Class Note for BME 416 with Professor Trouard at UA
Class Note for BME 416 with Professor Trouard at UA
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Date Created: 02/06/15
BME 4165167 Principles of Biomedical Engineering Biomedical Imaging The Essential Physics of Medical Imaging Bushberg Seiben Leidlioldt and Boone Williams and Wilkins Introduction to Biomedical Engineeng Enderle Blanchard and Bronzino Academic Press Linear Systems Fourier Transform and Optics J Gaskill Wiley Overview of biomedical imagigg systems WAVELENGTH 9 5 u 2 105 m5 103 I 039 quot7395 anumewm I I I I a I I I 1 I I 1 I I I J I I I I 1 I I I I I I a I2 50 95 w 2 1015 1c mEQUENCY henz Ullra A viola Televismn Radar MFll lt Gammarays in m xaavs Hamim Heal dawns If kur Vsible b ENERGY lll1llllllll lllllllI Elecm nvons inquot 10 5 we 7393 W i 10 105 1C9 l l r I Had Orange Yallaw Green Bins nala Figure 11 The electromagnetic spectrum LMEDML Mrmm swgmv M E Z9r MMmgw leMm HAM TRANSMISSION A 10 P l 10quot I V I 10 2 I 39 I 39 I 39 I I 3 gtI re DIAGNOSTIC XRAY 0 39 SPECTRUM I 10 4 I I I I 10 5 I l I 39 l 106 10 7 10quot8 3910399 10 10 I J 39 l 2 39 39 I 39 4 WAVELENGTH 100m 10 m 10 rn 10cm 10 cm 10 A 0113 001 A 00001 A 0001 A gt 2 x 1022 37 X 10 FREQUENCY AND PHOTON ENERGY FIG 11 Transmissionof EM waves through 25 cm of soft tissue Image preliminaries Display Digital Images are arrays of numbers related to some physical measurement These arrays can be in a variety offorms eg integers oating point numbers etc and can be displayed by a variety ofinstruments eg CRT at panel LCD printer etc LUT Contrast Brightness Quality Signal to noise ratio SNR 810 600 810 601 810602 810 605 810 610 Sl0620 Contrast to noise ratio CNR 810600 810601 810 605 810610 810 615 810620 Spatial resolution Mathematical preliminaries Special Functions Step function 0 i lt E b b stepx 0 J 0 5x 1 gt E b Rectangle rect Function 0 x x0 gt i b 2 rect xx 05 xx 1 b b 2 1 x x0 lt l b 2 Ramp Function Li S xo b b ramp x x0 b x Xe i gt xo b b b Triangle tn function 0 x x0 2 1 b tri x x0 b 1 x xo x xo lt1 Sinc function 2 39 Sznc function Gaussian funCtl39Oi l Delta function 1 x 5 x lim Gaus H b b txzx JIbI6ltxxogt 5x x01x x0 5x x00x x0 Comb function Mathematical preliminaries Convolution fit 2 o 3 a Aim a Ex 0 a A h1 o f g I Ix l o I I A h2 lt1 539 quotI T l H 2 01 th oz P l j h4 01 gl 39 x a I l A h5 a 1 r 7 l gtIx 5 7 quot fx hos J fx hx x39d s gx l aha Area g0 all 0 A fczh1 a 3 Area g1 0 a oaha 01 Area g2 0 a A mm 0 Area g3 0 a f a l 0 Area g4 o a A fah5 0 Area g5 0 01 Figure 62 Graphical method for convolving functions of Fig 61 quotAreasquot from Fig 6 2 g4 r85 80 43 5 f Ds X Figure 63 Resulting convolution of functions shown in Fig 61 Smoothing In general gx is Wider than fx or hx In general Width gx width fx Width hX Example 1 Noise suppression in a noisy spectrum 2 M 2 fx 40 so so 100 6 20 40 so so I 0 o 20 4 A 2 o 2 fgtlthqgtlt 6 20 40 60 80 1 0 4 2 0 m 2 fXh2X 6 20 4O 60 80 1 0 4 2 o M 2 fXh3X 0 20 4O 60 80 100 Convolution with delta functions 5x x0l x x0 5x x00 xixo fx5x j fx396x x39dx39 fx Example Impulse response function Point spread function psf 5fx gee when fx 5x 5fx 500 hx where hx is the impulse response of LSI system 0 The output of a LSI system is given by the convolution of the input with the impulse response of the system fr b hm gm hr f I l l 05 05 1 r 7 0 2 4 t 2 0 2 4 6 t Figure 69 An example involving a linear shift invariant system 39 ht 2 hr 1 hr 3 a b Figure 610 Output of system shown in Fig 69 a Responses to individual input delta functions b Overall response to combined input delta functions Two dimensional convolution fltxyhltxy j j fxy39hxixlyy dx dy gov y fKy fGay hXy1 fKy hXy2 fXy fXynoise fXy hXy1 fXy hXy2 Complex numbers caib where i J c Ae Ac0s isin where A a2 172 tan 1 2 a Jean Baptiste Joseph Fourier 1768 1830 Fourier transforms Time and frequency fw J39 fneimdt Space and spatial frequency Fkx J39fxeazmgxdx Inverse Fourier transform fx 5Fkxf1 IFkXe Z XkX dkx Example 1 Fourier transform of a cosine fx A c0s27zx Example 2 Fourier transform of a rectangular pulse f x x fx A rect EJ Example 3 Fourier transform of a narrower rectangular pulse f x x fx A rectB 2 As fx get narrow F x gets broad As fx gets broad F x gets narrow Example 4 Fourier transform of a shifted rectangular pulse x x0 fx A rect B j A shift in one domain is equivalent to a linear phase in the other Fourier transform of a convolution gx f 90 hx Fact Wm Ham Wm Gltkx j fx39HkxequotWk dx39 G023gx5fxhx 00 fx39hx x39dx39 Hag 1 fxle 12 x39kxdx39 7w FUR HUCX T jifxlhxx39dx39 eixmkxdx T fx39T hx x39 e39 mkx dxdx39 The Fourier transform of a convolution is the product of the individual transforms Transform of a product 5fx X h96 FkxHkx The Fourier transform of a product is the convolution of the individual transforms hx k x CONVOLUTI ON hX fx 2A2 xo xo 2x0 0100 fx v wx v HMO F tux ZAXO PAXO MULTIPLICATION L wquot T X X0 x 0 Figure 24 Convolution theorem 7fx hx F mxHmx gx fxhx GkxFkxgtltHkx Example fx A cos27rk0x 1 x h 39 2 x bsmc b b xv A no sin xx gxfxhxjcos2nk0x39 x dxv bj 2 Two dimensional Fourier transforms Faulty gt Smx ygt fXyequot quot dx m y 5quot mm gt Hwy 6 de iffxy is separable ie fxy xfzy then Fkkay5fxy Sfxf2y I39ikx139ky Example 1 2D FT of fx y A mct rectxy rectx 4y 4 re ctx 8y 8 sinc gvky 16 sinc4lg4ky 64 sinc81 8ky Example 2 2D FT ofan mags fxy Fagyky ifvyyk quk wdxdy fog y fx y rect fx y sinc16kx 1613 k k Fkxky Fkxkysinc4kx4ky Fkxkyxrect
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