Introduction to Medical Imaging Science

Derek Hill, Radiological Sciences, King's College London

Aims

To introduce medical images, and digital medical images in particular, to assist students in understanding subsequent material in the summer school.

Objectives By the end of this lecture,the students should be able to:

classify common medical images
define spatial resolution and contrast
state the sampling theorem and describe its implications for medical imaging
explain the origin of aliasing
relate features in the spatial domain to the k-space domain
describe how images are displayed on a monitor
explain the motivation for image segmentation, and the extent to which this can be achieved using intensity alone.

Types of image

Medical images come in a wide variety of forms: much more than the standard still and moving pictures produced with optical cameras. Medical images can take the form of projections, slices or volumes.

Projections

Many medical images are a projection of the three dimensional (3D) patient onto a two dimensional (2D) plane. Projections can be orthogonal or perspective. Considering object coordinates to be defined using (x,y,z),and image coordinates as (u,v), the projections can be drawn as:

Slices

Most medical images that are not projections, are made up of one or more slices. The orientation of the slices are defined anatomically:

transaxial - plane normal to a vector from head to toe.
coronal - plane normal to a vector from front to back
sagittal - plane normal to a vector from left to right.
oblique - a slice that is not (at least approximately) one of the above.

A medical image might be made of

a single slice
a series of parallel slices with uniform spacing
a series of slices with varying spacing and/or orientation
Volumes
In some types of medical image, data is acquired from an entire volume in one go (note this is different from acquiring multiple slices, one after the other).

Dimensionality

In medical imaging,the subject being imaged has three spatial dimensions, and changes with time. The dimensionality of the images is, however, variable.

Two spatial dimensions - eg: standard x-ray,single ultrasound image
Three spatial dimensions - multiple slices or volume acquisition.
Time dimension -In many cases, it take time to acquire information about the object, so for many images with 2D or 3D spatial information, it is assumed that the patient is still - if they move the image gets degraded by artefact. Othertimes, it is desirable to image dynamic change eg: ultrasound of beating heart
Higher dimensionality eg: scale space, different modalities
One dimensional: a single line representing a projection through all or part of the subject.

Spatial resolution

Fourier analysis

An extremely useful tool for simplifying a complicated function, is to break it down into a linear combination of sinusoidal components. This approach is known as Fourier Analysis. Joseph Fourier originally used it to study distribution of heat in a solids. It is equally applicable to the study of the distribution of brightness in a medical image.

Each sinusoidal component has characteristic frequency and phase. Because the image contains spatial information,the frequency components are called spatial frequency , and have units (1/length), eg: mm -1

An image can, therefore, be thought of as a sum of different spatial frequency component.

Low spatial frequency is about slowly changing parts of image.
High spatial frequency components provide information about fast changing parts of image ie: the fine detail.

The resolution of an image is a measure of how much fine detail can be seen, ie: the highest spatial frequency in the image. It is often measured using a pattern of parallel lines of different separation, measured in line-pairs per millimetre. The smallest separation (highest line-pair per mm) that can be correctly seen as two lines gives the resolution.

The resolution of medical images is normally different in different directions (anisotropic) and often varies with position in the image (heterogeneous).

Modulation transfer function

Real imaging systems are not equally sensitive to all spatial frequencies.

The modulation transfer function describes this relationship
The modulation transfer function is simply the Fourier Transform of the impulse response function (the appearance in an image of an object that is a point).

More formally:

If f(x) is the object and F(k) is the object in the spatial frequency domain.

The imaging process involves convolving the object with the impulse response function of the imaging system g , giving

The equivalent operation in the Fourier (k-space) domain is multiplication of the objects spectrum by the G , the modulation transfer function, ie:

Contrast Resolution

Adjacent image features that have very different brightness are said to have high contrast . Contrast, C , between features x and y of brightness B x and B y is defined as:

The contrast resolution of an imaging system is the lowest value of C that can be discerned. This depends on the noise.

Digital Images

Medical images stored on a computer are discrete .

They are stored as an array of numbers each corresonding to a different position in the patient.

In two dimensions (2D) the elements in the array are called picture elements (pixels).

Pixels are not necessarily square, but all pixels have the same dimensions.

In three dimensions (3D), the pixel becomes the volume element (Voxel)

The third dimension can be spatial as above, (eg: MR or CT) or temporal (eg: fluoroscopy). We can have a fourth dimension, most commonly a 3D image volume acquired over time.

Dimensions of images

An image has dimensions in pixels or voxels (eg: 256x256x124 for a typical MR image volume)
Each pixel or voxel has spatial dimensions (eg: 0.8mm x 0.9 mm x 2mm).
Image data is normally stored as a list of raw values, with a separate header (at the beginning of the file, or in a separate file) to describe the image.
The third dimension is often treated differently from the first two. A 3D image is commonly treated as a sequence of 2D images.

Spatial Sampling

The process of forming a discrete image from a continuous object is called sampling .
The gap between the centre of adjacent samples in a given direction is the sampling interval . The sampling interval is often anisotropic.
sampling interval in the through-slice direction in tomographic images is often larger than in the slice plane.
The sampling can be at regular intervals,or irregular intervals.
normally (but not always) regular in a 2D image, or in the slice plane of a multi-slice image.
sometimes irregular in the through-slice direction

Fourier Transforms (FT)

The FT is mathematical tool to alter a difficult problem into one that can be solved.
The FT converts a function (eg: an image) from one domain to another. For images, it converts between the spatial domain (the axes are distances, x ) and the spatial frequency domain (the axes are 1/distance, k ), i =
(an imaginary number) .

Convolution

the action of a sensor (eg: a camera) when it takes a weighted mean of a physical quantity (eg: light arriving at the camera) over a narrow range of some variable (distance).
A convolution in one domain reduces to multiplication in the inverse domain.

Digital images are discrete:

If an image is correctly sampled, it is possible to recover image values between the sample points with full accuracy.
The sampling interval must not exceed the semiperiod of a sinusoid of the maximum frequency present in the function being sampled.

Sampling theorem

A function whose Fourier transform is zero for frequencies |k| greater than a cut off frequency kc is fully specified by values spaced at equal intervals not exceeding 1/2 kc-1 save for any harmonic terms with zeros at the sampling points . - Whittaker - Shannon sampling theorem

It can be helpful to consider the sampling theorem using the following diagram, which is one-dimensional for simplicity. The left column is in the spatial domain, and the right column the spatial-frequency domain, or k-space.

The continuous function f(x) we wish to sample is is shown in panel (a). It has the continuous spectrum F(k) shown in (b).
The function f(x) being sampled is band limited, as can be seen by the restricted range of spatial frequencies in the spectrum (maximum spatial frequency k c ).
The function is sampled by multiplying it by the continuous "comb" of delta functions (c). The separation of the spikes on the comb is dx . The sampled function is shown in (e).
The Fourier transform of the sampling function is shown in (c).
The k-space equivalent of multiplication in the image domain is convolution. The sampling operation is, therefore, equivalent to convolution of the F(x) with the Fourier transform of the sampling function. The result is shown in (f).
dx needs to be small enough that the tails of the spectrum F(k) do not overlap in panel (f), ie: (satisfied in this case).

Aliasing

If the sampling interval is too large ⁽ , then frequencies that are too high to be sampled correctly
⁽ are aliased to lower frequencies.This phenomenon is very important in medical imaging. Remembering that an image can be decomposed into a sum of sinusoidal components, we can illustrate aliasing by considering a single continuous sinusoid. The thick black line is the signal. The sampling function is the series of rounded-spikes. The sampling interval is more than a half period of the sinusoid, and the resulting aliased signal is shown as the black dotted lin

Interpolation

If the object is correctly sampled, then intermediate values betweem sample points can be calculated. From the sampling theorem diagram above, it is clear that (b) can be recovered from (f) by multiplying (f) by the top-hat function (g). The equivalent operation in the spatial domain is convolution of (e) with the FT of (g), which is a sinc ⁽ ⁾ function.The correct interpolation function is, therefore, the sinc function.

Note:

In practical terms, a sinc function has infinite extent, so cannot be used.
Where speed is the priority, nearest neighbour, or linear interpolation is used.
Where accurate interpolation is important, higher order interpolants are used eg: cubic, approximation to sinc function.
Interpolation errors increase with spatial frequency (largest for sharp edges or points, smallest for uniform parts of image).
Interpolation is essential when images are translated by fractions of voxels or rotated, eg: to align two images acquired at different times.
Sinc interpolation is correct for uniformly sampled data. To interpolate irregularly sampled data onto a regular grid, inverse sinc interpolation is required.

k -Space

k -Space is the term given to the inverse spatial domain. The units are 1/distance. k x is 1/ x and k y is 1/ y .

Historical note : The terminology originally comes from solid state physics, where it is common to describe waves incident on a crystal lattice using the wave vector k , and to carry out calculations in the reciprocal lattice or k -space. We have not used the solid state notation here.

MRI is the imaging modality for which it is most important to consider k -Space.

Properties of K Space

Every point in k-space effects the entire image
The centre of k -Space (usually the point just up and right from the centre) is 0 frequency (DC).
The edges of k -Space correspond to high spatial frequencies
D kx and D ky determine the pixel separation D x and D y: Increasing the number of points in k -space for a given field of view increases the maximum spatial frequency content of the image, and hence the image resolution
D k x and D k y determine the image field of view D x and D y .: increasing the separation of points in k -space while maintaining the number of points decreases the field of view of the image

Contrast of digital images

Just as a digital image is sampled at discrete positions in space, so the image brightness can take discrete values. The number of discrete values needed to correctly store a digital image depends on the ratio between brightess ranges required and the spread of the noise.

For example, if the brightness range to be stored is 1000 times the standard deviation of the noise, then 1000 discrete brightness levels are needed. The number of bits needed to store 1000 brightness values is 10 (this would, in fact store values 0 - 1023). This corresponds to saying that the least significant bit samples only noise.

Pixel or voxel values are normally stored at byte boundaries, or perhaps nibble boundaries (1 byte = 2 nibble)
Many medical images need between 10 and 12 bits per pixel or per voxel.

Display

The digital image is displayed onto a computer monitor (CRT or LCD).

Displaying brightness values

Displays normally support 8 bits of brightness (0-255)
The image brightness values have to be mapped onto the range of brightnesses supported by the screen. This is often called "windowing".
The minimum and maximum image brightness values to be displayed are chosen, as is the intensity transformation (eg: linear or logarithmic).

Displaying pixel values

Displays typically have 1000+ pixels in each direction
The image pixels or voxels have to be re-sampled for display
image pixels or voxels may have non-uniform aspect ratio, but screens have square pixels.
interpolation : if the portion of the image to be viewed has fewer pixels than the part of the display being used, then intermediate image values need to be interpolated.
sub-sampling . If the portion of the image to be viewed has more pixels than the part of the display being used, then the image is subsampled, eg: by displaying alternate pixels, or by re-sampling using a Gaussian kernel that does local averaging.

Simple image segmentation Motivation

Measuring volumes (eg: MS lesions, hippocampus)
Quantifying abnormalities eg: percentage stenosis
Visualisation of structures eg: tumours
Automatic identification of lesions eg: computerised mamogram reporting
Measure "shape" of structures, eg: to characterize abnormality.

Thresholding Consider the following example CT and MR slices

These images have very different distributions of intensity

Histograms

Thresholded results

Region Growing Consider an image with two separate structure with similar intensities, eg: the brain and scalp in this image below.

Thresholding segments the brain + scalp (approximately)
Region growing can be used to segment a single connected region, eg: brain.
Region growing needs: UPPER THRESHOLD, LOWER THRESHOLD, SEED POINT.
Define connectivity (eg: 4 connect or 8 connect in 2D)
Start at seed point, and check each neighbour in turn, to see if it is within valid intensity range. If yes, add to list of valid pixels.
Choose next valid pixel, and repeat.
Small connections between regions cause problems.

Bibliography

The Fourier transfom and its applications. 2nd ed (revised) R.N. Bracewell, McGraw-Hill, 1986

Digital Image Processing. R.C. Gonzalez and R.E. Woods, Addison-Wesley 1992

The Physics of medical imaging. S. Webb, IoP Publishing 1988

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