TwinTree Insert


Chapter Seven
Image Data Transformation

07-01 Introduction

ts flexibility distinguishes MR imaging from all other medical imaging mo­da­li­ties. The ultimate reason for this is the unique handling of MR raw data in an abstract data collection matrix called k-space, where the data stay to be deciphered. This space consists of the raw data that have been collected during image ac­qui­si­tion but have not yet been converted into the final anatomical im­age.

The motto in the foreword to this book fits very nicely with this chapter:
"Why, some­times I’ve be­lieved as many as six im­pos­sib­le things before break­fast."

spaceholder redThe easiest way to deal with k-space is seeing and believing; this, however, is not very helpful when one wants to understand how some ima­ging techniques func­tion and what their pitfalls are (Figure 07-01).

Of course, k-space behaves differently from cat eyes, but there are some si­mi­la­ri­ties as will be explained in the text.

Figure 07-01:
There is something wrong here: we do not talk about CAT-scans in CAT-space — or do we?
spaceholder smallredThe picture on the left was taken during daytime, the picture on the right at night. Look at the cat’s eyes: the pupils are small when there is a lot of light, but then they are wide with little light.
 The central part of the retina displays extraordinary visual discrimination, thanks to the tiny size of the closely packed, light sensitive cones located there.
 This area with maximum resolution covers only 1° of the eye’s field-of-view. At night, the periphery of the retina is used; it has an incredible sensitivity to light but a very poor ability to distinguish de­tails.

spaceholder redFirst and foremost, a k-space is a mental concept. There is no hardware in an MR machine corresponding to it. It is a platform to collect, store, and pro­cess com­plex data. These data represent thousands of sine and cosine waves which build the MR image.

The term k-space is mathematical. The letter ‘k’ is used by mathe­ma­ti­cians and phy­si­cists to describe spatial frequency, for instance, in the propagation of sound, light, or, in general, electromagnetic waves.

07-02 The Optical Equivalent

One way of understanding the concepts and mechanisms of k-space is looking at a dif­­fe­rent physical property which, perhaps, is simpler to imagine: the col­lec­tion and pro­­ces­sing of light by a lens, as Mezrich ex­plains in his introduction to k-space [⇒ Mezrich 1995].

The processing of the incoming light to an image by the lens determines to a great ex­tent its resolution, size, and contrast. The light passing through the lens is bent slightly in the center, increasingly towards the edges. In a perfect lens, the light will meet in one point, the focus, and then cre­ate an inverted image (Figure 07-02).

Figure 07-02:
Image processing by a lens.

The processing of the light data by a lens is more complicated than generally thought: there is no point-to-point corre­spondence between points within the lens — or within a center plane in the middle of the lens — and the final image created by the lens. All points within the lens process data from all points of the ori­gi­nal ob­ject. How­ever, for our purposes we could imagine such a center plane as the lo­ca­tion where processing takes place (Figure 07-03).

Figure 07-03:
Image processing by a lens with a fictitious image-processing plane.

Visible light actually consists of differ­ent frequencies. As we have already seen in Chapter 2, a prism can make a frequency analy­sis. A lens is more sophisticated. We can consider it as a special filter which, depend­ing on its characteristics, lets some or all of these frequencies pass. It accepts signals, analyses them, pro­ces­ses them, and creates an image; basically, it performs a Fourier transform. We have assumed that the Fourier transform is accomplished in a fic­titious central plane of the lens. In front of the lens, we can set instruments performing optical functions, for in­stan­ce an iris, or we can change the size of a lens (Figure 07-04).

Figure 07-04:
Increasing the size of a lens with the same focus improves image resolution because the individual image points are smaller — the same holds for k-space: larger k-space with the same field-of-view means better spa­t­ial resolution of the image.

Changing the size of the lens or an iris in­fluences the size of our processing plane. The steeper the angle the light makes within the lens, the sharper the focus will be; the larger the lens, the better image reso­lution will be. The sharp­ness of the final ima­ge is determined by the outer parts of our 'Fourier' plane. Points in the outer re­­gions of the plane contribute more to image resolution than points close to the cen­ter because they allow higher spatial frequen­cies to pass through.

Lower spatial frequencies are closer to the center. Their main responsibility is the dis­tri­bu­tion of brightness and darkness. This means that they are re­spon­si­ble for im­a­ge contrast.