What is the Radon transfrom definition? [related to CT scan]

Question

What is the equation of Radon Transform?

What does it mean? Does it mean that one projection at one angle?

What if the angle is zero, will that mean we only integrating with respect now to only one variable, and I think that is y. Is that correct? How the integral will look like compared to the definition of Radon Transform?

Answer 1

Hi Ali A.,

The name is Radon Transform. You can certainly look it up on Google. Essentially, for CT it means that you started clinically with a 3-D density "function" (such as, a variably x-ray opaque solid mass, maybe a tumor, within the body), and you looked through it at all possible angles, on well-defined lines-of-sight.

That gives you an enormous dataset of single-value x-ray opacity data. "Integrate" here is used in a rather odd way; it's not the usual mathematical function integration! Rather, "integrate" here means, the absorption (opacity) value you get along a particular line-of-sight through the mass, which just is the result of the absorption of each little piece of the mass that lies along that line-of-sight (including contributions from parts of the body outside the mass!). It's the same as if you tried to shine an automobile headlight through some fog: each little bit of fog that the beam travels through knocks a little bit of the remaining beam away, and what gets through the other side eventually is the remainder. The weaker is the transmitted beam, the more fog was in the way -- in the same way, in CT, the weaker the transmitted x-ray signal, the denser the total "slice" of tissue it went through.

The fun on this, of course, is being efficient in writing the reverse Radon transformation coding, since the dataset is useless unless it is reversed to "make" a 3-D mass object.

Note: when the term "mass" is used above, frequently absorbance of the X-rays by a radioopaque dye is what is being imaged. The dye is chemically or biologically targeted to the tissue mass of interest.

One other note: the 'radon' part comes from radon gas, which (being a heavy element) is quite X-ray absorptive: it can be used to indicate lung ventilation (after an inhaled dose, it will linger in lobes with poor ventilation on exhalation, indicating bronchial obstruction, pneumonia, collapsed lung portions, etc.). (It is also radioactive, but that is a minor concern in comparison to its diagnostic utility).

-- Cheers, -- Mr. d.