Index. p, lip, 78 8 function, 107 v, 7-8 w, 7-8 i,7-8 sine, 43 Bo,94-96
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1 p, lip, 78 8 function, 107 v, 7-8 w, 7-8 i,7-8 sine, 43 Bo,94-96 B 1,94-96 M,94-96 B oro!' BIro!' I/r, 79 2D linear system, 56 2D FFT, 119 2D Fourier transform, 1, 12, 18,91 2D sinc, 107, 112 2D slice imaging, 108 3D FFT, 108 3D spectroscopic image, 121 3D volume imaging, 113 admissibility condition, 141 air pressure, 50 aliasing, 36, 106, 107, 175 angular frequency, 9 angular momentum, 91 anti-aliasing filter, 38 approximation coefficients, 163 attenuation coefficient, 55 back-projection, 82, band-limited, 23, 136, 178 band-pass, 41 bandwidth, 107, 141 basis Functions, 2, 132 beat, 134 bell-shaped curves, 2 bit-reversed, 187 black box, 25 blob, 2,3 blurring, 20-21, 63, building block, 133 butterfly, 189 cancellation, 7 Cartesian grid, 100, 114 caveats to using the DFT/FFT, 189 central section/slice Theorem, 83, 85, 100 chemical shift, 119 coherent state, 132, 134, 135 Coiflet, 145 comb function, 175 commute, 84 complex exponential, 9, 132, 137 numbers, 7 plane, 184
2 196 sinusoid, 176 wavelets, 141 composite waves, 3 computed Tomography, 54, 78, 85, 100 continuous Fourier transform pair, 180 continuous waves, 4 contraction, 141 convolution, 20,38-41,47,59, 178 cosine bell, 43 cosines, 3 critical frequency, 36 cross-sectional images, 74 curve-fitting, 33 cut-off frequency, 83 cyclic, 190 buffer, 179 shift, 192 data readout, 104 Daubechies, 145, 147, 155 decomposition, 57 deconvolution, delta function, 175 detecting element, 61 detector, 59 detector aperture function, 67 diffraction pattern, 16 digital filter, 41, 70, 163 dilation, 141 discrete correlation, 159 discrete Fourier transform, 33, 175, 181 discrete wavelet transform, 165 e iwt, 10 echo-planar imaging, 115 EEG,175 EKG,l72 electromagnetic energy, 7 electromyogram, 45 EM I-Scanner, 79 enhancement, 72 equilibrium, 96 Faraday's law, 97 fast Fourier transform, 33, 100, 175, 182 fast Wavelet transform, 165 field-of-view, 107, 109, 117 filtered back projection, 84-85, 100 filtering, 107, 193 flow, 50 Fourier basis functions, 131 domain, 18 domain filter, 107 sampling, 106 series, 157 transform, 1, 11, 18, 29, 83, 102 vs. Wavelet, 171 frame, 133 free parameters, 30 free precession, 97 frequency (k-space) sampling, 106 analysis, 141 demodulation, 101 domain, 49 -encoding gradient, 105 response, 33 fulcrum plane, functional MR imaging, 117 Gaussian, 138, 67, 138 Gaussian window, 135 gradient, 102 gradient performance, 117 gradient strength, 115 gradient waveforms, 103 gyromagnetic ratio, 94 Haar, 145, 147 Haar multiresolution analysis, 152 Haar scaling function, 151 Haar wavelet, 155 Hanning window, 43-45, 193 harmonics, 30 high-pass, 41
3 higher-order polynomials, 29 homogeneous magnetic field, 98 Hounsfield, Godfrey, 79 hydrogen nucleus, 93 i, 7 image compression, 172 noise, 53 processing, 70, 166 imaginary, 7-8, 186 imaging, 101 impulse, 38 response function, 20 infinite extent, 178 inhomogeneities, 98 inner product, 132 insufficient sampling, 106 integra1,4 interpolating, 117 isotropic high resolution, 109 k -space, 1, 102 interpretation, 103 trajectory, 108 Larmor equation, 94 frequency, 96 1ayergram, 78, 82 leakage, 43 least squares approximation, 49 lens, 16-19,58 light rays, 7 linear dynamic system, 26 longitudinal tomography, 77 lung tissues, 50 magnetic field, field gradients, 98 flux lines, 93 moment, 93 shielding, 119 susceptibility, magnetic resonance imaging, 1, 98 raw data, 100 signal equation, 100 spectroscopy, 119 spectroscopic imaging, 119 magnetization vector, 95 magnification, 63, 68 mammogram, matrix equation, 184 maximum intensity projection, 125 mechanical impedance, 50 modulation transfer function, 53, 65 modulo-n, 179 multi-slice imaging, 113 multidimensional wavelet transforms, 166 multiply-and-add operation, 189 multiresolution multiresolution analysis, 130, 141, 155, 165 multiresolution signal decomposition, 156 music, 131, 134, 135 neutron, 91 noise, 86 nonlinear quantity, 57 notch filters, 41 NMR,97 Spectroscopy, Nyquist criterion, 104, 175 folding frequency, 193 sampling rate, 36, 178 theorem, 137 object plane, 62 optical Density, optical transmittance, 57 optimal filter, 49 optimum magnification, 68 orthogonal, 4, 151, 154 overshoot, 67
4 198 Parseval theorem, 133 partial reconstructions, 171 pass-band, 38 periodic, 130 Pes signal, 45 phase, 3 -contrast velocity imaging, 124 cycle, 102 encoding direction, 106 encoding gradient, 103, 121 shift, 4, 14 variation, 104 piecewise constant, 147 pinhole, 62 point spread function, 20, 38, 58 power spectrum, 43 precession, 97 projection, 59 projection slice theorem, 100 proton, 93 subspectroscopy, 119 quantitative velocity map, 125 quantum selection rules, 93 radial Sampling, 114 radio signals, 7 raw k-space data, 105 readout gradient, 106 real, 7-8 real numbers, 185 reconstruction, 85, 117, 164 reinforcement, 7 resolution, 104, 11 0, 112, 117 resonant gradient systems, 116 respiratory system, 50 restoration, 72 RF pulse, 98 rho-filter, 82 rho-filtered layergram, 78 rotating frame, 96 rotationally symmetric, 65 sample spacing, 180 sampling, 177 function, 144 rate, 105 theorem, 35 scale, 142 scaling equation, 151 function, 143, 165 relation, 148 Shannon's sampling theorem, 36 signal-flow-graph, 184 signal-to-noise,49, 109, 119 simultaneous equations, 30 simultaneous frequency and time localization, 134 sines, 3 sinusoid, 11, 28, sliding window, 135 source aperture function, 66 spatial convolution, 193 encoding, 123 frequency, 83, 106 spatially varying, 58 spectroscopic imaging, 118 speed advantages of FFT, 187 spin, 91 -lattice relaxation, 97 -spin relaxation, 97 -warp imaging, 103, 124 spiral k-space trajectory, 117, 126 square integrable, 147 square window, 44 subband decomposition, 164 superposition, 25, 38 surface rendering, 112 symmetric, 15, 18 system identification, 49 Tl,97 T2, 97, 99 temporal frequency spectrum, 97 Tesla,98 time -domain, 33 -domain sampling, 106
5 localization, 135 window, 134 -frequency analysis, 130 -frequency plane, 131 -frequency resolution, 137, 171 tomography, 75 transition matrix, 183 transmittance, transverse magnetization, lo3 transverse section tomography, 78 uncertainty principle, 134, 137 under-sampling, 107 uniform sampling, 110 upsampling, 164 wavelet, 165 analysis, sequence, 155 series, 157 transform, 130, 132, 137, 140 weighting factor, 11 function, 41 Wiener filter, 49, 70, 83 window, 41 windowed Fourier transform, 141 windowed Fourier transform, 130, 132, 135, 141, 171 wrap-around, 107, 193 x-ray imaging system, 61 zero-padding, 195 zero-frequency, 32
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