Image- and k-space: Here we will be concerned with 2 spaces: (1) Image space and (2) k-space. By “space” we are taking about an organized way to describe an object. The object is the real thing we are interested in, such as a patient’s brain or heart, while the spaces (image and k) are ways to describe it that have rules for organizing and visualizing the information.

Image Space: Image space, is a way to describe the object in terms of pixels. For a complete image space description we specify a list of pixels, their intensities and where they are relative to one another. To visualize them we organize them into an image by arranging the pixels and coloring them in with their intensities.

k-space: k-space is a different way to describe the same object. Instead of pixels we are using a sum of stripe patterns. Mathematically these stripe patterns are sine waves, but you can think of them as smooth stripe patterns. To describe an object in this way we need to provide a list of the waves, how wide they are, how much of each one to use to use, their direction and how much to shift them by.

To get the image from k-space we just add up the waves on an imaging grid - that is we construct the pixel description in image space from the k-space description. Formally this process is called the inverse Fourier transform. Breaking up the image into waves from the pixel description is called the Fourier transform. Thus k-space is sometimes called Fourier space. The Fourier transform has a nice compact mathematical description but you don’t need to understand it to have an intuitive idea that it takes you between these 2 descriptions of our object - the image and k- space description. The Fourier transform is just a translation between the two.

k-space geography, visualizing k-space: k-space can be visualized as a collection of pixels just as you can visualize an image in terms of its pixels. Just like in image space the pixels location is important. Unlike image space, however, which (typically) has only 1 value, an intensity, the k-space pixel needs to have 2 values associated with it an intensity and a phase: each has a function as specified below.

1: The distance from the center determines the breadth of the stripe pattern.

2: The direction from the pixel to the center determines the direction of the stripe pattern.

3: The intensity of the pixel tells you how much of that strip pattern you need to make the image.

4: The phase of the pixel tells you how much to shift the stripe pattern by.

Given these general features of our visualization of k-space we give the following generalization about k-space: The details of the image are represented in the outskirts of k-space and the contrast and general features are in the center.

The Pulse Sequence: A pulse sequence is a way to describe and visualize the instructions given to an MRI machine to acquire the object’s k-space data. It consists of excitation, encoding and data acquisition:

1. Excitation pulses are transmitted radio frequency (RF) energy that select the slice or volume to image, provide contrast, and provide the energy that we receive later when we collect the data.

2. Encoding pulses are magnetic field gradient pulses that select which parts of k-space we will measure. They are what help us navigate k-space. They select the stripe patterns that we will measure.

3. Data acquisition or readout is when we record data. In this step we listen for the energy that we put in during excitation. The amount of energy we hear back reflects the amount of the stripe pattern pertaining to the k-space location we are reading out that must be used to build the object.

We will be concerned with only the most basic pulse sequence here: a simple 2D gradient echo spin warp sequence. In this pulse sequence excitation is a simple slice excitation, encoding consists of scanning across one straight line of k-space at a time, and data acquisition is turned one while traversing the line. This pulse sequence builds a 2D image of a single slice of the object.

Parameters: Using the spin warp sequence we will explore the effects of basic MRI imaging parameters on the image quality and explain these effects using our k-space description of the object.

Resolution: The image is a collection of pixels. Higher resolution means smaller pixels and hence a more detailed description of the object. The further out in k-space we sample the higher the resolution. Intuitively this is because the finer stripe patters live out in the outskirts of k-space and to make a detailed description of something you need to use finer stripe patterns. Since gradient pulses are used in the pulse sequence to traverse k-space, you need higher gradients to achieve higher resolution.

FOV: The field-of-view (FOV) of an image is inversely proportional to the spacing of our points in k-space. This is a bit tricky to understand intuitively but one way to imagine it is that since our resolution is determined by the maximum distance we go in k-space then, given a certain resolution, the other thing we can change is the amount of measurements we make between these maximum k-space points. The more measurements we make the more imaging pixels we will have since the number of unknowns that we can solve for has to be equal to the number of measurements we make. So, more measurements = more pixels, more pixels = bigger FOV.

SNR: Signal to noise ratio is a measure of how trustworthy your estimate of the object is. It is a measure of how much information about the object you have (signal) compared to junk (noise). The junk comes from random thermal-related effects in the object itself as well as from the hardware - in either case the noise is variable every you measure. It is random. It is this randomness that allows you to improve your information about the object by measuring many times. The object will be the same but the noise will change, so the information about the object (the signal) will accumulate faster than the random noise as you make more measurements.

The consequence of this is that the longer you scan, the higher your SNR will be. The SNR can be thought of a limited resource. The more pixels you have the lower you overall SNR because you have to distribute it into all of the pixels. So high resolution means low SNR. You can compensate for this by scanning longer.

Bandwidth: Another way of increasing SNR is to lower the bandwidth which is essentially another way of scanning longer. By lowering bandwidth you lower the speed at which you move through k-space effectively collecting data longer at each point in k-space. The main limitations of this are a result of sequence timing, signal decaying away (i.e. due to T2* decay), and blurring caused by off-resonance conditions. Some of these will be discussed in more detail.

k-space is where MRI measurements happen. The pulse sequence tells the MRI scanner how to move through k-space to get the measurements. The concept of k-space is helpful in understanding how the parameters of a pulse sequence affect the properties of an image.