Lecture 3: Imaging Theory (1/6) – Linear systems and convolutionLinearity ConditionsLinearity Example:Slide 4Example in medical imaging: Analog to digital converterLinearity allows decomposition of functionsThe delta function as a limit of another functionSifting property of the delta functionImaging Analyzed with System OperatorsSlide 10System response to a two-dimensional delta functionExample in medical imaging:Time invarianceSlide 14Space or shift invarianceOne-dimensional convolution example:One-dimensional convolution example, continued:One-dimensional convolution example, continued(2):One-dimensional convolution example, continued(3):PowerPoint PresentationTwo-dimensional convolution2D Convolution of a square with a rectangle.2D Convolution of letter E - 3D plots2D Convolution of letter E - Grayscale images2D Convolution of letter E - circle vs. square.2D Convolution of letter E with a large square – 2D plots2D Convolution of letter E with a circle – 2D plotsLinear Systems – why study them?•Develop general analysis tools for all modalities•Applicable beyond medical imaging•Tools provide valuable insights for understanding and design •Basis for further improvement of systems•Build upon your knowledge of one-dimensional theoryCommunications: time ↔ frequency (1 dimension)Imaging: space ↔ spatial frequency We will work in two dimensions. Human body is three-dimensional.Extension of 2D theory to three dimensions is straightforward.Lecture 3: Imaging Theory (1/6) – Linear systems and convolutionLinearity ConditionsLet f1(x,y) and f2(x,y) describe two objects we want to image.f1(x,y) can be any object and represent any characteristic of the object.(e.g. color, intensity, temperature, texture, X-ray absorption, etc.)Assume each is imaged by some imaging device (system).Let f1(x,y) → g1(x,y) f2(x,y) → g2(x,y) Let’s scale each object and combine them to form a new object. a f1(x,y) + b f2(x,y)What is the output? If the system is linear, output is a g1(x,y) + b g2(x,y)Linearity Example:Is this a linear system?xx Linearity Example:Is this a linear system? 9 → 3 16 → 4 9 + 16 = 25 → 5 3 + 4 ≠ 5 Not linear.xx Example in medical imaging: Analog to digital converterDoubling the X-ray photons → doubles those transmittedDoubling the nuclear medicine → doubles the reception source energyMR:Maximum output voltage is 4 V.Now double the water:The A/D converter system is non-linear after 4 mV.(overranging)Linearity allows decomposition of functions Linearity allows us to decompose our input into smaller, elementary objects. Output is the sum of the system’s response to these basic objects.Elementary Function:The two-dimensional delta function δ(x,y)δ(x,y) has infinitesimal width and infinite amplitude.Key: Volume under function is 1.δ(x,y)1δ (x,y)dxdyThe delta function as a limit of another function Powerful to express δ(x,y) as the limit of a functionA note on notation: Π(x) = rect(x) Gaussian: lim a2 exp[- a2(x2+y2)] = δ(x,y) 2D Rect Function: lim a2 Π (ax) Π(ay) = δ(x,y) where Π(x) = 1 for |x| < ½ For this reason, δ(bx) = (1/|b|) δ(x) aaSifting property of the delta function The delta function at x1 = ε, y1= η has sifted out f (x1,y1) at that point. One can view f (x1,y1) as a collection of delta functions, each weighted by f (ε,η). 1111)δ,(),(d)d,y(xfyxfImaging Analyzed with System OperatorsLet system operator (imaging modality) be , so that 1111)δ,(),(d)d,y(xfyxf][ )()g(1122,yxf,yxFrom previouspage,Why the new coordinate system (x2,y2)?Imaging Analyzed with System OperatorsLet system operator (imaging modality) be , so that Then, Generally System operating on entire blurred object input object g1 (x1,y1)By linearity, we can consider the output as a sum of the outputs from all the weighted elementary delta functions. Then, 1111)δ,(),(d)d,y(xfyxf][][ 1111δ),()(d)d,y(xf,yxf 1122δ ),()g(dd),y(xf,yx ][][ )()g(1122,yxf,yxFrom previouspage,System response to a two-dimensional delta function Output at (x2 , y2) depends on input location ().Substituting this into yields the Superposition Integral ][),y(x,yxh1122δ),;( 1122δ ),()g(dd),y(xf,yx ][ 2222),;( ),()g(dd,yxhf,yxExample in medical imaging:Consider a nuclear study of a liver with a tumor point source at x1= , y1= Radiation is detected at the detector plane.To obtain a general result, we need to know all combinationsh(x2, y2; ,)By “general result”, we mean that we could calculate the image I(x2, y2) for any source input S(x1, y1)Time invarianceA system is time invariant if its output depends only on relative time of the input, not absolute time. To test if this quality exists for a system, delay the input by t0. If the output shifts by the same amount, the system is time invariant i.e. f(t) → g(t)f(t - to) → g(t - to) input delay output delayIs f(t) → f(at) → g(t) (an audio compressor) time invariant?Time invarianceA system is time invariant if its output depends only on relative time of the input, not absolute time. To test if this quality exists for a system, delay the input by t0. If the output shifts by the same amount, the system is time invariant i.e. f(t) → g(t)f(t - to) → g(t - to) input delay output delayIs f(t) → f(at) → g(t) (an audio compressor) time invariant?f(t - to) → f(at) → f(a(t – to)) - output of audio compressor f(at – to) - shifted version of output (this would be a time invariant system.)So f(t) → f(at) = g(t) is not time invariant.Space or shift invarianceA system is space (or shift) invariant if its output depends only on relative position of the input, not absolute position.If you shift input → The response shifts, but in the plane, the shape of the response stays the same.If the system is shift invariant,h(x2, y2; ,) = h(x2 y2) and the superposition integral becomes the 2D convolution function: Notation: g = f**h
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