# Thermography time evolution

I have an imaging problem. I essentially have a metallic plate with a number of holes through it. I am looking at the number of holes with a thermal camera. While looking at the plate I have two phases of this analysis, 1.a source of hot fluid is flowing through the holes and behind the plate 2.a source of cold fluid flowing behind the plate and through the holes.

During each phase I take a number of images N. So in effect I have a set of random variables for each pixel in the image. $H[x,y] = (x_1, x_2 .... x_n)$ and similarly $C[x,y]=(x_1, x_2 .... x_n)$ where H is the hot frames, C is the cold frames , x-y are the focal plane position of that particular pixel and 1,2,3 are the discrete time of the reading x.

As it stands I'm just taking the mean value for H and C then subtracting them to generate an image ($I=\bar{H} - \bar{C}$ where the bar denotes the expected value) to gain an image that has significantly decreased noise and removes much of the background effects.

This is really just to decrease the noise but I think there is more information there that I'm not utilizing considering this is not a wide sense stationary process, but is evolving with time as the temperatures reach equilibrium for each phase.

I'm looking for a way to utilize this data and for any recommendations on processing this time evolution to form an image. Any suggestions?

I have tried Finite Difference Operator to approximate first and second derivatives, but this and various other algorithms have failed me.

First thing I would probably do is view it as a pseudo-video in matlab or something. Plot each time image and step through them to get a qualitative idea of if/how it changes in time.

Effectively what you have is a 3D matrix with dimensions ($n \times m \times t$). So for visualization purposes you are not restricted to the $xy$ plane you could for example view the time evolution along one row of you data. This is probably the best bet to visulise stuff as viewing 4D data is hard...

For quantitative ways to describe the data I would guess you want to fit some model to the data. This should really be done on the whole dataset. You are probably in the best position to decide on an appropriate model, but I guess the heat equation would be a good place to start.

• So in a nutshell "view the time evolution along one row of you data." you are thinking a linear heat equation in a row for instance to be a time evolved 2D function over a "rod"? Then it would essentially be a solving and evaluating the partial diff eq. I like it! I will give it a shot. – Felix Castor May 3 '14 at 15:41
• Yes, that's a good starting point. Ideally you solve it for the entire plane, but I expect that's significantly more difficult. – nivag May 6 '14 at 8:27

I've had this post up with no takers so I will just put what I have found so far.

The algorithm that has resulted in the least noise for my application that I have tried is:

Let $X =[x1,x2,....,xn]$ be a sample of a function over discrete time intervals where x is a real number and represents the output of a particular pixel at $\delta t*i$ where $0<i<n$ .

Find a linear function $f$ that best approximates the data in the form (I tried polynomials of varying degrees but a linear function performed the best):

$f(x) = m *x + b$

Evaluate the function at ${t }= {{n*\delta t} \over{2}}$

So $f({{n*\delta t} \over{2}}) \approx Median$

My motivation was to find the median but it may be a somewhat bold assertion. I wanted the median value of each pixel but the chances of picking a pixel due to noise was too great. Plus because of the physical constraints I know that the data is either decreasing or increasing over the intervals with respect to the phase of data collection.

I do this for both hot and cold images to form the image using the notation above $I=\bar{H} - \bar{C}$.