Spatial resolution is measured in line pairs per centimeter (lp/cm) or line pairs per millime-ter (lp/mm). Spatial frequency is measured in cycles per unit distance (e.g., cycles/cm or cycles/mm). The two are nearly identical in meaning and intuition because line pairs per unit distance applies to cycles per unit distance as well.
The ability to image high spatial frequencies is the ability to image very small objects. Five line pair patterns and five spatial frequency patterns are shown in Figure 14-10. The larger Figure 14-6 A, A continuous free induction decay
(FID). B, A sampled version of the FID in which the samples are taken too far apart in time. C, The con-nect-the-dots representation of the undersampling shows that the higher frequency has been aliased into a lower frequency.
Figure 14-7 The apparent backward rotation of wagon wheels in Wild West movies is due to aliasing. The frame rate is not fast enough to record the actual motion. Aliasing in mag-netic resonance imaging results in a “wraparound” artifact.
A
B
C
patterns are represented by smaller numbers.
As spatial frequency increases, object size decreases and becomes more difficult to image.
The spatial resolution of the best clinical MRI system is approximately 10 lp/cm or 1 lp/mm.
Table 14-3 shows the approximate spatial resolu-tion for some other medical imaging modalities.
Space—Spatial Frequency
cm = 1/2 lp/cm = 1/2
Question: An MRI system is capable of 10 lp/cm resolution. What pixel size does this represent?
Answer: 10 lp/cm = 20 objects/cm, therefore 1 object = 1/20 cm
= 0.05 cm
= 0.5 mm
Question: A 256 × 256 MR image is acquired over a 10-cm field of view (FOV). What is the limiting spatial frequency?
Answer: pixel size = = 0.4 mm 2 pixels = 1 lp therefore 0.8 mm = 1 lp
and = 1.25 lp/mm Figure 14-8 This is a computed tomography
image of a plastic cylinder containing an air-filled triangular section. The dark streaks appearing off the edges of the triangles are the result of aliasing.
(Courtesy Edward Nickoloff, New York, NY.)
Figure 14-9 This patient was positioned too far anteriorly in the head coil; the result was wrap-around posteriorly. (Courtesy Errol Candy, Dallas, TX.)
One line pair Object
Spatial frequency Ip/mm Image
Image contrast
1 3 5
0.88 0.59 0.31 0.11 0.01
2 4
Figure 14-10 These five line pair patterns are spa-tial frequency patterns used to demonstrate spaspa-tial resolution.
100 mm 256
1 lp 0.8 mm
1 lp/cm
1 cm
A high spatial frequency reflects many changes in the intensity of the MR signal. This is a func-tion of locafunc-tion within the patient. A low spa-tial frequency arises from few or no changes in the intensity of the MR signal across the patient.
An example of different spatial frequencies can be illustrated by the three businessmen shown in Figure 14-11: a used car salesman, an undertaker, and a banker. The used car sales-man is wearing a loud plaid jacket, the under-taker is wearing a plain black suit, and the banker is wearing a pinstripe suit.
The pattern of the fabric of the used car salesman’s plaid jacket has many abrupt changes in any square centimeter. Thus there are many high spatial frequencies in that jacket because there are many changes per centime-ter, whether the jacket is sampled horizontally across the cloth or vertically. Whether the operator samples vertically or horizontally, there are no changes in the pattern of the fab-ric of the undertaker’s solid black suit, and thus the jacket has very low spatial frequencies.
Between these extremes is the banker’s pin-stripe suit. A vertical sample on the back of the jacket reveals no change. Therefore the jacket has very low spatial frequencies vertically. On the other hand, when the jacket is sampled horizontally across the fabric, the pinstripes are crossed, and some relatively high spatial frequencies are observed. The more closely
spaced the pinstripes, the higher the horizon-tal spatial frequency.
Image Matrix
Because the computer has a limited amount of memory in which an image can be stored, the image must be sampled in space as well as in time. For the sampling of an image in space, the values are selected at the intersections of an imaginary grid, which is superimposed on the image (Figure 14-12), and the values rep-resent a small region of the image.
TABLE14-3 Approximate Spatial Resolution Capability of Several Medical Imaging Modalities
Imaging Modality Object Size (mm) Spatial Frequency (lp/mm)
Radioisotope scan 10.0 0.05
Ultrasound 2.0 0.25
Computed tomography 0.5 1.0
Magnetic resonance imaging 0.5 1.0
Intensified fluoroscopy 0.15 3.0
Screen-film 0.05 10
Human eye 0.05 10.0
Direct exposure film 0.02 25.0
Figure 14-11 Three entrepreneurs and their work-ing attire demonstrate the concept of spatial fre-quency. The used car salesman’s plaid jacket contains high spatial frequencies both horizontally and vertically. The undertaker’s plain black jacket has zero spatial frequency. The banker’s pinstripe suit has zero vertical spatial frequency but higher horizontal spatial frequency.
The term image matrix refers to a layout of rows and columns, usually containing num-bers representing intensity in boxes or cells.
Figure 14-13 shows a 10 × 10 matrix of cells, a 5 × 5 matrix of cells, and a 5 × 5 matrix of numbers in imaginary cells. Each MR image consists of a matrix of imaginary cells, each having various brightness levels (the gray scale). The brightness of a cell is determined by the computer-generated number in that cell.
The size of the image matrix is determined by characteristics of the imaging system and
the capacity of the computer. Most MRI sys-tems provide image matrix sizes of 256 × 256, although often other sizes such as 192 × 256 and 512 × 512 are available. Sometimes, an arbitrary acquisition matrix, such as 384 × 512, is placed in the center of the nearest larger power-of-2 matrix, 512 × 512 in this case, with the unused cells at minimum inten-sity.
The spatial resolution of any digital image is limited by pixel size. Therefore spatial resolu-tion is improved with a larger image matrix, Figure 14-12 A, The original image with a sampling grid superimposed. B, The sampled values of the original image. Because of sampling in space, the resulting computer image may look blocky. numbers in imaginary cells.
A B
A B C
assuming that the FOV of the image is held constant.
Spatial Resolution Pixel size =
Figure 14-14 illustrates the influence of matrix size on image quality when the FOV remains fixed. A 64 × 64 cell matrix appears definitely
“blocky,” whereas a 128 × 128 matrix is a fairly good representation of the original image.
In theory, the sampled value represents the intensity of the original continuous image at an infinitesimally small point. Such a point would be invisible, so each point is represented by a pixel of the displayed image.
Spatial resolution is limited by pixel size.
A sampled image is just like a mosaic. Each pixel is a small tile having a brightness equal to that of the sampled value at that point in the image.
This accounts for the possible blocky appear-ance of some computer-displayed digital images.
The spacing of the samples must be suffi-ciently close so that there are no surprises between samples. For example, if there were a small structure in the patient and the samples were far apart, it would be difficult to say pre-cisely where the structure should be located in the image and the precise shape of the struc-ture. Sharp edges and small structures contain high spatial frequencies.
Frequency Domain Map
The spatial frequency content of an image is displayed in two dimensions, just the way the image itself is displayed. Low frequencies are near the center of the spatial frequency display and higher spatial frequencies are toward the edges (Figure 14-15).
Horizontal spatial frequencies appear along horizontal lines, and vertical spatial frequencies appear along vertical lines. The intensity of an
arbitrary point represents the strength of a par-ticular combination of horizontal and vertical spatial frequencies. For example, the banker’s pinstripe suit was a combination of low vertical
FOV Matrix size
Original 128 128
64 64
16 16
4 4
32 32
8 8
22 Figure 14-14 The small features and sharp edges in the image that contain much of the high spatial fre-quency information are progressively obscured as the sampling rate and image matrix are reduced. The fine structures become blurred as their high spatial fre-quencies are aliased into lower spatial frefre-quencies.