Non-uniqueness, ghosts, and ill-posedness
I would like to explain why and how the EIT image formation problem is ill-posed, or extremely sensitive to modelling errors and measurement noise. To explain ill-posed we need to understand what is well-posed.
This simple simulated example illustrates the ill-posedness of EIT. We start with two circular, flat patients seen on the left. The top patient is OK, but the bottom patient has something wrong with the left lung.
For mathematical convenience we use voltage-to-current measurements instead of more practical current-tovoltage measurements in our simulated example. Here we impose the same electric voltage around both patients. I used the Finite Element Method for computing the resulting electric potential inside the patients.
We measure current flowing through the boundary after applying voltage on the boundary. The current distributions are shown on the right. They look very similar.
Let us overlap the currents measured. They have very little difference between them, although the patients are very different. This is our first hint about the ill-posedness of EIT.
Perhaps we were just unlucky with the choice of voltage distribution at the boundary in the previous slide? Let's try something else: a more oscillatory voltage.
Again, there seems to be only little difference between the current measurements.
Overlapping the two current distributions shows that the big difference in the two patients shows up as only a small difference in these measurements.
The previous choices of variously oscillating cosine functions had a hidden agenda. As some of you certainly know, one can represent general functions as so-called Fourier series, where building blocks are sines and cosines with different frequencies. Here you see a collection of measurements based on such increasingly oscillatory inputs. It seems that the big difference in the two patients shows up as only a small difference in all of these measurements. That's ill-posedness.
Now if we simulate measurement noise, we get these slightly jagged curves as our data. The difficulty of EIT imaging is seen here: we should design a computer algorithm that produces the upper patient image with the blue curves as input, and the lower patient image with the red curves as input. That is a hard problem indeed as there is only a little difference between the data but a big difference between the images that should be recovered.
The previous slides showed that Hadamard's third condition, continuity, fails for EIT. Also, the practically measured data is never the Dirichlet-to-Neumann map of any conductivity, so the first condition, existence, fails as well. In the next two slides we show a couple of problems with uniqueness as well.
In their insightful article, Chesnel, Hyvö and Staboulis derive nonconstant conductivity distributions that produce the same EIT data than a homogeneous background. This is non-uniqueness.
It is well-known in the mathematical inverse problems community that several matrix-valued (anisotropic) conductivities produce the same EIT data.