A.G.)


A.G. © 2016. All Rights Reserved.