10x Localization Project

This repository contains code developed as one of Innolitic's 10x mini-projects.

Introduction

• problem has shown up on multiple client projects
• sensors; integration vs sampling
• background
• noise
• relative size of the object

Problem: find the location of the object (usually a fiducial used for image registration) as accurately as possible.

Formalized 1D Problem

We mode the underlying object that we are imaging as a continuous, 1D function over the domain [0, 100].

This function has the form:

I(x) = rect((x - c)/w) + b(x)

Where

• c = "Center of the object"
• w = "Width of the object
• b(x) = "Background function".

For now, we assume that the background function is linear, that is:

I(x) = rect((x - c)/w) + b0 + x*b1

Now that we have modeled the underlying object, we need to model the processing of converting this object into a image.

Assume an image of this object is formed using an imager with 100 sensor elements. For now we assume that each sensor element acts as a perfect integrator. I.e.,

I_integrated[k] = \int_k^{k+1} I(x) dx

for k = 0 to 99.

In the other extreme, each sensor element acts as a perfect sampler. I.e.,

I_sampled[k] = I(k + 0.5) * alpha

where alpha is an arbitrary constant.

In practice, one would expect image sensors to be a combination. If s is a number between 0 and 1, representing how full the image sensor is:

I_b[k] = \int_{k + s/2}^{k + 1 - s/2} I(x) dx

The bit-depth of the Analog to Digital converter also plays a roll in image formation.

Finally, we expect a certain amount of noise to be introduced by the sensor, readout machinery, etc. For now, we assume that all of the noise is zero-mean Gaussian noise (this is certainly not the case).

For now, we model these effects using:

• N = "number of gray scale levels available in our sensor's analog to digital converter"
• A_max = "maximum measurable value"
• A_min = "minimum measurable value"
• s = "relative size of the sensor elements, [0, 1]"
• sigma = "standard deviation of the gaussian noise".

Problem

We are interested in developing an algorithm that can determine object's center, c, as accurately as possible given an I[k].

We know N, A_min, A_max, and s before hand, since they are attributes of the imaging system. We may also know the approximate ranges that the image w, b0, b1, and sigma will take.

We can also assume that c is in the middle of the image, i.e., c \in [25, 75].

If our algorithm guesses c_expected, then we define the localization error to be:

E = |c - c_expected|

Questions to investigate

• What algorithm have the lowest mean localization error?
• For a given algorithm, what is the relationship between mean localization error an X? (where X is a paramater of our model, such as w, N, b0, b1, s, or sigma).
• Generalize to 2D
• Investigate the effect of different types of background functions

References

• Wikipedia article about super-resolution imaging