Chapter 6

CT, MRI and Image Processing Problems

Image processing has become one of the most important components in medical imaging modalities such as magnetic resonance imaging, computed tomography, ultrasound and other functional imaging modalities. Image processing techniques such as image restoration and sparse sensing are being used to deal with various imperfections in the data acquisition processes of the imaging modalities. Image segmentation, referring to the process of partitioning an image into multiple segments, has numerous applications, including tumor detection, quantification of tissue volume, computer-guided surgery, study of anatomical structure and so on. In this chapter, we review the basic mathematics behind X-ray computed tomography (CT) and magnetic resonance imaging (MRI), and then discuss some image processing techniques.

6.1 X-ray Computed Tomography

X-ray computed tomography (CT) is the most widely used tomographic imaging technique, which uses X-rays passing through the body at different angles. It visualizes the internal structures of the human body by assigning an X-ray attenuation coefficient to each pixel, which characterizes how easily a medium can be penetrated by an X-ray beam Hounsfield (1973). The idea is to visualize the imaging object in a slice by taking X-ray data at all angles around the object based on mathematical methods suggested by Cormack (1963). They shared the 1979 Nobel Prize. Indeed, some of the ideas of CT (reconstructing cross-sectional images of an object from its integral values along lines in all directions) were previously developed by Radon (1917). Bracewell (1956) had applied his theory to radioastronomy, but unfortunately little attention was paid to it at that time. Cormack was unaware of Radon's earlier work, and Radon himself did not know the even earlier work by the Dutch physicist Lorentz, who had already proposed a solution of the mathematical problem for the three-dimensional case (Cormack 1992). We refer to Kalender (2006) for an excellent review of CT.

6.1.1 Inverse Problem

The corresponding inverse problem to X-ray CT can be described roughly as follows.

  • Quantity to be imaged. The distribution of linear attenuation coefficients, denoted by a function f(x) at point x = (x1, x2) or x = (x1, x2, x3). See Figure 6.1.
  • Input data. An incident X-ray beam is passed through a patient placed between an X-ray source and a detector. These beams are transmitted in all directions Θ: = (cosθ, sinθ), 0 ≤ θ ≤ 2π. Assuming a fixed angle θ, a number of X-ray photons are transmitted through the body along projection lines images, images. See Figure 6.1.
  • Output data. Detectors measure X-ray intensity attenuation Iθ(s) along the individual projection lines Lθ, s at all angles. These measured data Iθ(s) provide an X-ray image Pθ(s) that can be expressed roughly by

6.1 6.1

where images, called the Radon transform of f, is the integral along the line images in the direction Θ = (cosθ, sinθ):

6.2 6.2

where images is the length element. See Figure 6.1.
  • Inverse problem. Recover f from the series of X-ray data images, where θn = 2nπ/N.

Figure 6.1 Illustration of the Fourier slice theorem

6.1

6.1.2 Basic Principle and Nonlinear Effects

We begin with understanding the relationship between X-ray intensity attenuation Iθ(s) and the X-ray data Pθ(s) in (6.1). For simplicity, we ignore scattering and metal shadowing effects for the moment.

Fix θ and s. The incident X-ray beam is composed of a number of photons of different energies. Hence, the incident intensity of the X-ray beam, denoted by I0, can be viewed as a function of the photon energy level E (Brooks and Chiro 1976). Imagine that an X-ray passes through a patient along Lθ, s. Let Iθ(E, s) be the measured attenuated intensity along the line Lθ, s. Then I0(E) indicates the source X-ray at energy level E, and Iθ(E, s) indicates the detected X-ray after passing through the body along the line Lθ, s. Denoting by fE(x) the attenuation coefficient at point x and at energy level E, the relation between Iθ(E, s) and fE is dictated by the Beer–Lambert law (Beer 1852; Lambert 1760):

6.3 6.3

where Emax and Emin, respectively, are the maximum and minimum energy levels of the X-ray beam. Soft tissues have roughly fE = 0.38 at E = 30 keV cm−1 and fE = 0.21 at E = 60 keV cm−1. If the beam does not interact with any medium, then the unattenuated beam is Iθ(E, s) = I0(E).


Remark 6.1.1 (Beam hardening)
The lower-energy photons tend to be absorbed more rapidly than higher-energy photons. As a result, the mean energy of the incident X-ray beam is lower than the mean energy of the X-rays reaching the detectors after passing through an object, that is,

images

This effect is called beam hardening, since the mean energy of Iθ (after passing through the object) becomes greater than that of I0 (before passing through the object). In a disk-like object, as shown in Figure 6.2, the reconstructed CT image of this homogeneous object appears brighter near the boundary than in its interior as a result of the beam hardening process.

Figure 6.2 Illustration of metal artifacts

6.2

The measured attenuation intensity Iθ(s) along the projection line Lθ, s is given by

6.4 6.4

This Iθ(s) provides the X-ray data Pθ(s) given by

6.5 6.5

We try to reconstruct an image f such that

6.6 6.6

We hope that f is related to images for some energy level E0.


Example 6.1.2
In the special case when the X-ray beam is monochromatic, the relation between images and the measured intensity along the line Lθ, s can be expressed by Beer's law,

6.7 6.75

In this case, there is no nonlinear effect in (6.1), and f can be reconstructed directly by the inverse Radon transform algorithm described in the next section.

 


Example 6.1.3
In the case when the X-ray beam is bichromatic, having two dominant energy levels E1 and E2 with E1 < E0 < E2, the measured intensity along the line Lθ, s can be expressed by

6.8 6.76

Assume thatfE/∂E < 0. Then,

6.9 6.77

From the fundamental theorem of calculus, (6.9) can be expressed as

6.10 6.78

Then, the difference between the Radon transform of the tomographic image f and the X-ray image Pθ(s) = ln[I0/Iθ(s)] is given by

6.11 6.79

The right-hand side of the above identity is the nonlinear effect described in (6.1).

6.1.3 Inverse Radon Transform

Recall that the Radon transform images of an attenuation coefficient f(x) is the one-dimensional projection of f(x) taken at an angle θ. Let images denote the projection map in (6.2) as a function of θ and s, which can be expressed as

6.12 6.7

The back-projection operator associated with the projection map images is defined by

6.13 6.8

The Hilbert transform of a function ϕ(t) is defined as

6.14 6.9


Theorem 6.1.4 (Inverse Radon transform and back-projection)
Given images, its inverse Radon transform is

6.15 6.10

which can be expressed as

images

On the other hand, the back-projection images provides a blurred image of f(x, y) given by

6.16 6.11

where * denotes the two-dimensional convolution in Cartesian coordinates.

In order to prove this theorem, we need to know the following Fourier slice theorem.


Theorem 6.1.5 (Fourier slice theorem)
For a fixed θ, the one-dimensional Fourier transform with respect to s of the projection images is equal to the line in the two-dimensional Fourier transform of f taken at θ:

6.17 6.12

where images and images are the one- and two-dimensional Fourier transforms, respectively.

 


Proof.
The proof goes as follows:

6.18 6.13


Now, we are ready to prove the inverse Radon transform.


Proof.
(Inverse Radon transform) The inverse Fourier transform

images


can be changed into the polar coordinate form

images

Using (6.17), we can change the form again into the following:

6.19 6.14

Denoting images, we obtain

6.20 6.15

because

images

Hence, it follows from (6.19) and (6.20)) that

6.21 6.16

This completes the proof of the inverse Radon transform.


Remark 6.1.6
The back-projection operator images is not the inverse of the Radon transform images, but the adjoint of images in the following sense:

6.22 6.17

Since images, the image of images can be viewed as a blurred version of f.

 


Remark 6.1.7
There is also the inverse Radon transform without derivative, which is called the filtered back-projection. To be precise, let images be supported in the interval [ − Γ, Γ] (called band-limited). Then, according to (6.19), we have

images

Using the fact that images, we have

images

where h is the inverse Fourier transform of images:

images

Hence, f can be computed from

images

This is called the filtered back-projection method, which was first described by Bracewell and Riddle (1967). From the Nyquist sampling criterion, the bandwidth Γ and projection sampling interval images should satisfy the relation, images.

 


Exercise 6.1.8
Consider the inverse Radon transform of Pθ(s) in (6.11) described in Example 6.1.3:

6.23 6.18

Show that the relation between f and Pθ is images for all θ and s.

We refer to Faridani (2003) and Natterer (2008) for detailed explanations on the Radon inversion formula, and refer to Tuy (1983), Feldkamp et al. (1984), Kak and Slaney (1988) and Grangeat (1991) for the inversion algorithm in cone-beam CT, which uses a cone-shaped X-ray beam rather than a conventional linear fan beam.

6.1.4 Artifacts in CT

In X-ray CT, the incident intensity I0(E) and the attenuated intensity Iθ(E, s) are functions of the energy level E, which ranges over 10–150 keV, and the attenuation coefficient fE differs with the energy level E. Mostly, ∂fE/∂E < 0, which means that the lower-energy photons are absorbed more rapidly than higher-energy photons. Hence, there exist fundamental artifacts in the reconstructed image f(x) as in Figure 6.2 by the inverse Radon transform since f can be regarded as a distribution of the linear attenuation coefficient f(x) ≈ f(x) at a mean energy level E0.

The presence of metal objects in the scan field can lead to severe streaking artifacts owing to the above-mentioned inconsistencies by beam hardening and severely high contrast in the attenuation coefficient distribution between the metal and the surrounding subjects. Figure 6.3 shows such metal artifacts. Additional artifacts due to beam hardening, partial volume and aliasing are likely to compound the problem when scanning very dense objects. We refer to Kalender et al. (1987), Meyer et al. (2010) and Wang et al. (1996) for metal artifacts reduction.


Example 6.1.9
Figure 6.2 illustrates how beam hardening produces artifacts in CT images in the simplest case of the Example 6.1.3 with I0 = I0(E1) + I0(E2), I0(E1) = I0(E2) = 1 and

images

  • images
  • images
  • images
We try to reconstruct images, but the reconstructed image f using the standard inversion formula can be very different from images for any α.

Figure 6.3 Examples of metal artifacts

6.3

6.2 Magnetic Resonance Imaging

MRI uses magnetic fields to visualize the quantity of hydrogen atoms inside biological tissues by creating magnetization; hydrogen atoms can interact with the external magnetic field B because a nucleus with spin has a local magnetic field around it. In the human body, the amount of hydrogen would be the major factor of the net magnetization vector M. In MRI, we use various techniques to localize M in such a way that we provide a cross-sectional image of the density of M inside the human body. These techniques are based on the nuclear magnetic resonance (NMR) phenomenon, which is determined by the interaction of a nuclear spin M with the external magnetic field B and its local environments, including relaxation effects.

The idea behind this imaging modality was published by Lauterbur (1973), and the first cross-sectional image of a living mouse was published by Lauterbur (1974). Damadian (1971) discovered that NMR can distinguish tumors from normal tissues due to their relaxation time. Lauterbur and Mansfield were awarded the Nobel Prize for developing the mathematical framework and some MRI techniques, but the award was denied by Damadian, who claimed that his work was earlier than those of Lauterbur and Mansfield. We refer to Filler (2010) for the history of MRI. We will describe the NMR phenomenon in the next section, taking account of measurable signals in an MRI system.

6.2.1 Basic Principle

We consider a human body occupying a domain Ω inside an MRI scanner with its main magnetic field B0 + δB. Throughout this section, we shall assume that the field inhomogeneity δB is negligible, that is, δB = 0 and images is constant. The strong uniform main field produces a distribution of net magnetization M(r, t) in Ω by aligning protons inside the human body, where images depends on time t and position r. A stronger B0 produces a larger images. The interaction of images with the external magnetic field B0 is dictated by the Bloch equation:

6.24 6.19

where γ is the gyromagnetic ratio1. From (6.24), we have

6.25 6.20

which means that the vector ∂M/∂t is perpendicular to both M and B0. To be precise, (6.24) can be written as

6.26 6.21

Writing M = Mx + iMy, we express the above identity as

6.27 6.22

The solution of the above ODE is

6.28 6.23

which means that the transverse component M rotates clockwise at the angular frequency images. The above expression explains how B0 causes M to precess around the z axis at the angular frequency of images. For a 1.5 T MRI system, the frequency of images is approximately 64 MHz. According to (6.28), images causes M to precess clockwise about the images direction at the angular frequency images. If we apply a magnetic field B(r), then M(r, t) satisfies the Bloch equation

images

and M(r, t) precesses at the angular frequency images.

6.2.2 k-Space Data

If we could produce a magnetic flux density B that is localized at each position r (in such a way that images), we could encode the positions independently. However, it is impossible to localize the magnetic field B because ∇ · B = 0. Instead of a point localization, we could make a slice localization by applying a magnetic field gradient that varies with respect to one direction. The following is the one-dimensional magnetic field gradient along the z axis with the magnetic field increasing in the z direction:

6.29 6.24

where B0 and Gz are constants. The first component B0 is the main magnetic field, which causes individual M(r) to precess around the z axis, and the second term images is the gradient field. This Gz is said to be the magnetic field gradient in the z direction. With this B, the vector M in the body is essentially vertically aligned and rotates at the Larmor frequency γ(B0 + Gzz) around the z axis. Note that the frequency changes with z.

To extract a signal of M, we flip M toward the transverse direction to produce its xy component. Flipping M toward the xy plane requires a second magnetic field B1 perpendicular to B0 = (0, 0, B0). We can flip M over the xy plane by using a radio-frequency (RF) magnetic field B1 that is generated by RF coils through which we inject sinusoidal current at the Larmor frequency images.

After terminating the RF pulse, we apply a phase encoding gradient in the y direction, which makes the spin phase change linearly in the phase encoding direction. Then we apply the frequency encoding gradient field as

images

and the signal during frequency encoding in the x direction becomes

6.30 6.25

Through multiple phase encodings to get multiple signals of (6.30) for different values of ky, we may collect a set of k-space data

6.31 6.26

for various kx and ky. See Figure 6.4 (top left) for an image of S(kx, ky). We refer the reader to the book by Haacke et al. (1999) for detailed explanations on MRI.

Figure 6.4 The two images at the top are fully sampled k-space data and the inverse Fourier transform. The two images at the bottom are subsampled k-space data by the factor of 2 and the inverse Fourier transform

6.4

6.2.3 Image Reconstruction

In the Cartesian sampling pattern, we select a cross-sectional slice in the z direction using the gradient coils and get spatial information in the xy plane using the frequency encoding (x direction) and phase encoding (y direction). Let m be a spin density supported in the region {(x, y): − FOV/2 < x, y < FOV/2}, where FOV is the field of view, and let m be the N2 column vector representing the corresponding N × N image matrix with pixel size Δx × Δy = FOV/N × FOV/N. The inverse Fourier transform provides the image of m(x, y) (see Figure 6.4 (top right)). Let us quickly review the inverse discrete Fourier transform (DFT). Assume that the image of m(x, y) is approximated by an N × N matrix m:

images

Then, the relation (6.26) between the MR data and the image of m(x, y) can be expressed as

6.32 6.27

for some scaling constant c and j, images = 0, 1, …, N − 1. Here, the inverse DFT with a k-space sampling that is designed to meet the Nyquist criterion provides a discrete version of the image m(x, y). Using the two-dimensional inverse DFT, we have

6.33 6.28

for some scaling constant images.

In MRI, the data acquisition speed is roughly proportional to the number of phase encoding lines due to the time-consuming phase encoding, which separates signals from different y positions within the image m(x, y). Hence, to accelerate the acquisition process, we need to skip phase encoding lines in k-space. Definitely, reduction in the k-space data violating the Nyquist criterion is associated with aliasing in the image space. If we use subsampled k-space data by a factor of R in the phase encoding direction (y direction), according to the Poisson summation formula, the corresponding inverse Fourier transform produces the following fold-over artifacts (as shown in Figure 6.4 (bottom right) and Figure 6.5):

images

Figure 6.5 The images at the top are subsampled k-space data sets. The images at the bottom are the corresponding inverse Fourier transforms

6.5

Parallel MRI (pMRI) is a way to deal with this fold-over artifact by using multiple receiver coils and supplementary spatial information in the image space. Parallel imaging has received a great deal of attention since the work by Sodickson and Manning (1997), Sodickson et al. (1999) and Pruessmann et al. (1999). In pMRI, we skip phase encoding lines in k-space during MRI acquisitions in order to reduce the time-consuming phase encoding steps, and in the image reconstruction step, we compensate the skipped k-space data by the use of space-dependent properties of multiple receiver coils. Numerous parallel reconstruction algorithms such as SENSE, SMASH and GRAPPA have been suggested, and those aim to use the least possible data of phase encoding lines, while eliminating aliasing, which is a consequence of violating the Nyquist criterion by skipping the data. We refer to Jakob et al. (1998), Kyriakos et al. (2000), Heidemann et al. (2001), Pruessmann et al. (2001), Bydder et al. (2002) and Griswold et al. (2002) for pMRI algorithms and to Larkman and Nunes (2001) for a review.

6.3 Image Restoration

Numerous algorithms for image restoration and segmentation are based on mathematical models with partial differential equations, level set methods, regularization, energy functionals and others. In these models, we regard the intensity image as a two-dimensional surface in a three-dimensional space with the gray level assigned to the axis. We can use the distribution of its gradient, Laplacian and curvature to interpret the image. For example, Canny (1986) defined edge points as points where the gradient magnitude assumes a local maximum in the gradient direction. The noise is usually characterized as having a local high curvature. Using these properties, one may decompose the image roughly into features, background, texture and noise. Image processing techniques are useful tools for improving the detectability of diagnostic features. We refer the reader to the books by Aubert and Kornprobst (2002) and Osher et al. (2002) for PDE-based image processing.

For clear and easy explanation, we restrict ourselves to the case of a two-dimensional grayscale image that is expressed as a function uL2(Ω) in the rectangular domain Ω = {x = (x, y):0 < x, y < 1}. Throughout this section, we assume that the measured (or observed) image fL2(Ω) and the true image u are related by

6.34 6.29

where η represents noise and H is a linear operator including blurring and shifting. For image restoration and segmentation, we use a priori knowledge about the geometric structure of the surface {(x, u(x)):x ∈ Ω} in the three-dimensional space.

The goal of denoising is to filter out noise while preserving important features. If the noise η is Gaussian random noise satisfying ∫η = 0, we can apply the Gaussian kernel

images

to get an approximation

images

with an appropriate choice of variance σ > 0. Note that Gt * f(x) is a solution of the heat equation with the initial data f(x):

images

Hence, we expect that this smoothing method eliminates highly oscillatory noise that results in Gσ * (Hu + η) ≈ Gσ * (Hu), while edges having high frequencies are also smeared at the same time. Several schemes have been proposed to deal with this blurring problem.

A typical way of denoising (or image restoration) is to find the best function by minimizing the functional:

6.35 6.30

where the first term images, called the fidelity term, forces the residual Huf to be small, the second term images, called the regularization term, enforces the regularity of u, and λ, called the regularization parameter, controls the tradeoff between the residual norm and the regularity. We can associate (6.35) with the Euler–Lagrange equation:

6.36 6.31

where H* is the dual of H. The above equation is nonlinear except for p = 2. When p = 2, the minimizer u satisfies the linear PDE,

images

which has computational advantages in computing u. But the Laplace operator has very strong isotropic smoothing properties and does not preserve edges because it penalizes the strong gradients along edges. This is the major reason why many researchers use p = 1 since it does not penalize the gradients along edges with alleviating noise:

images

We can solve (6.35) using the gradient descent method by introducing the time variable t:

6.37 6.32

To understand the above derivation, we present the following simple example.


Example 6.3.1 (Gradient descent)
Let images be a symmetric and positive definite matrix. For a given vector images, consider the minimization problem

images

where ||u||2 = 〈u, uand 〈 ·, · 〉 is the usual vector inner product. Then

images

If u is a minimizer, then u satisfies the Euler–Lagrange equation

images

where I is the identity matrix in images. Since − ∇Φ(u) is the direction of steepest descent at a vector images, we have the following iteration scheme:

images

where α is the step size. Introducing the time variable t and taking

images

we derive

images

Expecting limt→∞du(t)/dt = 0, u(∞) is the minimizer of Φ(u).

6.3.1 Role of p in (6.35)

Let us investigate the role of the diffusion term ∇(|∇u|p−2u) in (6.35). The image u(x) can be viewed as a two-dimensional surface {(r, u(r)):(x, y) ∈ Ω}. To understand the image structure, we look at the level curve images, the curve along which the intensity u(x) is a constant c. Let x(s) represent a parameterization of the level curve images, that is, images. The normal vector and tangent vector to the level curve images, respectively, are

images

where ux = ∂u/∂x and uy = ∂u/∂y. Let us investigate the double derivative of u in the tangent and normal directions:

images

where (D2u) is the Hessian matrix of u. Direct computation shows that

images

This leads to

images

Substituting ux = − y′|∇u|/|x′| and uy = x′|∇u|/|x′| into the above identity and with 0 = ∇u · x′, we obtain

images

where ∇ · n = κ is the curvature. Hence,

6.38 6.33

Similarly, we have

6.39 6.34

  • In the case p = 2, we have the following isotropic diffusion:

6.40 6.35

Here, we use the fact that ∇u = |∇u|n. Indeed, the rotation invariance property of the Laplace operator yields (6.40).
  • In the case of p ≠ 2, it follows from (6.38), (6.39) and (6.40) that

images

Hence, the diffusion term in (6.35) can be expressed as

6.41 6.36

The diffusion rate in the T direction is |∇u|p−2, while the diffusion rate in the n direction is (p − 1)|∇u|p−2. Since n is normal to the edges, it would be preferable to smooth more in the tangential direction than in the normal direction. In order to preserve edges, p = 1 would be best, since it annihilates the coefficient of unn.

6.3.2 Total Variation Restoration

In one dimension, the total variation (TV) of a signal f defined in an interval [0, 1] means the total amplitude of signal oscillations in [0, 1]. The classical definition of the total variation of f is

6.42 6.37

where the supremum is taken over all partitions in the interval [0, 1]. The space of bounded variation functions in [0, 1], denoted by BV([0, 1]), is the space of all real-valued functions fL1([0, 1]) such that ||f||BV([0, 1]) < ∞. In the special case when fC1([0, 1]) is differentiable, it can be defined by

6.43 6.38


Example 6.3.2
If f is the well-known Cantor function, then ||f||BV(Ω) = 1 = f(1) − f(0) and it is in BV(Ω). However, the Cantor function f has the special property that f′ = 0 almost everywhere. Since images, the definition (6.43) is not appropriate for general functions in BV([0, 1]).

The classical definition (6.42) of BV in one dimension is not suitable in the two- or three-dimensional cases. Hence, we use the following definition that is closely related to (6.43):

6.44 6.39

Now, we come back to the two-dimensional grayscale image u(r) defined in the rectangular domain Ω = {r = (x, y):0 < x, y < 1}.


Definition 6.3.3
The total variation of a function u on Ω is defined by

6.45 6.40

A function u in Ω is a bounded variation on Ω if ||u||BV(Ω) < ∞:

6.46 6.41


 


Theorem 6.3.4
If images, then

images


 


Proof.
Let images. From the divergence theorem,

images

Hence, for all images,

images

This leads to

images

It remains to prove images. We can make a sequence images such that

images

For example, we may choose

images

where Gσ is the Gaussian kernel and ϕn is a positive function in images such that ϕ(x) = 1 if images. Hence, we have

images

This leads to images. This completes the proof.

For a smooth domain D, the bounded variation of the characteristic function χD is

images

In general, according to the co-area formula, the total variation of images can be expressed as

images

Now, we are ready to explain the TV restoration that is the case for p = 1 in (6.35):

6.47 6.42

Assuming that H is a linear operator with its dual H*, this is associated with the nonlinear Euler–Lagrange equation:

6.48 6.43

Hence, the minimizer u can be obtained from the standard method of solving the corresponding time marching algorithm:

6.49 6.44

To understand TV denoising effects, consider a smooth function u satisfying

6.50 6.45

For a fixed value images, let images be a simply closed curve in Ω such that u(x, t) = α on images and u(x, t) > α inside images. Since u satisfies the nonlinear diffusion equation (6.50),

images

Note that the outer normal vector n on images can be represented as

images

where images denotes the sign of u(x, t) − α in images. Hence, we have

6.51 6.46

and therefore

images

or

images

Hence, the TV regularization quickly removes smaller-scale noise where the level set ratio images is very large.

According to (6.48), the noisy image f can be decomposed into

6.52 6.47

where u is a minimizer of the TV-based ROF (Rudin–Osher–Fatemi) model (Rudin et al 1992)

images

With the use of the G-norm introduced by Meyer (2002), we can understand the role of the parameter λ more precisely (see Figure 6.6):

  • images and v = f, where images is the G-norm of v given by

images

Here, G, called the G-space, is the distributional closure of the set

images

  • images and images

The G-space is somehow very close to the dual of the BV space because of the following inequality:

images


Example 6.3.5
Numerous variations of the ROF model have been proposed. We will present some of them.
1. Meyer (2002) suggested

images

2. Vese and Osher (2002) proposed

images

with p ≥ 1.
3. Osher et al. (2005) used the Bregman distance function for making an iterative regularization procedure:

images

where D(u, v) is the Bregman distance between u and v associated with the functional Φ.
4. Osher et al. (2002) suggested

images

Its Euler–Lagrange equation is

images

The L2-fitting in ROF is replaced by the H−1-fitting. In this case, we have
a. images
b. images and images.

Figure 6.6 Role of the parameter λ: f is the noisy image; u is the image reconstructed by the TV-based ROF model; and v = fu

6.6

6.3.3 Anisotropic Edge-Preserving Diffusion

Diffusion PDE can act as a smoothing tool for removing high-frequency noise in the surface {(x, u(x)):x ∈ Ω}. In the previous section, we learnt that the TV restoration performs an anisotropic diffusion process by smoothing the surface except cliffs and preventing diffusion across cliffs. This type of edge-preserving diffusion technique has been widely used in the image processing community after Perona and Malik (1990) proposed an anisotropic diffusion model.

We begin by studying the general Perona–Malik model:

6.53 6.48

where ut = ∂u/∂t. This is a nonlinear PDE expressing the fact that the time change of u is equal to the negative of the divergence of the current − P(|∇u|2)∇u. It follows from (6.38) and (6.39) that

images

where Q(|∇v|2) = P(|∇v|2) + 2|∇v|2P′(|∇v|2), n(x) = ∇u/|∇u| and T = ( − uy, ux)/ |∇u|. Hence, the Perona–Malik model in (6.53) can be expressed as

6.54 6.49

From the standard arguments about diffusion equations, we conclude the following:

  • Q(|∇u|) > 0 images forward diffusion along ∇u direction images edge blurring.
  • Q(|∇u|) < 0 images backward diffusion along ∇u direction images edge sharpening.

Based on this observation, Perona and Malik (1990) proposed the diffusion coefficient

images

which corresponds to the concave energy functional

images

Note that the corresponding Perona–Malik PDE (forward–backward diffusion equation) has the following property:

  • |∇u| < K images forward diffusion along ∇u direction images blurring.
  • |∇u| > K images backward diffusion along ∇u direction images edge sharpening.

Hence, the Perona–Malik PDE permits forward diffusion along the edges while sharpening across the edge (Perona and Malik 1990). However, this Perona–Malik PDE raises issues on the discrepancy between numerical and theoretical results in that it is formally ill-posed in the sense of Hadamard, whereas its discrete PDE is nevertheless numerically found to be stable by Calder and Mansouri (2011). Kichenassamy (1997) investigated this phenomenon, termed the Perona–Malik paradox, in which the Perona–Malik PDE admits stable schemes for the initial value problem without a weak solution.


Exercise 6.3.6
The behavior of solutions of the forward heat equation ut = ∇2u as t increases is very different from that of the backward heat equation ut = − ∇2u. Consider the one-dimensional forward–backward heat equation:

images

where H(x) is the Heaviside function. Explain why this PDE process causes blurring for x > 0 but sharpening for x < 0.

 


Remark 6.3.7
The Perona–Malik PDE can be unstable with respect to small perturbations of the initial condition (You et al. 1996). The Perona–Malik process may produce a piecewise constant image (staircasing effect) even with a very smooth initial image (Weickert 1997).

6.3.4 Sparse Sensing

Fast imaging in MRI by accelerating the acquisition is a very important issue since it has a wide range of clinical applications such as cardiac MRI, functional MRI (fMRI), MRE, MREIT and so on. To reduce the MR data acquisition time, we need to skip as many phase encoding lines as possible (violating the Nyquist criterion) in k-space during MRI data acquisitions to minimize the time-consuming phase encoding step. In this case, we need to deal with the under-determined linear problem such as

6.55 6.50

We know that this under-determined linear system (6.55) has an infinite number of solutions since the null space has dim(N(A)) ≥ Nm. Without having some knowledge about the true solution, such as sparsity (having a few non-zero entries of the solution), there is no hope of solving the under-determined linear system (6.55).

Imagine that the true solution, denoted by xtrue, has sparsity. Is it possible to reconstruct xtrue by enforcing the sparsity constraint in the under-determined linear system in (6.55)? If so, can the solution be computed reliably? Surprisingly, this very basic linear algebra problem was not studied in depth until 1990.

Donoho and Elad (2003) found the following uniqueness result by introducing the concept of the spark of A:

images

where images indicates the number of non-zero entries of x.


Theorem 6.3.8 (Donoho and Elad 2003)
If the under-determined linear system (6.55) has a solution x obeying images, this solution is necessarily the sparsest possible.

 


Proof.
Assume that x′ satisfies Ax = Ax′ and images. Since A(xx′) = 0, it follows from the definition of spark that

images

Noting that

images

we must have x = x′.

For S = 1, 2, 3, …, define the set

images


Exercise 6.3.9
Let images. Show that, for x, x′ ∈ WS, xxif and only if AxAx′.

Although the under-determined linear system (6.55) has infinitely many solutions in images, according to Theorem 6.3.8, it has at most one solution within the restricted set WS for images. Hence, one may consider the following sparse optimization problem:

6.56 6.51

Let x0 be a solution of the images0-minimization problem (P0). Unfortunately, finding x0 via images0-minimization is extremely difficult (NP-hard) due to lack of convexity; we cannot use Newton's iteration.

Admitting fundamental difficulties in handling the images0-minimization problem (P0), it would be desirable to find a feasible approach for solving the problem (P0). One can consider the relaxed images1-minimization problem that is the closest convex minimization problem to (P0):

6.57 6.52

Let x1 be a solution of the images1-minimization problem (P1). Then, can the sparsest solution of (P0) be the solution of (P1)? If yes, when? Donoho and Elad (2003) observed that it could be x0 = x1 when ||x0||0 is sufficiently small and A has incoherent columns. Here, the mutual coherence of A measures the largest correlation between different columns from A. For more details see Donoho and Huo (1999).

For robustness of compressed sensing, Candès and Tao (2005) used the notion of the restricted isometry property (RIP) condition: A is said to have RIP of order S if there exists an isometry constant δS ∈ (0, 1) such that

6.58 6.53

If A has RIP of order 2S, then the under-determined linear system (6.55) is well-distinguishable within the S-sparse set WS:

images

If δ2S < 1, then the map A is injective within the set WS. If δ2S is close to 0, the transformation A roughly preserves the distance between any two different points.

Candès et al. (2006b) observed that the sparse solution of the problem (P0) can be found by solving (P1) under the assumption that A obeys the 2S-restricted isometry property (RIP) condition with δ2S being not close to one (Candès and Tao 2005; Candès et al. 2006a).


Theorem 6.3.10 (Candès et al. 2006b)
Let xexact be a solution of the under-determined linear system (6.55). Assume that A has RIP of order 2S with the isometry constant images. Then there exists a constant C0 such that

6.59 6.54


 


Proof.
For ease of explanation, we only prove (6.59) in the case where N = 3 and S = 1. Denote

images

We may assume |a| ≥ |b| ≥ |c|. Then

images

which leads to

images

Hence, it is enough to prove that

6.60 6.55

since the above estimate implies that

images

Without loss of generality, we may assume that |h2| ≥ |h3|. Denoting images, images and images, we have

images

This completes the proof of (6.60) since δ2/(1 − δ2) < 1 for images.

From the above theorem, if the true solution xexact is S-sparse, then x1 = xexact = x0, where x0 is a solution of (P0). Theorem 6.3.10 also provides robustness of compressed sensing. Consider the case where the measurements b are corrupted by bounded noise in such a way that

images


Corollary 6.3.11
Let images be a solution of the modified images1-problem:

6.61 6.56

Under the assumption of Theorem 6.3.10, if xS is the best S-sparse approximation of xexact, then

images

for some constants C0 and C1 > 0.

Lustig et al. (2007) applied these sparse sensing techniques for fast MR imaging. They demonstrated high acceleration in in vivo experiments and showed that the sparsity of MR images can be exploited to reduce scan time significantly, or alternatively to improve the resolution of MR images.

6.4 Segmentation

Image segmentation of a target object in the form of a closed curve has numerous medical applications, such as anomaly detection, quantification of tissue volume, planning of surgical interventions, motion tracking for functional imaging and others. With advances in medical imaging technologies, many innovative methods of performing segmentation have been proposed over the past few decades, and these segmentation techniques are based on the basic recipes using thresholding and edge-based detection. In this section, we only consider edge-based methods, which use the strength of the image gradient along the boundary between the target object and the background.

6.4.1 Active Contour Method

The most commonly used method of segmentation in the form of a closed curve would be (explicit or implicit) active contour methods that use an application-dependent energy functional to evolve an active contour toward the boundary of the target region; the direction of the velocity of the active contour is the negative direction of the gradient of the energy functional.

To be precise, let u be a given image. We begin by considering the minimization problem of finding a closed curve images that minimizes the energy functional

6.62 6.57

where g(α) is a decreasing function, for example, g(α) = 1/(1 + α). For computation of a local minimum images of the functional, we may start from an initial contour images and consider a sequence images that converges to the local minima images:

images

For computation of images, imagine that the sequence of curves images is parameterized by images, 0 < s < 1:

images

Then, the energy functional images at images can be expressed as

images

To calculate the next curve images from images, the gradient descent method based on the Fréchet gradient images is widely used. To determine the Fréchet gradient images, it is convenient to consider a time-varying contour images instead of the sequence images. See Figure 6.7 for the time-varying contour.

Figure 6.7 Time-varying contour

6.7

Setting images, we have

images

The variation of the energy functional Φ is

images

where rt = ∂r/∂t, rs = ∂r/∂s, n = n(s, t) is the unit normal to the curve images, and T = T(s, t) is the unit tangent vector. Hence, the direction for which Φ(t) decreases most rapidly is given by

6.63 6.58

Decomposing images, equation (6.63) becomes

6.64 6.59

which means that the curve r(s, t) moves along its normal with speed

6.65 6.60

Since

images

we can determine the update images by

6.66 6.61


Exercise 6.4.1
The energy functional Snake (Kass et al. 1987) is defined as

images

where the first two terms are regularization terms and g(u(r)) is used for attracting the contour toward the boundary of the target object. Here, we can set g(u(r)) = − |∇u(r)|2 or g(u(r)) = 1/[1 + |∇u(r)|p]. Explain why the first term suppresses the forces that shrink Snake and the second term keeps the forces that minimize the curvature. Show that the time evolution equation of Snake is

images


 


Exercise 6.4.2
The time evolution equation of deformable models by gradient vector flow (GVF) is expressed as

6.67 6.62

where v is the minimizer of the energy functional:

6.68 6.63

where μ is a regularization parameter and the gradient operator ∇ is applied to each component of v separately. Show that v can be achieved by solving the following equation:

images


6.4.2 Level Set Method

The active contour scheme using the explicit expression images is not appropriate for segmenting multiple targets whose locations are unknown. Using an auxiliary function ϕ(r, t), the propagating contour images changing its topology (splitting multiple closed curves) can be expressed effectively by the implicit expression of the zero level set

images

With the level set method (Osher et al. 2002), the motion of the active contour with the explicit expression (6.64)) is replaced by the motion of level set. Using the property that

images

we have the equation of ϕ containing the embedded motion of images; for a fixed s,

images

Since

images

the above identity leads to

6.69 6.64

Consider the special model:

images

This can be expressed as

6.70 6.65

The convection term 〈∇g, ∇ϕ 〉 increases the attraction of the deforming contour toward the boundary of an object. Note that (6.70) is related to the geodesic active contour model (6.64).

The level set method is one of the most widely used segmentation techniques; it can be combined with the problem of minimizing the energy functional of a level set function φ:

6.71 6.66

where Fitφ is a fitting term for attracting the zero-level contour Cφ: = {φ = 0} toward the target object in the image, Regφ is a regularization term of φ for penalizing non-smoothness of the contour, and μ is the regularization parameter. There exist a variety of fitting models, such as edge-based methods (Caselles et al. 1993, 1997; Goldenberg et al. 2001; Kichenassamy et al. 1995; Malladi et al. 1995), region-based methods (Chan and Vese 2001; Paragios and Deriche 1998; Yezzi et al. 1999), methods based on prior information (Chen et al. 2002) and so on. These fitting models are mostly combined with the standard regularization term penalizing the arc length of the contour Cφ. Among the variety of fitting models, the best-fitting model has to be selected depending on characteristics of the image. The selected fitting model is usually combined with an appropriate regularization term.

For example, Chan and Vese (2001) used the following energy functional (Chan–Vese model):

6.72 6.67

where λ1 and λ2 are non-negative parameters, ϕ is the level set function, and ave{ϕ≥0} and ave{ϕ<0} are the average values of u(r) in the two-dimensional regions {ϕ < 0} and {ϕ > 0}, respectively. Here, H(ϕ) is the one-dimensional Heaviside function, with H(s) = 1 if s ≥ 0, and H(s) = 0 if s < 0. To compute a minimizer ϕ for the minimization problem (6.72), the following parabolic equation is solved to the steady state:

6.73 6.68

with an appropriate initial level set function. After the evolution comes to a converged state, the zero level set of ϕ becomes the contour that separates the object from the background, as shown in Figure 6.8.

Figure 6.8 Segmentation of different objects with different intensities, with ϕ initialized as a signed distance function. The images at the top show the evolution of the zero level set. The images at the bottom show the evolution of the corresponding level set function. These are the converged results after 323 iterations

6.8

Example 6.4.3
Sarti et al. (2005) proposed the energy functional using the maximum likelihood estimator in ultrasound images with the Rayleigh distribution as the intensity distribution. Using the level set formulation, the energy functional can be expressed as

6.74 6.80

Then, the associated flow equation is

6.75 6.81

where

images


6.4.3 Motion Tracking for Echocardiography

Ultrasound imaging is widely used because of its non-invasiveness, real-time monitoring, cost-effectiveness and portability compared with other medical imaging modalities such as CT or MRI. In particular, owing to its high temporal resolution, cardiac ultrasound (echocardiogram) has been very successful in providing a quick assessment of the overall health of the heart. For quantitative assessment of cardiac functions, wall motion tracking and left ventricle (LV) volume quantification at each time are needed. Since manual delineation from echocardiography is extremely labor-intensive and time-consuming, the demands for automated analysis methods are rapidly increasing.

The motion tracking of LV is carried out by observing the speckle pattern associated with deforming tissue. Speckle pattern is inherent appearance in ultrasound imaging and its local brightness reflects the local echogeneity of the underlying scatterers. Since it is a difficult task to automatically track the motion of the endocardial border in ultrasound images because of the poor image quality, low contrast and edge dropout by weak signals, some user intervention is required for stable and successful tracking of the endocardial border.

The most commonly used method for motion tracking is optical flow methods, which use the assumption that the intensity of a moving object is constant over time. If I(r, t) represents the intensity of echocardiography at the location r = (x, y) and the time t, a voxel at (r, t) will move by Δr between two image frames that are taken at times t and t + Δt:

images

From Taylor's expansion, the time-varying images I(r, t) approximately satisfy

images

Hence, the displacement vector (or velocity vector) u(r, t) to be estimated is governed by

6.76 6.69

Based on (6.76), numerous approaches for estimating the velocity vector u(r, t) have been proposed and have been applied to LV border tracking in echocardiography (Duan et al. 2009; Linguraru et al. 2008; Suhling et al. 2005; Veronesi et al. 2006).

Horn and Schunk (1981) proposed an optical flow technique that incorporates the smoothness of the motion vector in the entire image as a global constraint. In their model, the velocity u(r, t) at each time t is determined by minimizing the energy functional:

6.77 6.70

where Ω is the image domain and λ is a regularization parameter that controls the balance between the optical flow term and the smoothness on u. The velocity u(r, t) at each time t can be computed by solving the corresponding Euler–Lagrange equation, which is a reaction–diffusion equation. Barron et al. (1994) observed that this global method with the global smoothness constraint is significantly more sensitive to noise than the local method used by Lucas and Kanade (1981).

Lucas and Kanade (1981) used the assumption of locally constant motion to compute the velocity u(r0, t) at a target location r0 = (x0, y0) and time t by forcing a constant velocity in a local neighborhood of a point r0 = (x0, y0), denoted by images. They estimated the velocity u(r0, t) by minimizing the weighted least-squares criterion in the neighborhood images:

6.78 6.71

where w is a weight function that enables more relevance to be given to central terms rather than the ones in the periphery. Since this method determines u(r0, t) at each location r0 by combining information from all pixels in the neighborhood of r0, it is reasonably robust against image noise.

Let us explain the numerical algorithm for the Lucas–Kanade method. We denote the endocardial border traced at an initially selected frame (for example, end-systole or end-diastole frame) by a parametric contour images that can be identified as its n control points images. Here, we have 0 = s1 < s2 < ··· < sn = 1. Let images be the contour deformed from images at time t. The motion of the contour images will be determined by an appropriately chosen velocity Ut indicating a time change of control points (r1(t), …, rn(t)):

images

Here, we identify the contour images with control points (r1(t), …, rn(t)).

In the Lucas–Kanade method, U(t) for each time t is a minimizer of the energy functional:

6.79 6.72

To derive the Euler–Lagrange equation from (6.79), we take the partial derivative of images with respect to each uj:

6.80 6.73

For a numerical algorithm, we replace the integral over images by summation over pixels around rj(t). Assuming that the neighborhood images consists of m pixels rj1, …, rjm, equation (6.80) becomes

6.81 6.74

where Aj = [∇I(rj1, t), …, ∇I(rjm, t)]T, images and bj = [It(rj1, t), …, It(rjm, t)]T. As we mentioned before, there often exist some incorrectly tracked points due to weak signals on the cardiac wall, since echocardiography data are acquired through transmitting and receiving ultrasound signals between the ribs, causing considerable shadowing of the cardiac wall (Leung et al. 2011). Owing to these incorrectly tracked points, the Lucas–Kanade method may produce a distorted LV shape.


Example 6.4.4
Suhling et al. (2005) improved the Lucas–Kanade method (6.78) by introducing a linear model for the velocity along the time direction, and the displacement u(r0, t) is obtained by evaluating u such that images and the 2 × 2 matrix A minimizes the following energy functional:

6.82 6.82

Since this method uses multiple frames between t − Δt and t + Δt, it is more robust than the Lucas–Kanade method (6.78) using a single frame at t.

 


Example 6.4.5
Compared with the approaches based on the Lucas–Kanade method, Duan et al. (2009) used the region-based tracking method (also known as the block matching or pattern matching method) with the cross-correlation coefficients as a similarity measure. Given two consecutive images I at times t and t + Δt, the velocity vector u = (u, v) for each pixel r = (x, y) ∈ Ω is estimated by maximizing the cross-correlation coefficients:

6.83 6.83

The block matching method uses similarity measures that are less sensitive to noise, fast motion and potential occlusions and discontinuities (Duan et al. 2009).

 

 

1 For hydrogen 1H, γ = 2π × 42.576 × 106 rad s−1 T−1.

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Further Reading

Aubert G and Vese L 1997 A variational method in image recovery. SIAM J. Numer. Anal. 34(5), 1948–1979.

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