3. CT IMAGE RECONSTRUCTION  Hounsfield envisioned dividing a slice into a matrix of 3-dimensional rectangular boxes (voxels) of material (tissue). Conventionally, the X and Y directions are within the plane of the slice, whereas the Z direction is along the axis of the subject (slice thickness direction). The Z dimension of the voxels corresponds to the slice thickness. The X and Y voxel dimensions, depend on the size of the area over which the x-ray measurements are obtained as well as on the size of the matrix (the number of rows and columns) into which the slice is imagined to be divided.  The objective of CT image reconstruction is to determine how much attenuation of the narrow x-ray beam occurs in each voxel of the reconstruction matrix. These calculated attenuation values are then represented as gray levels in a dimensional image of the slice.

4. CT Image Reconstruction The Radon Transform is widely applicable to Tomography, the creation of an image from the Projection Data associated with cross-sectional scans of an object. The Radon Transform represents the projection data obtained as the output of a Tomographic scan. Hence the Inverse of the Radon Transform can be used to reconstruct the original density from the Projection Data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as Image reconstruction. Ray, Ray Sum, View & Attenuation Profile  Ray – Imaginary line between Tube & Detector.  Ray Sum – Attenuation along a Ray.  View – The set of Ray Sums.

5. For example, consider a simple 2-row by 2-column reconstruction Matrix. Views are collected at 4 angles, 0 (left to right), 90 (top to bottom), 45 (diagonal), and 135 (diagonal), and each measurement is expressed as the sum of the voxel attenuation values along each ray. In this case, there are 10 equations:U1U2U3U4 U1U133U1U1333 U2U244U2U2444 U3U311U3U3111 U4U488U4U4888 No 45 90 135 o X 1 = U 1 + U 2 X 2 = U 3 + U 4 X 3 = U 1 + U 3 X 4 = U 2 + U 4 X 5 = U 2 X 6 = U 1 + U 4 X 7 = U 3 X 8 = U 1 X 9 = U 2 + U 3 X 10 = U 4

6. U1U133U1U1333 U2U244U2U2444 U3U311U3U3111 U4U488U4U4888 11 4 412 1 8 5 3 7 9

7. 7799 11 1 418111020 4 + 7 = 11 11 + 7 = 18 11 + 9 = 20 1 + 9 = 10

8. 18111020 124 4 + 18 = 22 4 + 10 = 14 12 + 11 = 23 12 + 20 = 3222231432

9. 25281940 22231432 8 3 5 3 + 22 = 25 5 + 23 = 28 5 + 14 = 19 8 + 32 = 40

10. 7 + 9 = 16 Initial x-ray scan combined combined 25 - 16 = 9 28 - 16 = 12 we subtract the total initial x-ray 19 - 16 = 3 combined from the value of each box. 40 - 16 = 24 9 / 3 = 3 because we have the scan not once 12 / 3 = 4 but three times we returned others. 3 / 3 = 1 24 / 3 = 8 2528 1940 912324 3418

11. The three images demonstrate a haemoperitoneum, shattered right kidney and a lacerated spleen in Axial (A), Sagittal (B) and Coronal (C) Planes. (A) (C)(B)

13. Shaded Surface Displays (SSD). (A) Coronal Multiplanar volume rendering and (B) minimum intensity projection

14. (A) Maximum Intensity Projection and (B) Volume Rendered, Images of the Abdomen.

15. Virtual Endoscopy (VE)

16. Dental CT reconstructions. By drawing a manual trace on an axial CT scan (A), it is possible to obtain shaded surfaced display volume rendered (B), curved Multiplanar reconstruction

22. Beam-hardening artifact caused by unusually severe hardening of x-rays passing though thick Bone.

23. Ring Artifact Partial Ring Artifact

24. Motion Artifacts Metal Artifacts

25. Partial Volume Artifact

26. A.60 mA, 120 kVp, slice thickness 5 mmB. 440 mA, 120 kVp, slice thickness 5 mm

27. Out of Field Artifacts