Assignment for next week

Transcription

1 Assignment for next week Due: 6:30 PM Monday April 11 th (2016) -> PDF by only 5 points: Pose two questions based on this lecture 5 points: what (and why) is sinc(0)? 5 points: ketch (by hand okay) rect(x,y) and sinc(x,y) and label axes 5 points each: Determine if these are linear functions g(x) = f (x) +1 g(x) = x f (x) g(x) = ( f (x)) 2 5 points: If the input to a 2D imaging system is an impulse, why is the output called a point spread function (PF)

2 2D ignals and ystems

3 ignals A signal can be either continuous f i, j,k f (x), f (x, y), f (x, y,z), f (x) or discrete etc. where i,j,k index specific coordinates Digital images on computers are necessarily discrete sets of data Each element, or bin, or voxel, represents some value, either measured or calculated

4 Digital Images Real objects are continuous (at least above the quantum level), but we represent them digitally as an approximation of the true continuous process (pixels or voxels) For image representation this is usually fine (we can just use smaller voxels as necessary) For data measurements the element size is critical (e.g. hannon's sampling theorem) For most of our work we will use continuous function theory for convenience, but sometimes the discrete theory will be required

5 Important signals - rect() and sinc() functions 1D rect() and sinc() functions both have unit area 1, for x < 1/ 2 (a) rect(x) = 0, for x > 1/ 2 sin(π x) (b) sinc(x) = π x

6 Important signals - 2D rect() and sinc() functions 2D rect() and sinc() functions are straightforward generalizations (a) rect(x, y) = (b) sinc(x, y) = 1, for x < 1/ 2 and y < 1/ 2 0, otherwise sin(π x)sin(π y) π 2 xy Try to sketch these 3D versions exist and are sometimes used Fundamental connection between rect() and sinc() functions and very useful in signal and image processing

7 Important signals - Impulse function 1D Impulse (delta) function A 'generalized function' operates through integration has zero width and unit area has important 'sifting' property can be understood by considering: Ways to approach the delta function δ(x) = 0, x 0, δ(x) dx = 1 f (x)δ(x) dx = f (0) f (x)δ(x t) dx = f (t) δ(t) = lim arect(at) δ(t) = lim asinc(at) δ(t) = lim ae πa 2 t 2 a a a

8 Exponential and sinusoidal signals Recall Euler's formula,which connects trigonometric and complex exponential functions ( j = sqrt(-1) ) e jθ = cos(θ) + j sin(θ) (not i) The exponential signal is defined as: e j 2π x = cos(2π x) + j sin(2π x), where j 2 = 1 u 0 and v 0 are the fundamental frequencies in x- and y- directions, with units of 1/distance e(x, y) = e j 2π (u 0 x+v 0 y) We can write e(x, y) = e j 2π (u 0x+v 0 y) ( ) = cos 2π u 0 x + v 0 y ( ) + j sin 2π u 0 x + v 0 y real and even imaginary and odd

9 Exponential and sinusoidal signals Recall that sin(2π x) = 1 2 j ( e j 2π x e j 2π x ) cos(2π x) = 1 ( 2 e j 2π x + e j 2π x ) so we have ( ) sin 2π u 0 x + v 0 y = 1 2 j ( e j 2π( u 0x+v 0 y ) e j 2π ( u x+v y ) 0 0 ) ( ) cos 2π u 0 x + v 0 y ( ( )) = 1 2 e j 2π ( u 0x+v 0 y) + e j 2π u 0 x+v 0 y Fundamental frequencies u 0, v 0 affect the oscillations in x and y directions, E.g. small values of u 0 result in slow oscillations in the x- direction These are complex-valued and directional plane waves

10 Exponential and sinusoidal signals ( ) Intensity images for s(x, y) = sin 2π u 0 x + v 0 y y x

11 ystem models ystems analysis is a powerful tool to characterize and control the behavior of biomedical imaging devices We will focus on the special class of continuous, linear, shiftinvariant (LI) systems Many (all) biomedical imaging systems are not really any of the three, but it can be useful tool, as long as we understand the errors in our approximation "all models are wrong, but some are useful" -George E. P. Box Continuous systems convert a continuous input to a continuous output ome notation examples: g(x) = [ f (x)] g(t) = [ f (t)] ( ) f (x) g(x)

12 Linear ystems A system is a linear system if: we have then [ f (x)] = g(x) [ a 1 f 1 (x) + a 2 f 2 (x)] = a 1 g 1 (x) + a 2 g 2 (x) or in general K k=1 K K w k f k (x) = w [ k f k (x)] = w k g k (x) k=1 k=1 Is this a linear system? g(x) = e π f (x)

13 Linear ystems A system is a linear system if: we have then [ f (x)] = g(x) [ a 1 f 1 (x) + a 2 f 2 (x)] = a 1 g 1 (x) + a 2 g 2 (x) or in general K k=1 K K w k f k (x) = w [ k f k (x)] = w k g k (x) k=1 k=1 Is this a linear system? Yes! g(x) = e π f (x) [ a 1 f 1 (x) + a 2 f 2 (x)] = e π ( a 1 f 1 (x) + a 2 f 2 (x)) = a 1 e π f 1 (x) + a 2 e π f 2 (x) = a 1 g 1 (x) + a 2 g 2 (x)

14 2D Linear ystems Now use 2D notation Example: sharpening filter In general f (x, y) g(x, y) K K K w k f k (x, y) k=1 = w [ k f k (x, y) ] = w k g k (x, y) k=1 k=1

15 hift-invariant ystems tart by shifting the input f x0 y 0 (x, y)! f (x x 0, y y 0 ) then if g x0 y 0 (x, y) = f x0 y 0 (x, y) = g(x x, y y ) 0 0 the system is shift-invariant, i.e. response does not depend on location hift-invariance is separate from linearity, a system can be shift-invariant and linear shift-invariant and non-linear shift-variant and linear shift-variant and non-linear

16 hift invariant and shift-variant system response scanner object shift f (x, y) image g(x, y) FOV unshifted response shift invariant shift variant (shape, location)

17 hift invariant and shift-variant system response scanner object shift f (x, y) image g(x, y) FOV unshifted response shift invariant shift variant (value)

18 Impulse Response Linear, shift-invariant (LI) systems are the most useful First we start by looking at the response of a system using a point source at location (ξ,η) as an input point object input f ξη (x, y)! δ(x ξ, y η) y x output g ξη (x, y)! h(x, y;ξ,η) The output h() depends on location of the point source (ξ,η) and location in the image (x,y), so it is a 4-D function ince the input is an impulse, the output is called the impulse response function, or the point spread function (PF) - why?

19 Impulse Response of Linear hift Invariant ystems For LI systems o the PF is f (x x 0, y y 0 ) [ ] = g(x x 0, y y 0 ) δ(x x 0, y y 0 ) [ ] = h(x x 0, y y 0 ) Through something called the superposition integral, we can show that g(x, y) = And for LI systems, this simplifies to: g(x, y) = f (ξ,η)h(x, y;ξ,η) dξ dη f (ξ,η)h(ξ x,η y) dξ dη The last integral is a convolution integral, and can be written as g(x, y) = f (x, y) h(x, y) (or f (x, y) h(x, y))

20 Review of convolution Illustration of h(x) = f (x) g(x) = f (u)g(x u)du original functions g(x-u), reversed and shifted to x x curve = product of f(u)g(x-u) area = integral of f(u)g(x-u) = value of h() at x x