Projection Data, Image Matrices and Voxel Sampling

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Raw Data vs. Image Data & The Sinogram

Computed tomography transforms physical attenuation measurements gathered by an array of detectors into cross-sectional maps of tissue attenuation. Understanding the distinction between raw data and image data is fundamental to CT physics and protocol design.

Raw Data (Scan Data / Pre-Reconstruction Space)

Raw data consists of the direct, unprocessed electrical signals generated by the detector photodiode-scintillator channels during the exposure. When the x-ray beam traverses patient anatomy, its intensity diminishes in accordance with the idealized Beer–Lambert law for a monochromatic beam; clinical polychromatic data require spectral and beam-hardening corrections:

$$I = I_0 e^{-\int \mu(x,y,z)\, dl}$$

where:

• $I_0$ represents the unattenuated reference x-ray beam intensity measured by reference detectors or calibration channels.
• $I$ represents the transmitted photon intensity measured by a specific detector channel.
• $\mu(x,y,z)$ represents the distribution of linear attenuation coefficients along the ray path $l$.

The Data Acquisition System (DAS) samples these analog transmission measurements tens of thousands of times per rotation, amplifies them, digitizes them with analog-to-digital converters (ADCs), and performs a logarithmic transformation to extract the linear attenuation profile (the ray sum, $p$):

$$p = \ln\left(\frac{I_0}{I}\right) = \int \mu(x,y,z)\, dl$$

A single projection (or view) represents the set of all parallel or fan-beam ray sums acquired across the entire detector array at a single gantry angular position ($\theta$). As the gantry rotates $360^\circ$, the scanner acquires hundreds to thousands of individual projections.

The Sinogram Representation

When all raw projections gathered during a $360^\circ$ rotation are arranged in a 2D mathematical coordinate space where the vertical axis represents gantry projection angle ($\theta = 0^\circ\text{ to }360^\circ$) and the horizontal axis represents detector channel position ($r$), the resulting data array is termed a sinogram.

A single off-center high-attenuation point in the patient (such as a surgical clip or compact bone fragment) traces a sinusoidal curve in the parallel-beam representation (fan-beam coordinates modify the trajectory) across this coordinate plane as the gantry orbits. The reconstruction computer operates directly on this sinogram to synthesize anatomical images.

Image Data (Post-Reconstruction Space)

Image data results only after the reconstruction engine processes the raw data. Image data consists of a two-dimensional grid of numerical CT numbers (Hounsfield Units) assigned to discrete pixels. Once reconstructed, image data is stored in the standard Digital Imaging and Communications in Medicine (DICOM) format.

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Digital Image Architecture: Matrix, DFOV, Pixel & Voxel Mechanics

The reconstructed CT image is composed of a two-dimensional array of square picture elements called pixels. When the physical slice thickness ($T$) is incorporated along the longitudinal ($z$-axis), each pixel represents a three-dimensional volume element termed a voxel.

Matrix Size & Display Field of View (DFOV)

• Matrix Size: The number of rows and columns forming the digital image grid. In routine clinical CT, the standard matrix size is $512 \times 512$ ($262,144\text{ pixels}$). Ultra-high-resolution (UHR) scanners and dedicated cardiovascular/temporal bone systems frequently provide $1024 \times 1024$ ($1,048,576\text{ pixels}$) or $2048 \times 2048$ matrices to resolve microscopic vascular or osseous structures.
• Scan Field of View (SFOV): The physical diameter of the circular area within the gantry bore from which the detector channels actively acquire raw projection data (typically $50\text{ cm}$ for adult body imaging, $25\text{--}30\text{ cm}$ for pediatric or head imaging). Calibration and beam-hardening corrections depend directly on correct SFOV selection.
• Display Field of View (DFOV): The physical dimension of the square anatomical region reconstructed and presented on the display monitor. The DFOV can be equal to or smaller than the SFOV, and some systems offer extended reconstruction beyond the standard measured field, with different accuracy and limitations.

Mathematical Formulation: Pixel Size Calculation

The physical in-plane dimension ($d$) of a single pixel along the $x$- and $y$-axes is defined by the ratio of the DFOV to the matrix dimension:

$$Pixel\ Size\ (mm) = \frac{DFOV\ (mm)}{Matrix\ Size\ (pixels)}$$

Notice that pixel area is $(Pixel\ Size)^2$. For example, a $512 \times 512$ matrix covers the selected DFOV evenly across both dimensions.

Voxel Dimensions & Volume

A voxel represents a rectangular cuboid of patient tissue. Its volume is given by:

$$Voxel\ Volume\ (mm^3) = (Pixel\ Size)^2 \times Slice\ Thickness = \left(\frac{DFOV}{Matrix}\right)^2 \times T$$

where $T$ is the nominal reconstructed slice thickness along the $z$-axis in millimeters.

Cubic voxel dimensions provide isotropic sampling, which supports reformations with similar geometric sampling in different directions. Physical resolution can still differ because of detector response, focal spot, slice sensitivity and reconstruction. In the example, 120/1024 = 0.1171875 mm pixels with 0.5 mm thickness are anisotropic. Selecting a 0.117 mm slice does not create that longitudinal resolution if the acquired data cannot support it. A supported 0.5 mm slice with 256 mm DFOV on a 512 matrix would instead have 0.5 mm cubic sampling, though actual resolution must still be evaluated.

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