Biosignal and Biomedical Image Processing phần 11 pps

388 Chapter 13 % % Reconstruct image using Ram-Lak filter I_RamLak = iradon(p,delta_theta,‘Ram-Lak’); % Display images Radon and Inverse Radon Transform: Fan Beam Geometry The MATLAB routines for performing the Radon and inverse Radon transform using fan beam geometry are termed fanbeam and ifanbeam , respectively, and have the form: fan = fanbeam(I,D) where I is the input image and D is a scalar that specifies the distance between the beam vertex and the center of rotation of the beams. The output, fan ,isa matrix containing the fan bean projection profiles, where each column contains the sensor samples at one rotation angle. It is assumed that the sensors have a one-deg. spacing and the rotation angles are spaced equally over 0 to 359 deg. A number of optional input variables specify different geometries, sensor spac- ing, and rotation increments. The inverse Radon transform for fan beam projections is specified as: I = ifanbeam(fan,D) F IGURE 13.9 Original MR image and reconstructed images using the inverse Radon transform with the Ram-Lak derivative and the cosine filter. The cosine filter’s lowpass cutoff has been modified by setting its maximum relative fre- quency to 0.4. The Ram-Lak reconstruction is not as sharp as the original image and sharpness is reduced further by the cosine filter with its lowered bandwidth. (Original image from the MATLAB Image Processing Toolbox. Copyright 1993– 2003, The Math Works, Inc. Reprinted with permission.) TLFeBOOK Image Reconstruction 389 where fan is the matrix of projections and D is the distance between beam vertex and the center of rotation. The output, I , is the reconstructed image. Again there are a number of optional input arguments specifying the same type of information as in fanbeam. This routine first converts the fan beam geometry into a parallel geometry, then applies filtered back-projection as in iradon . During the filtered back-projection stage, it is possible to specify filter options as in iradon . To specify, the string ‘Filter’ should precede the filter name ( ‘Hamming’ , ‘Hann’ , ‘cosine’ , etc.). Example 13.3 Fan beam geometry. Apply the fan beam and parallel beam Radon transform to the simple square shown in Figure 13.4. Reconstruct the image using the inverse Radon transform for both geometries. % Example 13.3 and Figure 13.10 % Example of reconstruction using the Fan Beam Geometry % Reconstructs a pattern of 4 square of different intensities % using parallel beam and fan beam approaches. % clear all; close all; D = 150; % Distance between fan beam vertex % and center of rotation theta = (1:180); % Angle between parallel % projections is 1 deg. % I = zeros(128,128); % Generate image I(22:54,22:52) = .25; % Four squares of different shades I(76;106,22:52) = .5; % against a black background I(22:52,76:106) = .75; I(76:106,76:106) = 1; % % Construct projections: Fan and parallel beam [F,Floc,Fangles] = fanbeam (I,D,‘FanSensorSpacing’,.5); [R,xp] = radon(I,theta); % % Reconstruct images. Use Shepp-Logan filter I_rfb = ifanbeam(F,D,‘FanSensorSpacing’,.5,‘Filter’, ‘Shepp-Logan’); I_filter_back = iradon(R,theta,‘Shepp-Logan’); % % Display images subplot(1,2,1); imshow(I_rfb); title(‘Fan Beam’) subplot(1,2,2); imshow(I_filter_back); title(‘Parallel Beam’) TLFeBOOK 390 Chapter 13 The images generated by this example are shown in Figure 13.10. There are small artifacts due to the distance between the beam source and the center of rotation. The affect of this distance is explored in one of the problems. MAGNETIC RESONANCE IMAGING Basic Principles MRI images can be acquired in a number of ways using different image acquisi- tion protocols. One of the more common protocols, the spin echo pulse sequence, will be described with the understanding that a fair number of alternatives are commonly used. In this sequence, the image is constructed on a slice-by-slice basis, although the data are obtained on a line-by-line basis. For each slice, the raw MRI data encode the image as a variation in signal frequency in one dimen- sion, and in signal phase in the other. To reconstruct the image only requires the application of a two-dimensional inverse Fourier transform to this fre- quency/phase encoded data. If desired, spatial filtering can be implemented in the frequency domain before applying the inverse Fourier transform. The physics underlying MRI is involved and requires quantum mechanics for a complete description. However, most descriptions are approximations that use classical mechanics. The description provided here will be even more abbre- viated than most. (For a detailed classical description of the MRI physics see Wright’s chapter in Enderle et al., 2000.). Nuclear magnetism occurs in nuclei with an odd number of nucleons (protons and/or neutrons). In the presence of a magnetic field such nuclei possess a magnetic dipole due to a quantum mechani- F IGURE 13.10 Reconstruction of an image of four squares at different intensities using parallel beam and fan beam geometry. Some artifact is seen in the fan beam geometry due to the distance between the beam source and object (see Problem 3). TLFeBOOK Image Reconstruction 391 cal property known as spin.* In MRI lingo, the nucleus and/or the associated magnetic dipole is termed a spin. For clinical imaging, the hydrogen proton is used because it occurs in large numbers in biological tissue. Although there are a large number of hydrogen protons, or spins, in biological tissue (1 mm 3 of water contains 6.7 × 10 19 protons), the net magnetic moment that can be pro- duced, even if they were all aligned, is small due to the near balance between spin-up ( 1 ⁄ 2 ) and spin-down (− 1 ⁄ 2 ) states. When they are placed in a magnetic field, the magnetic dipoles are not static, but rotate around the axis of the applied magnetic field like spinning tops, Figure 13.11A (hence, the spins themselves spin). A group of these spins produces a net moment in the direction of the magnetic field, z, but since they are not in phase, any horizontal moment in the x and y direction tends to cancel (Figure 13.11B). While the various spins do not have the same relative phase, they do all rotate at the same frequency, a frequency given by the Larmor equation: ω o =γH (11) F IGURE 13.11 (A) A single proton has a magnetic moment which rotates in the presence of an applied magnet field, B z . This dipole moment could be up or down with a slight favoritism towards up, as shown. (B) A group of upward dipoles create a net moment in the same direction as the magnetic field, but any horizon- tal moments (x or y) tend to cancel. Note that all of these dipole vectors should be rotating, but for obvious reasons they are shown as stationary with the as- sumption that they rotate, or more rigorously, that the coordinate system is ro- tating. *Nuclear spin is not really a spin, but another one of those mysterious quantum mechanical proper- ties. Nuclear spin can take on values of ±1/2, with +1/2 slightly favored in a magnetic field. TLFeBOOK 392 Chapter 13 where ω o is the frequency in radians, H is the magnitude of the magnitude field, and γ is a constant termed the gyromagnetic constant . Although γ is primarily a function of the type of nucleus it also depends slightly on the local chemical environment. As shown below, this equation contains the key to spatial localiza- tion in MRI: variations in local magnetic field will encode as variations in rota- tional frequency of the protons. If these rotating spins are exposed to electromagnetic energy at the rota- tional or Larmor frequency specified in Eq. (11), they will absorb this energy and rotate further and further from their equilibrium position near the z axis: they are tipped away from the z axis (Figure 13.12A). They will also be syn- chronized by this energy, so that they now have a net horizontal moment. For protons, the Larmor frequency is in the radio frequency (rf) range, so an rf pulse of the appropriate frequency in the xy-plane will tip the spins away from the z-axis an amount that depends on the length of the pulse: θ=γHT p (12) where θ is the tip angle and T p pulse time. Usually T p is adjusted to tip the angle either 90 or 180 deg. As described subsequently, a 90 deg. tip is used to generate the strongest possible signal and an 180 deg tip, which changes the sign of the F IGURE 13.12 (A) After an rf pulse that tips the spins 90 deg., the net magnetic moment looks like a vector, M xy , rotating in the xy-plane. The net vector in the z direction is zero. (B) After the rf energy is removed, all of the spins begin to relax back to their equilibrium position, increasing the z component, M z , and decreas- ing the xy component, M xy . The xy component also decreases as the spins de- synchronize. TLFeBOOK Image Reconstruction 393 moment, is used to generate an echo signal. Note that a given 90 or 180 deg. T p will only flip those spins that are exposed to the appropriate local magnetic field, H. When all of the spins in a region are tipped 90 deg. and synchronized, there will be a net magnetic moment rotating in the xy-plane, but the component of the moment in the z direction will be zero (Figure 13.12A). When the rf pulse ends, the rotating magnetic field will generate its own rf signal, also at the Larmor frequency. This signal is known as the free induction decay (FID) signal. It is this signal that induces a small voltage in the receiver coil, and it is this signal that is used to construct the MR image. Immediately after the pulse ends, the signal generated is given by: S(t) =ρsin (θ)cos(ω o t) (13) where ω o is the Larmor frequency, θ is the tip angle, and ρ is the density of spins. Note that a tip angle of 90 deg. produces the strongest signal. Over time the spins will tend to relax towards the equilibrium position (Figure 13.12B). This relaxation is known as the longitudinal or spin-lattice relaxation time and is approximately exponential with a time constant denoted as “T 1 .” As seen in Figure 13.12B, it has the effect of increasing the horizontal moment, M z , and decreasing the xy moment, M xy . The xy moment is decreased even further, and much faster, by a loss of synchronization of the collective spins, since they are all exposed to a slightly different magnetic environment from neighboring atoms (Figure 13.12B). This so-called transverse or spin-spin relaxation time is also exponential and decays with a time constant termed “T 2 .” The spin-spin relaxation time is always less than the spin lattice relaxation time, so that by the time the net moment returns to equilibrium position along the z axis the individual spins are completely de-phased. Local inhomogeneities in the applied magnetic field cause an even faster de-phasing of the spins. When the de-phasing time constant is modified to include this effect, it is termed T* 2 (pronounced tee two star). This time constant also includes the T 2 influences. When these relaxation processes are included, the equation for the FID signals becomes: S(t) =ρcos(ω o t) e −t/T * 2 e −t/T 1 (14) While frequency dependence (i.e., the Larmor equation) is used to achieve localization, the various relation times as well as proton density are used to achieve image contrast. Proton density, ρ, for any given collection of spins is a relatively straightforward measurement: it is proportional to FID signal ampli- tude as shown in Eq. (14). Measuring the local T 1 and T 2 (or T* 2 ) relaxation times is more complicated and is done through clever manipulations of the rf pulse and local magnetic field gradients, as briefly described in the next section. TLFeBOOK 394 Chapter 13 Data Acquisition: Pulse Sequences A combination of rf pulses, magnetic gradient pulses, delays, and data acquisi- tion periods is termed a pulse sequence. One of the clever manipulations used in many pulse sequences is the spin echo technique, a trick for eliminating the de-phasing caused by local magnetic field inhomogeneities and related artifacts (the T* 2 decay). One possibility might be to sample immediately after the rf pulse ends, but this is not practical. The alternative is to sample a realigned echo. After the spins have begun to spread out, if their direction is suddenly reversed they will come together again after a known delay. The classic example is that of a group of runners who are told to reverse direction at the same time, say one minute after the start. In principal, they all should get back to the start line at the same time (one minute after reversing) since the fastest runners will have the farthest to go at the time of reversal. In MRI, the reversal is accom- plished by a phase-reversing 180 rf pulse. The realignment will occur with the same time constant, T* 2 , as the misalignment. This echo approach will only cancel the de-phasing due to magnetic inhomogeneities, not the variations due to the sample itself: i.e., those that produce the T 2 relaxation. That is actually desirable because the sample variations that cause T 2 relaxation are often of interest. As mentioned above, the Larmor equation (Eq. (11)) is the key to localiza- tion. If each position in the sample is subjected to a different magnetic field strength, then the locations are tagged by their resonant frequencies. Two ap- proaches could be used to identify the signal from a particular region. Use an rf pulse with only one frequency component, and if each location has a unique magnetic field strength then only the spins in one region will be excited, those whose magnetic field correlates with the rf frequency (by the Larmor equation). Alternatively excite a broader region, then vary the magnetic field strength so that different regions are given different resonant frequencies. In clinical MRI, both approaches are used. Magnetic field strength is varied by the application of gradient fields ap- plied by electromagnets, so-called gradient coils, in the three dimensions. The gradient fields provide a linear change in magnetic field strength over a limited area within the MR imager. The gradient field in the z direction, G z , can be used to isolate a specific xy slice in the object, a process known as slice selection.* In the absence of any other gradients, the application of a linear gradient in the z direction will mean that only the spins in one xy-plane will have a resonant frequency that matches a specific rf pulse frequency. Hence, by adjusting the *Selected slices can be in any plane, x, y, z, or any combination, by appropriate activation of the gradients during the rf pulse. For simplicity, this discussion assumes the slice is selected by the z- gradient so spins in an xy-plane are excited. TLFeBOOK Image Reconstruction 395 gradient, different xy-slices will be associated with (by the Larmor equation), and excited by, a specific rf frequency. Since the rf pulse is of finite duration it cannot consist of a single frequency, but rather has a range of frequencies, i.e., a finite bandwidth. The thickness of the slice, that is, the region in the z-direc- tion over which the spins are excited, will depend on the steepness of the gradi- ent field and the bandwidth of the rf pulse: ∆z ϰ γG z z(∆ω) (15) Very thin slices, ∆z, would require a very narrowband pulse, ∆ω, in com- bination with a steep gradient field, G z . If all three gradients, G x , G y , and G z , were activated prior to the rf pulse then only the spins in one unique volume would be excited. However, only one data point would be acquired for each pulse repetition, and to acquire a large volume would be quite time-consuming. Other strategies allow the acquisition of entire lines, planes, or even volumes with one pulse excitation. One popular pulse sequence, the spin-echo pulse sequence, acquires one line of data in the spatial frequency domain. The sequence begins with a shaped rf pulse in con- junction with a G z pulse that provides slice selection (Figure 13.13). The G z includes a reversal at the end to cancel a z-dependent phase shift. Next, a y- gradient pulse of a given amplitude is used to phase encode the data. This is followed by a second rf/G z combination to produce the echo. As the echo re- groups the spins, an x-gradient pulse frequency encodes the signal. The re- formed signal constitutes one line in the ferquency domain (termed k-space in MRI), and is sampled over this period. Since the echo signal duration is several hundred microseconds, high-speed data acquisition is necessary to sample up to 256 points during this signal period. As with slice thickness, the ultimate pixel size will depend on the strength of the magnetic gradients. Pixel size is directly related to the number of pixels in the reconstructed image and the actual size of the imaged area, the so-called field-of-view (FOV). Most modern imagers are capable ofa2cmFOVwith samples up to 256 by 256 pixels, giving a pixel size of 0.078 mm. In practice, image resolution is usually limited by signal-to-noise considerations since, as pixel area decreases, the number of spins available to generate a signal dimin- ishes proportionately. In some circumstances special receiver coils can be used to increase the signal-to-noise ratio and improve image quality and/or resolu- tion. Figure 13.14A shows an image of the Shepp-Logan phantom and the same image acquired with different levels of detector noise.* As with other forms of signal processing, MR image noise can be improved by averaging. Figure *The Shepp-Logan phantom was developed to demonstrate the difficulty of identifying a tumor in a medical image. TLFeBOOK 396 Chapter 13 F IGURE 13.13 The spin-echo pulse sequence. Events are timed with respect to the initial rf pulse. See text for explanation. 13.14D shows the noise reduction resulting from averaging four of the images taken under the same noise conditions as Figure 13.14C. Unfortunately, this strategy increases scan time in direct proportion to the number of images aver- aged. Functional Magnetic Resonance Imaging Image processing for MR images is generally the same as that used on other images. In fact, MR images have been used in a number of examples and prob- lems in previous chapters. One application of MRI does have some unique im- TLFeBOOK Image Reconstruction 397 F IGURE 13.14 (A) MRI reconstruction of a Shepp-Logan phantom. (B) and (C) Reconstruction of the phantom with detector noise added to the frequency do- main signal. (D) Frequency domain average of four images taken with noise simi- lar to C. Improvement in the image is apparent. (Original image from the MATLAB Image Processing Toolbox. Copyright 1993–2003, The Math Works, Inc. Re- printed with permission.) age processing requirements: the area of functional magnetic resonance imaging (fMRI). In this approach, neural areas that are active in specific tasks are identi- fied by increases in local blood flow. MRI can detect cerebral blood changes using an approach known as BOLD: blood oxygenation level dependent. Special pulse sequences have been developed that can acquire images very quickly, and these images are sensitive to the BOLD phenomenon. However, the effect is very small: changes in signal level are only a few percent. During a typical fMRI experiment, the subject is given a task which is either physical (such a finger tapping), purely sensory (such as a flashing visual stimulus), purely mental (such as performing mathematical calculations), or in- volves sensorimotor activity (such as pushing a button whenever a given image appears). In single-task protocols, the task alternates with non-task or baseline activity period. Task periods are usually 20–30 seconds long, but can be shorter and can even be single events under certain protocols. Multiple task protocols are possible and increasingly popular. During each task a number of MR images TLFeBOOK [...]... distributions Bruce, E N Biomedical Signal Processing and Signal Modeling, John Wiley and Sons, 409 TLFeBOOK 410 Bibliography New York, 2001 Rigorous treatment with more of an emphasis on linear systems than signal processing Introduces nonlinear concepts such as chaos Cichicki, A and Amari S Adaptive Bilnd Signal and Image Processing: Learning Algorithms and Applications, John Wiley and Sons, Inc New York,... in noise Excellent coverage of Fourier analysis, and autoregressive methods Good introduction to statistical signal processing concepts Sonka, M., Hlavac V., and Boyle R Image processing, analysis, and machine vision Chapman and Hall Computing, London, 1993 A good description of edge-based and other segmentation methods Strang, G and Nguyen, T Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley,... 314–316 variance, local, 353 window (see Rectangular window filter) Wiener, 213, 223, 225, 226 Wiener-Hopf, 215, 216, 218, 222, 223 Yule-Walker, 118 Filter design, 94, 311 313 three-stage FIR, 111 117 three-stage IIR, 119 –123 two-stage FIR, 109 111 two-stage IIR, 118 119 Filtered back-projection, 380, 380, 383–386 Finite data considerations, 53–57 (see also Edge effects) Finite impulse response (FIR), (see... surrounds and includes the potentially active pixels Normally this area would be selected interactively by an operator Reformat the images so that each frame is a single row vector and constitutes one row of an ensemble composed of the different frames Perform both an ICA and PCA analysis and plot the resulting components % Example 13.5 and Figure 13.18 and 13.19 % Example of the use of PCA and ICA to... mapping), others in c-language such as AFNI (analysis of neural images) Some packages can be obtained at no charge off the Web In addition to identifying the active pixels, these packages perform various preprocessing functions such as aligning the sequential images and reshaping the images to conform to standard models of the brain Following preprocessing, there are a number of different approaches to identifying... by a non-signal processing friend Ingle, V.K and Proakis, J G Digital Signal Processing with MATLAB, Brooks/Cole, Inc Pacific Grove, CA, 2000 Excellent treatment of classical signal processing methods including the Fourier transform and both FIR and IIR digital filters Brief, but informative section on adaptive filtering Jackson, J E A User’s Guide to Principal Components, John Wiley and Sons, New York,... (mri.tif) Construct parallel beam projections of this image using the Radon transform with two different angular spacings between rotations: 5 deg and 10 deg In addition, reduce spacing of the 5 deg data by a factor of two Reconstruct the three images (5 deg unreduced, 5 deg reduced, and 10 deg.) and display along with the original image Multiply the images by a factor of 10 to enhance any variations... 13–14 average, 312 bandpass, 13–14 bandstop, 13 bandwidth, 13–15 banks, 188–194 Butterworth, 17, 107, 119 -121 Canny, 346, 350, 369 Cauer (elliptic), 107 Chebyshev, 17, 107, 119 , 121 TLFeBOOK 416 [Filters] cosine (window), 380, 382, 386 Daubechies, 191, 194, 198, 200 digital, 87–124 8-pole, 26 elliptic, 107, 119 , 121 finite impulse response (FIR), 87, 93–97, 98, 101, 103, 104, 107, 108 117 , 213, 220, 222... test pattern image, testpat1.png with noise added Reconstruct the image using the inverse Radon transform with two filter options: the Ram-Lak filter (the default), and the Hamming filter with a maximum frequency of 0.5 3 Load the image squares.tif Use fanbeam to construct fan beam projections and ifanbeam to produce the reconstructed image Repeat for two different beam distances: 100 and 300 (pixels)... Wigner-Ville Distribution, IEEE Trans Acoust Speech Sig Proc ASSP-35:1 611 1618, 1987 Practical information on calculating the Wigner-Ville distribution Boudreaux-Bartels, G F and Murry, R Time-frequency signal representations for biomedical signals In: The Biomedical Engineering Handbook J Bronzino (ed.) CRC Press, Boca Raton, Florida and IEEE Press, Piscataway, N.J., 1995 This article presents an exhaustive, . is not as sharp as the original image and sharpness is reduced further by the cosine filter with its lowered bandwidth. (Original image from the MATLAB Image Processing Toolbox. Copyright 1993– 2003,. used to increase the signal-to-noise ratio and improve image quality and/ or resolu- tion. Figure 13.14A shows an image of the Shepp-Logan phantom and the same image acquired with different levels of. Magnetic Resonance Imaging Image processing for MR images is generally the same as that used on other images. In fact, MR images have been used in a number of examples and prob- lems in previous