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258 Nuclear Medicine Physics

f such that

v = f (u)

Several types of image transformation can be implemented depending on the

function f . In general, f is monotonic and increasing. As a result, the ordering

of the gray level values between black and white is preserved [106]. This

mapping can be deﬁned by a look-up table (LUT), as the gray level values

are discrete and integer (the function is discrete and, therefore, it deﬁnes

a mapping between discrete values). This mapping can be represented by

Figure 6.26.

The use of the LUT enables the contrast and image intensity to be changed.

If the function is linear or piecewise linear, then changing the slope of the

line permits changes in image contrast. Figure 6.27 shows an example of a

piecewise linear change of contrast. This transformation can be deﬁned by

v = f (u) =

⎧

⎨

⎩

au 0 ≤ u < I

b(u −I) + I



I ≤ u < J

c(u −J) + J



J ≤ u < K

(6.35)

If the line slope is greater than 1, then the mapping increases the contrast.

Image histogram analysis enables the values of the parameters to be chosen,

namely, a, b, c and I and J. The coordinates of points (I, I



) and ( J, J



) deﬁne

the type of transformation. If I = I



and J = J



, the transformation is a linear

function that does not change the gray levels. If I = J, I



= 0, and J



= K − 1,

then the mapping performs an image binarization. In this case, the image has

only two gray level values, in general 0 and 1.

White

Black

0 255

FIGURE 6.26

Gray level mapping (Adapted from A. K. Jain. Fundamentals of Digital Image Processing, 2nd

Edition, Prentice-Hall, Englewood Cliffs, NJ.).

Imaging Methodologies 259

0 IJ

J′

K′

I′

FIGURE 6.27

Changing the image contrast using piecewise linear transformations. (From A. K. Jain.

Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989. With

permission.)

6.3.2.3 Gamma Correction

In general, images with high contrast are easier to visually analyze. In many

cases, the nonlinear adjustment of the contrast offers better results. One of

the commonest nonlinear adjustments of the contrast is known as “gamma

correction.” In this case, the function f(u) is deﬁned as follows:

v = f (u) = cu

(6.36)

Values of γ > 1 increase the contrast whereas values of γ < 1 decrease the

contrast. However, saturation does not occur for gray level values within the

variation range [107]. These transformations are also known as mappings

according to the power law. Many devices used for printing and visualizing

images operate according to the power law (Figure 6.28).

It is, in fact, the process by which these devices correct their response obey-

ing the power law that is called gamma correction. For example, cathode ray

tubes have a relationship between light intensity and voltage that follows the

power law. Gamma correction is important for the correct visualization of the

images. Another type of correction that can be used is based on a sigmoid

function:

v = f (u) =

1 +e

−αu+β

(6.37)

This nonlinearity has two degrees of freedom and can, therefore, produce

a more balanced contrast improvement.

260 Nuclear Medicine Physics

– 1

γ = 0.04

γ = 0.10

γ = 0.20

γ = 0.40

γ = 0.67

γ = 1

γ = 1.5

γ = 2.5

γ = 5.0

γ = 10.0

γ =25.0

3L/4

L/2

L/4

L/4 L/2 3L/4 L

– 1

FIGURE 6.28

Gamma correction for several gamma values. (From R. C. Gonzalez, R. W. Woods, Digital Image

Processing, 2nd Edition, Prentice-Hall, Englewood Cliffs, NJ. With permission.)

6.3.2.4 Logarithmic Transformations

Logarithmic transformations are usually employed to increase the range

of gray level values corresponding to dark pixels while simultaneously

compressing the range of gray level values corresponding to the whitest pix-

els [107]. The function f (u) specifying the transformation is generally deﬁned

as follows:

v = f (u) = c log(1 +u) (6.38)

Transformations deﬁned according to the power law tend to be more

ﬂexible than this one, but logarithmic transformation provides a more effec-

tive compression of the dynamic range of the gray level values when the

image contains considerable variations in gray level values. Atransformation

deﬁned by the inverse of the logarithm has the opposite effect.

6.3.2.5 Histogram Processing

The histogram image represents the relative frequency of occurrence of gray

levels in one image. Histogram modeling methods change the image so

that the resulting histogram matches a predeﬁned type. This modiﬁcation

improves the image quality. One of the most widely used methods is his-

togram equalization [106]. The goal is to change the histogram so that the

Imaging Methodologies 261

ﬁnal histogram is uniform, that is, a histogram in which all the gray levels

occur with the same frequency. Let m

be the number of pixels in an image

with gray level k. If M is the total number of pixels in one image, the normal-

ized histogram is given by p(k) = m

/M. In the normalized histogram, the

sum of all its elements is 1. To obtain an equalized histogram, the following

value is computed for each gray level k:

r = T(k) =



i=0

p(i) =



i=0

(6.39)

The transformed image is obtained by replacing each gray level k by gray

level r. The image transformation has an inverse and is monotonic, which

means that it will preserve the gray level ordering from black to white. This

transformation ensures that one gray level will never be replaced by a darker

one. This type of histogram modiﬁcation is usually employed to increase the

image contrast. The procedure can be generalized to obtain other types of

histogram.

6.3.2.6 Correction of the Geometric Distortion

Geometric distortions often occur in image acquisition processes, leading to

changes in the spatial relationships between the pixels. Spatial transforma-

tions are used to suppress geometric distortions. The spatial transformations

change the relative positions between pixels. Once the geometric changes

have been suppressed, the new gray level values for the pixels have to be esti-

mated. These new values are estimated by means of interpolation. Geometric

transformations are usually expressed by means of a matrix. The commonest

geometrictransformations include translation and rotation (which are usually

called Euclidean because they preserve distances and angles).Aslightly more

general set of transformations includes rotations, translations, scaling, and

reﬂection(these arecalled afﬁnetransformations because they preserve paral-

lelism). Given a pixel of coordinates (x, y), the coordinates of the transformed

pixel (x



, y



) can be calculated by

















(6.40)

In the general case, the coordinates (x



, y



) may not be integers. It is, there-

fore, necessary to estimate the corresponding gray level (Figure 6.29). The

usual procedure consists of considering that the coordinates (x, y) correspond

to the pixel in the image with distortion and then estimate the coordinates of

the corresponding pixel (x



, y



) in the image without distortion. Since there

is an analytic model for the geometric distortion which implies that, in gen-

eral, the corresponding gray level value has to be estimated based on the

262 Nuclear Medicine Physics

(x, y)

(x′, y′)

FIGURE 6.29

Correction of geometric distortion and interpolation.

gray level values of the neighboring pixels. The simplest solution consists of

interpolating the value based on the gray level value of the nearest neigh-

bor. This kind of interpolation is usually called zero-order interpolation. It

involves choosing for the gray level value of the pixel in the image without

distortion (with coordinates [x, y]) the gray level value of the pixel closest

to (x



, y



). The coordinates for this pixel can be obtained by rounding off the

coordinates (x



, y



). A widely used solution is bilinear interpolation. Consider

four neighboring pixels whose coordinates are (x

, y

),(x

, y

i+1

),(x

i+1

, y

), and

i+1

, y

i+1

). Let g(x, y) be the gray level value of pixel (x, y). Then the gray

level value g(x



, y



) estimated using bilinear interpolation is given by

g(x



, y



) = (1 −t)(1 −s)g(x

, y

) +t(1 −s)g(x

i+1

, y

)

+(1 −t)sg(x

, y

j+1

) +tsg(x

i+1

, y

j+1

)

where x

< x



< x

i+1

, y

> y



< y

i+1

and

t =



−x

i+1

−x

s =



−y

i+1

−y

(6.41)

Other types of interpolation are possible, in particular, cubic interpola-

tion using splines. However, results obtained with bilinear interpolation are

generally good enough for most applications.

6.3.2.7 Color Preprocessing

Human perception of color is a complex phenomenon that is not yet fully

understood. It has been experimentally determined that for most viewers

only three colors (usually called primary colors) are required to obtain (by

combination) any other color. However, for this to be possible, it has to be

admitted that the combination can be subtractive and that the primary colors

are independent (i.e., mixing two of the primaries cannot produce the third).

This principle is usually known as the trichromatic principle. It is explained

Imaging Methodologies 263

by admitting that there are three types of photoreceptors in the retina. In gen-

eral, given the same primary colors and a test color, most viewers choose the

same mixture of primaries to obtain the test color. Based on these facts, one

of the ways most often used to represent a speciﬁc color is to deﬁne a set of

primary colors and express any visible color by means of three multiplying

weights corresponding to those that an average observer would employ to

get that speciﬁc color. The retina photoreceptors responsible for color percep-

tion are designated by cones. Experimental evidence shows that the 6 to 7

million retina cones can be divided into three types roughly corresponding

to red, green, and blue. About 65% of the cones are sensitive to red, 33% are

sensitive to green, and only about 2% are sensitive to blue (but they are the

most sensitive) [107]. The spectral sensitivities of the three types of cones are

represented in Figure 6.30. These are the colors known as additive primaries.

It is important to distinguish between light primary colors and pigment

primary colors (also known as subtractive primary colors). Pigment primary

colors are deﬁned as those colors that subtract or absorb a light primary color

and reﬂect or transmit the other two [107]. Therefore, the pigment primary

colors are cyan, magenta, and yellow. Additive primary colors are employed

in cathode ray tubes; and they are red, green, and blue. Pigments, however,

combine based on the subtraction of colors.

–1

–2

–3

–4

–5

–6

–7

–8

350 400

Wavelength (nm)

450 500 550 600 650 750 800 850700

FIGURE 6.30

Cone responses to visible radiation: logarithms of the spectral sensitivities of the three types

of cones in the human retina. The S curve corresponds to blue cones, the M curve corresponds

to green cones, and the L curve corresponds to red cones. (From R. C. Gonzalez, R. W. Woods,

Digital Image Processing, 2nd Edition, Prentice-Hall, Englewood Cliffs, NJ. With permission.)

264 Nuclear Medicine Physics

Colors can be represented in several types of spaces. One of the most com-

mon linear spaces for representing colors is RGB. This is a linear space using

primary colors. Each of the primary colors in this space corresponds to single

wavelength radiation (645.16 nm for red, 526.32 nm for green, and 444.44 nm

for blue). In this space, colors are represented in a cube, the RGB cube, whose

edges represent the weights of the primary colors (Figure 6.31). The cube is

represented in a Cartesian coordinate system. Black is located at the origin,

and white is located at the vertex farthest from the origin. The vertices on the

axes of the coordinate system correspond to the primary colors red, green, and

blue. In this model, different colors are points inside the cube and are deﬁned

by vectors that connect them with the origin [107]. If the pixels in each of the

images arerepresented by 8 bits, then each color pixel is represented by 24 bits.

The spaces used to represent the subtractive colors are different. The sim-

plest color space for subtractive colors is CMY made up of cyan, magenta,

and yellow. Cyan is obtained by subtracting red from white, magenta is

obtained by subtracting green from white, and yellow is obtained by sub-

tracting blue from white. However, in practice, printing systems require at

least four colors (cyan, magenta, yellow, and black), because the mixture of

the three subtractive primary colors does not generally produce good quality

black colors.

Additive colours can be represented in other, nonlinear spaces. In these

spaces, the goal is to represent color properties and their perceptual

FIGURE 6.31

The RGB cube. The cube has unit dimensions, and in this space all the colors can be obtained

combining the primary colors red, green, and blue, corresponding to the three axes. (From D.

Forsyth, J. Ponce, Computer Vision: A Modern Approach, Prentice-Hall, Engle-wood Cliffs, NJ. With

permission.)

Imaging Methodologies 265

Input

Images

I(m,n)

Feature extraction

Color coordinates

transformation

Display

(m, n)

FIGURE 6.32

Image transformation using pseudo-color.

relationships. One of these spaces is the HSV (Hue, Saturation, Value). Hue

is the pure color or the dominant wavelength. Saturation is the whiteness of

the color or the relative bandwidth of the color. A high saturation color is a

color with low bandwidth. Value is the brightness or intensity of the color.

This space can be readily obtained from RGB.

The use of pseudo-colour for image processing aims at increasing the per-

ception of some aspects or details of the information available in an image.

Visualization and interpretation of images become easier. This is because the

human visual system is able to discriminate many more colours than gray lev-

els. Pseudo-color processing deals also with mapping a set of images I

(m, n)

with i = 1, ..., k into a single color image. This mapping is usually deﬁned

so as to ensure that the different elements or structures in the dataset can be

distinguished by means of the colors [106]. This makes it easier to visualize

complex datasets (Figure 6.32).

The usual procedure to determine pseudo-colour mappings consists of

mapping the set of input images into three feature images (by means of the

functions V

(m, n), V

(m, n), and V

(m, n)) and then mapping those three

images into the primary colors. If the goal is to transform a gray level image

into a pseudo-color image, however, a mapping must ﬁrst be deﬁned between

the gray levels and the color space. This establishes a correspondence between

each gray level and a color. If the number of initial images is bigger than three,

then it will be necessary to generate three intermediate images. These images

have to be generated by processing them so that the relevant goals can be

achieved. After the three images have been generated, the color mapping can

then be applied. In general, pseudo-color mappings are deﬁned by trial and

error so that colors are selected to best ensure the visualization of the features.

6.3.3 Image Registration

Image registration is the process of placing different images in the same frame

of reference with the goal of integrating and comparing their information. It

consists of aligning images by ﬁnding correspondences between common

characteristics [109,110]. For this, it is necessary to determine the transforma-

tion that maps the points in an image in the corresponding points in the other

image.

266 Nuclear Medicine Physics

Image registration aims at beneﬁting from integrating information from

several images. Usually, it is possible to extract more useful information from

many images than can be obtained by separately analyzing each of these

images. This is the main reason for the interest in attempting to solve the

difﬁculties posed by the problem of image registration.

Registration can be performed for images of the same type or for images of

different types. The term multimodal registration is used in the second case. Dif-

ferenttypes of images add extra information to the ﬁnal result, thus increasing

the amount of useful information. For instance, CT images provide anatomi-

cal information, and PET images provide functional information. When these

images are registered, in addition to the existing information, we gain from

learning the precise location of the functional information. In many situa-

tions, it is advantageous to align images of the same type, either to analyze

the variations in time or to perform studies between different subjects. In the

case of NM, we should note that although one may use images of the same

modality and of the same individual, the information can vary if different

tracers are used.

The importance of registration in the medical ﬁeld is essentially based on

two circumstances: on one hand, diagnosis often implies the acquisition of

data from many image modalities; and on the other hand, an individual goes

through several exams in his or her lifetime. It is often up to the physician

to combine the information from the different acquired images in his or her

mind or to compare studies acquired at different times, searching for dif-

ferences between them. If the task of merging only two exams is already

complex enough, trying to combine information from a sample of subjects

with the goal of determining a pattern is almost impossible to be mentally

performed. Image registration creates a map between different images, allow-

ing the accumulation of point-to-point information or the search of small

differences, either in time or in a population.

Besides allowing the creation of better images, registration also helped new

technologies such as guided surgery to emerge. The images acquired before

intervention are usually registered (in real time) with the surgical device and

used as a guide by the surgeon.

The work in this ﬁeld is vast, and the algorithms found in the literature are

very diverse. The diversity is easily understood if we take into account the

different paths that can be taken to reach the ﬁnal goal, which is to establish

a correspondence between the points of the images to be coregistered.

The problem can be generically seen as the search for the optimal transfor-

mation, τ, between two images, where by transformation we consider a way

to map points between the two images. Thus, we have

arg min

S(A −τ(B)). (6.42)

where S represents a measure of the similarity between points in image A and

points in the transformed image B. If the two images were exactly the same, τ

Imaging Methodologies 267

would represent a unitary operation; and the measure of similarity, S, applied

to all points in the image would give zero.

Although the previous equation suggests that registration is always seen as

an optimization problem to which the usual optimization methods and tech-

niques are applied, there are examples where the transformation τ is obtained

byother means, for instance, bysolving adifferentialequation.However,even

in these cases, it is common to use iterative algorithms with a stopping cri-

terion deﬁned by a function of similarity. Except for the due differences, we

can say that the problem is still that of ﬁnding adequate mapping between

the two images.

Equation 6.42 allows us to check the various questions that are posed in

solving this problem. Summarizing, we can have different types of initial

data, different forms of considering the transformation that is inherent to

the correspondence, several ways of measuring the similarity or the proxim-

ity associated with the correspondence, and, ﬁnally, different approaches to

minimization. Apart from what is generically indicated by Equation 6.42, it

is necessary to take particular aspects into account that are proper to each

speciﬁc application, for example, an increase or decrease of the area in a zone

of tracer uptake does not usually have the same anatomical meaning.

One of the ﬁrst classiﬁcation criterion for registration methods was pro-

posed by Elsen [111], which was later returned to by Maintz [112,113]. The

criteria adopted in this work are found in subsequent literature, with small

variations [114–116]. Essentially, the following classiﬁable aspects of the

method are distinguished:

1. Dimension

2. Nature of the basis of registration

3. Nature and domain of the transformation

4. Interaction

5. Optimization

6. Modalities

7. Subject

6. Object of registration

These aspects will now be further described.

6.3.3.1 Dimension

The initial data can originate from 2D or 3D images. Registration can be

achieved through either 2D or 3D images and, in some cases, between 2D and

3D images. The type of image used at the start has direct implications for the

mapping of the data being searched for; at least, there is an implication regard-

ing the number of dimensions that need to be taken into account. However,