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Comp 665. Convolution. Questions?. Ask here first. Likely someone else has the same question. Modeling the imaging process. We know how to compute where points in the world will map on the image plane Now, how will they be changed?. Impulse Response Function. Point Spread Function

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comp 665

Comp 665

Convolution

questions
Questions?
  • Ask here first.
  • Likely someone else has the same question.
modeling the imaging process
Modeling the imaging process
  • We know how to compute where points in the world will map on the image plane
  • Now, how will they be changed?
impulse response function
Impulse Response Function
  • Point Spread Function
  • What is the image of a point?
    • Shape of pinhole for points at infinity
    • Typically a little blob for a good lens
    • Could have aberrations and are distance, color, or position dependent.
  • What happens as we enlarge the pinhole?
blurring as convolution irf s and apertures
Blurring as convolution:IRF’s and apertures
  • Blurring by convolution with impulse response function: Iblurred(x)= Iinput(y) h(x-y) dy
    • Replace each point y by y’s intensity times IRF h centered at y
      • h(x-y) is effect of a fixed y on x over various image points x
    • Sum up over all such points
  • Aperture in image space x:
    • effect of y on x over various y: h(-[y-x])
    • weighting of various input image points in producing image at a fixed point x
linear systems
Linear Systems
  • Favorite model because we have great tools
  • F(a+b) = F(a) + F(b), F(k a) = k F(a)
  • Shift Invariant
  • Time Invariant
  • Is camera projection linear?
properties of convolution i out x i in y h x y d y
Properties of convolution:Iout(x) =  Iin(y) h(x-y) dy
  • h(x) is called the convolution kernel
  • Linear in both inputs, Iin and h
  • Symmetric in its inputs, Iin and h
  • Cascading convolutions is convolution with the convolution of the two kernels: (I*h1)*h2 = I*(h1*h2)
    • Thus cascading of convolution of two Gaussians produces Gaussian with s = (s12 + s22)½
  • Any linear, shift invariant operator can be written as a convolution or a limit of one
linear shift invariant operators
Linear Shift-Invariant Operators
  • Blurring with IRF that is constant over the scene
  • Viewing scene through any fixed aperture
  • All derivatives D
    • So D(I*h1) = DI*h1= I*Dh1
  • Designed operations, e.g., for smoothing, noise removal, sharpening, etc.
  • Can be applied to parametrized functions of u
    • E.g., smoothing surfaces
convolution equation

Notorious flip!

Convolution Equation
  • For non-causal symmetric impulse responses it doesn’t matter
  • For causal or non-symmetric impulse responses it is critical!
  • Important for relating convolution to correlation
  • Some texts (web sites) reverse the input, don’t do that
  • Reverse the impulse response instead
1d convolution code
1D Convolution Code

Output side algorithm

for i indexing y

sum = 0

for j indexing h

sum += h[j] * x[i-j]

y[i] = sum

Input side algorithm

zero y

for i indexing y

for j indexing h

y[i+j] += x[i]*h[j]

derivative of gaussian weighting functions
Derivative of Gaussian Weighting Functions

Barness along u: Guu

Gaussian: G

Edgeness along u: Gu

properties of convolution
Properties of Convolution
  • Commutative: a*b = b*a
  • Associative: (a*b)*c = a*(b*c)
  • Distributive: a*b + a*c = a*(b+c)
  • Central Limit Theorem: convolve a pulse with itself enough times you get a Gaussian
properties of convolution continued
Properties of convolution,continued
  • For any convolution kernel h(x), if the input Iin(x) is a sinusoid with wavelength (level of detail) 1/n, i.e., Iin(x) = A cos(2pnx) + B sin(2pnx), then the output of the convolution is a sinusoid with the same wavelength (level of detail), i.e., Iin * h = C cos(2pnx) + D sin(2pnx), for some C and D dependent on A, B, and h(x)
sampling and integration digital images
Sampling and integration(digital images)
  • Model
    • Within-pixel integration at all points
      • Has its own IRF, typically rectangular
    • Then sampling
  • Sampling = multiplication by pixel area  brush function
    • Brush function is sum of impulses at pixel centers
    • Sampling = aliasing: in sinusoidal decomposition higher frequency components masquerading as and thus polluting lower frequency components
    • Nyquist frequency: how finely to sample to have adequately low effect of aliasing
fun with convolution
Fun with Convolution

http://www.eas.asu.edu/~spanias/convolution_demo.htm

http://www.isip.msstate.edu/projects/speech/software/demonstrations/applets/util/convolution/current/