BME 210 Homework 3: Medical Imaging Solutions
Page 2 of 10
Part 1: Introduction and Report Outline (10 points)
X-Ray Computed Tomography (the process of making a CT scan) is an imaging technique that
involves passing
...
BME 210 Homework 3: Medical Imaging Solutions
Page 2 of 10
Part 1: Introduction and Report Outline (10 points)
X-Ray Computed Tomography (the process of making a CT scan) is an imaging technique that
involves passing x-rays through the body from different directions and measuring the attenuation
of the x-ray energy along each x-ray beam pathway. This “projection” information can then be
used to reconstruct the x-ray absorption/attenuation properties of the body’s tissues. The figure on
the left, below, illustrates the concept by showing two locations/orientations of the x-ray source
and the x-ray detectors that would be involved in generating a CT image for a single slice of the
body. The figure on the right illustrates the relationship between a particular x-ray beam, the x-ray
absorption characteristics of individual pixels (denoted as the CT numbers 𝜇! through 𝜇"), and the
measured attenuation of the x-ray energy (denoted 𝑠).
The collection of 𝑀 x-ray projection measurements can be related to the 𝑁 CT numbers in an
imaged section as 𝑊𝜇 = 𝑠, where 𝑊 is a known 𝑀 × 𝑁 weighting matrix for the contribution of
each pixel to each x-ray projection measurement (𝑁 is on the order of 106 and 𝑀 is as big or bigger
than 𝑁). This formulation of the CT imaging problem represents the algebraic reconstruction
framework used by Hounsfield, the inventor of the x-ray computed tomographic scanner. The
system of equations 𝑊𝜇 = 𝑠 can be solved in several ways as will be illustrated in this report.
Major Findings
MATLAB functions implementing Crout’s algorithm and forward and backward substitution were
coded and applied to solve a system of five algebraic equations. CT projection data were used to
reconstruct a 50x50 image of the chest using both exact and noisy data, illustrating the effect of
both limited resolution and measurement error on image reconstruction. An MR image from a
DICOM file was displayed to illustrate the importance of different dynamic range selection on
image visualization.
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