I've wanted to have machines that could interpret and understand images of natural shapes, just like we now have recognition systems for speech and language. Nowadays every time you pick up the phone to dial an operator you're dealing with those kinds of recognition systems. But we have nothing like that, or very few systems, at least, that can recognize objects in images, that can see and interpret the world and take action. We have a lot of imaging devices—cameras. But they're ignorant. There's no interpretation, no global decision making, no context. What we want to do is to develop machines that could, for instance, really understand faces or shapes, or the language of faces and the language of shapes. Just as Chomsky made us very comfortable with the ideas behind an acoustic stream of words, with the global connections of words that form deeper structures, we would like to do the same with "the language of shape". There are much more global constructs that form our notion of anatomical shape beyond the dots on a screen. Take, for instance, the context of understanding biological shapes: if we meet, you and I, we will immediately recognize each other's age, within five or ten years; you'll know I'm a man; you won't mistake me for a dog or a baby. And yet all our bodies are so different. There's this global context that emerges. This is what computational anatomy is all about: getting machines to do computations that generate biological and anatomical structures.

  Is your aim getting machines to generate biological and anatomical structures or getting them to understand such structures and make decisions based on that understanding?

Alan Turing defined an intelligent machine as one that you can talk to and you can't tell, from its responses, that it’s a machine and not a human. That’s known as the Turing test. If we can get a machine to generate a face that we could not distinguish from a human face, then it's going to be an interesting machine. That's the first step. We are already seeing a lot of this technology in the movies. There's incredible animation available. If machines can compute structures that are equivalent to the structures we see in the world, then we can begin to understand them. That’s what computer anatomy is all about: getting machines to compute equations of motion that simulate biology. It's analogous to physics, in a sense. In the 1800s, we had Maxwell's equations, which told us about electricity and magnetism and how they propagate through space. In computational anatomy, we now have equations as well, that describe how tissues can grow and bend and morph and change. These equations seem to generate very realistic structures and we can use them to do recognition, as well.

  Who gets credit for inventing the field of computational anatomy?

Well, there have been many of us working in computational anatomy since the early 1980s, although we didn't call it computational anatomy at the time. In a sense, it goes all the way back to D'Arcy Thompson in 1917 who wrote On Growth and Form and literally laid out the vision, although at the time they didn't have computers to process high-dimensional data and perform difficult calculations. Now we have computers so we can do all this stuff. Ulf Grenander and I, working together, made some breakthroughs in the ‘90s that led to a paper formalizing the idea. We called it "Computational anatomy: an emerging discipline," (Quart. Appl. Math. 56[4]: 617-94, December 1998) and it's now one of our most-cited papers.

  How did that idea develop in the past decade?

Well, with Gary Christensen and Richard Rabbitt, who are colleagues of mine, we introduced some basic equations of motion from mechanics that we used to describe shape change and transformation. These equations of motion—they're called Eular-Lagrange equations—define the way particles move and transform, and they are very fundamental. And we thought we could use these equations of motions to compute growth and shape change in medical imagery. So now we can take medical imagery and grow one shape into another—a tumor, for instance—and we can use these equations to track a tumor's growth.

  What do you mean by "track a tumor"?

You have a series of images of it over time and you want to quantify how it moves and changes and grows or shrinks. In effect, you want to quantify this dynamic process. So you need some sort of computational equations that really describe what you're looking at. Then you can look at the equations and see what's really happening at a precise metric scale. Otherwise, it's just a picture; you're just looking at a fuzzy diagram. We started doing that in the early 1990s, and we had the use of tremendous supercomputers, so we could do computations that nobody could do before. And then with Grenander, who has really been very influential in creating new theories of patterns, we realized that what we were doing was part of a much deeper basic area termed Metric Pattern Theory, and that's what started to get rounded up into the general equations that are now fundamental to computational anatomy.

  What are the implications of your work? Where will this technology lead in the coming years?

We hope that it becomes part of the mathematic and algorithmic basis for making the analysis of shape and structure from imagery into a precise discipline. So we don’t just look at pictures, we can actually decode the structures in them. One area where researchers are really exploiting computational anatomy is in computational neuropsychiatry. People are using these techniques to look at brain shape and change in many neuropsychiatric illnesses. So we know now that schizophrenics, for instance, have an asymmetry between the left and right sides of their brain. We know that in Alzheimer's disease, there are all sorts of shape-related changes in the cortex that are associated with aging. They had all been qualitatively described, but now you can actually study them by computing metrics that describe specific aspects of growth and aging. In radiation oncology, this computational anatomy is very fundamental to tracking the effectiveness of therapies. You would like to be able to make statements about whether a drug therapy or an intervention is working and you'd like to be quantitative about it. Computational anatomy allows that to happen. You can use it for drug trials—to report efficacies of treatments. Then there's the whole burgeoning field of security biometrics. Computational anatomy is really the fundamental aspect of metrics for biological shapes. Since September 11th, there has been tremendous interest in coming up with metrics for facial images, so we can make statements about identification based on these metrics.

  What has to be done to get this technology to the next level of wide commercial application?

The problem at the moment with image analysis is the fragility of the technologies. They are very brittle, in the sense that we can compute metrics for pictures and across pictures and describe things that are similar and different, but they require a lot of very specialized preparation. In effect, they are adopted to specific instances. They're not robust and don't work well out of context. Remember in the early 1980s, when speech-recognition systems came out, you would have to train them on your own 200 words, and if you spoke quickly or had a cold they wouldn't recognize the word? They were fragile. The same is true in many of the image analysis technologies. They're highly tuned to certain parameters, with a lot of assumptions about context. You change the context and they fail and so you can't use them routinely.

  Are you satisfied with pace of research?

Yes, I'm satisfied. I'm amazed at how this recent line of attack has been received by the community. When we were talking about various parts of computational anatomy, people were very very cautious. They weren't sure what it was that we were doing. And now it’s really moving quickly. Recently, David Mumford made the connection of these equations of motion to the now-classic work of Arnold on Euler’s equations for incompressible fluids. These equations seem to be fundamental to a theory of shape. I'm pretty excited. The ideas are very powerful, and people are seeing that our community is moving forward.

  What are your research plans for the next few years?

Well, we are continuing to search for the equivalent of Maxwell's equations for computational anatomy. I would like to see more precise equations for growth and development that really describe the way biological shapes change and transform, and I'd like to have it actually make sense biologically, so that it's not just about good mathematics, good computational algorithms, but is actually relevant to the underlying biology.

  Do you work closely with biologists?

I tend to, yes. I work with neuroscientists, people who really work on cutting-edge methods for generating microscopic and macroscopic pictures of biology. That's what forms the basis for our applied mathematics of shapes, structure and change and growth.

  What would you like to convey to the general public about your work?

In image analysis, we have never had metric spaces and models like we've always had, say, for classical communication theory. In classical communication theory we have precise models that Qualcomm, for instance, and all the other companies use for signaling and signal transmission and capture. In image analysis, it's really been a free-for-all. It's the Wild West of science. We simply don't have tractable models for analysis of most natural images. If you go to the most fundamental level and say what is it that allows modern communications to be so effective, it's that there are some very fundamental concepts associated with metrics—with measuring the closeness of signals—allows the field to be so successful. This is what D'Arcy Thompson said shape was all about—it was about comparisons of things. In image analysis, one of the Holy Grails is this "metric pattern theory" of Grenander’s that will allow us to say in a rational way that this shape is close to something else or very different. You have to make a plan that's rational and logical. You'd like to have some quantitative measure that actually says this shape is moving closer to that shape, in therapy, for instance, declare that a shape is moving closer to a healthy shape or a healthy state. We don’t have any of that. We just look at things and roughly make decisions. If we could build metric spaces that accommodate the space of all observable imagery of biological shapes—all human faces, cells, mitochondria, nuclei, folding brains, hearts, sorts of heart diseases, etc.—if we could do that all quantitatively; if we had a precise ruler-like metric that allows us to take pictures and say that a shape here is very close to a shape or state there, then we would start to have real understanding. That's what we hope to accomplish in computational anatomy.