Bush__The_Essential_Physics_for_Medical_Imaging - Biomedical ...

would correspond to 1 cycle/mm. If il = 0.10 mm, then one complete cycle wouldoccur every 2il = 0.20 mm, and thus in the distance of 1 mm, five cycles would beseen, corresponding to a spatial frequency of 5.0 cycles/mm. The relationshipbetween the distance spanned by one-half cycle of a sine wave, il, and the spatialfrequency F, is given by:1F=- 2ilIn addition to the sine wave shown in Fig. 10-17, a square wave is shown aboveit. Whereas the spatial frequency domain technically refers to the frequency of sinewave-shaped objects, it is a common simplification conceptually to think of thesine wave as a square wave. The square wave is simply a pattern of alternating densitystripes in the image. With the square wave, each cycle becomes a line pair-thebright stripe and its neighboring dark stripe. Thus, the units of spatial frequency aresometimes expressed as line pairslmm Op/mm), instead of cycleslmm. A square objectof width il can be loosely thought of as corresponding to the spatial frequency givenby Equation 10-12. So, objects that are 50 /lm across (0.050 mm) correspond to 10line pairs/mm, and objects that are 0.250 mm correspond to 2 line pairs/mm, andso on.Spatial frequency is just a different way of thinking about object size. There isnothing more complicated about the concept of spatial frequency than that. Lowspatial frequencies correspond to larger objects in the image, and higher spatial frequenciescorrespond to smaller objects. If you know the size of an object (il), youcan convert it to spatial frequency (F = 1/2il), and if you know the spatial frequency(F), you can convert it to object size (il = 1/2F). With an understanding of spatialfrequency in hand, how it relates to the spatial resolution of an imaging system cannow be discussed.Let's start with a series of sine waves of different spatial frequencies, as shown in Fig.10-18. The six sine waves shown in Fig. 10-18 have spatial frequencies of 0.5, 1.0,1.5, 2.0, 2.5, and 3.0 cycles/mm, and these correspond (via Equation 10-12) toobject sizes of 1.0, 0.50, 0.333, 0.25, 0.40, and 0.167 mm, respectively. Each sinef = 0.5 f = 1.0 f = 1.5.rWJ;l" •• "•••..• I'85--.- -------.-------48 . 27-- -------- -------,J'00FIGURE 10-18. A series ofsine waves of differing spatialfrequencies (e.g., f = 0.5cycleslmm). The contrast (differencebetween peaks andvalleys) of the input sinewaves (solid lines) is 100%.After detection, the outputsine waves (dotted lines)have reduced amplitude (theoutput sine wave amplitudeis indicated for each sinewave). The reduction is moredrastic for higher frequencysine waves.