# Why are CT scans much smaller than the raw data?

I'm not too familiar with CT, but I worked with medical CT images long ago, and recall that the raw data, recorded by the scanner was way bigger than the 3D image itself (Both were 16-bit TIFF files, I believe)

Is my recollection correct? And if so, why is this the case?

It's my understanding that the processed result (the 3D image) is obtained by solving a system of equations (linear in log space?). Shouldn't we be able to get about as much information out of this process as we put in?

• I think we could use a tomography tag, but I don't have enough rep to put it in.
– MWB
Commented May 13, 2023 at 9:54
• CT faces a so called "inverse problem". While the mapping of the actual structure to the CT data is fairly simple, the "invers" reconstruction problem is ill-defined. There is no system of linear (or non-linear) equations that can be used to calculate a unique structure from the data. Commented May 13, 2023 at 10:03
• If you want to "get about as much information out of this process as we put in", then you would have to wade through all the noise, since most of the raw image data would be coming from where the beams are not even yet passing through the body. Commented May 13, 2023 at 13:01
• @FatterMann There isn't a serious inverse problem in CT. The projection theorem allows you to obtain the Fourier transform of the object from the Fourier transforms of its projections. Commented May 14, 2023 at 22:16

A single slice from a medical CT is typically a 512x512x2 byte image. A single CT study can contain anywhere from 10s of images to ~1000-2000 images depending on the scan length, reconstructed slice thickness and slice spacing interval.

The raw data on the other hand, consists of a series of projections at different angles around the patient. Depending on the detector array in the scanner, each projection consists of several hundred to a few thousand samples. For a modern CT scanner, there are around 4000 projections acquired for a single rotation. Scale this by the total length of the scan, scan pitch (for helical scanners) and you can easily get raw data files in the hundreds of GB.

While the reconstruction problem can be represented as a system of linear equations, this has never been a practical way of doing tomographic image reconstruction.

• Is the Radon transform equivalent to solving a system of linear equations?
– MWB
Commented May 15, 2023 at 14:37
• Your answer seems to assume that the scanner's pixel size must equal the voxel size of the 3D image, and that the hundreds of projections are required: "The sizes and numbers are as follows, and that's why one is bigger than the other" But they are actually chosen by the designers of the hardware and software, or by the user.
– MWB
Commented May 15, 2023 at 14:42

The previous answer is correct. To put it more concise if you view CT reconstruction as a linear inverse problem it is ill-posed in the sense that in most practical scanning scenarios it is overconstrained and due to noise the measurements are inconsistent with each other. When discretized this is equivalent to saying there are more equations than unknowns which contradict each other. This is often stated as a major difficulty but in reality instead of trying to solve Ax = y where y is the measured data instead min(||Ax-y||_2) is solved which has a unique minimum. However, this redundancy in the raw data is one theoretical argument why your reconstructed data must be smaller than the reconstructed data. Another reason is, as stated before that the reconstruction resolution is mostly smaller than the raw data resolution. Finally also the bit depth of the reconstruction may be chosen to be smaller than the raw data. As a final comment, I would say reconstruction always ends up implicitly solving a linear system of equations even if done analytical with an pseudo-inverse of the Radon transform.