3-d computational vision csc 83020
DESCRIPTION
3-D Computational Vision CSc 83020. Image Processing II - Fourier Transform. The Fourier Transform. Previous lecture: filtering in the spatial domain. A signal (i.e. scanline/audio/image) has equivalent representation in the Frequency Domain. Frequency domain. Spatial domain. - PowerPoint PPT PresentationTRANSCRIPT
3-D Computational Vision3-D Computational VisionCSc 83020CSc 83020
Image Processing II - Fourier Image Processing II - Fourier TransformTransform
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
The Fourier TransformThe Fourier TransformPrevious lecture: filtering in the spatial domain.
A signal (i.e. scanline/audio/image) hasequivalent representation in the Frequency Domain.
Spatial domain Frequency domain
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
1-D Continuous Fourier Transform1-D Continuous Fourier TransformSpatial Domain(x) => Frequency Domain (u)Spatial Domain(x) => Frequency Domain (u)
1,sincos
)()(
)()(
2
2
ikike
dueuFxf
dxexfuF
ik
uxi
uxi
Note that F(u) is generally COMPLEX.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
)()(
)2sin()()2cos()()(
uiIuR
dxuxxfidxuxxfuF
Real and imaginary part.
Integration with cos/sin waves of different frequencies.
Magnitude |F(u)| : Fourier Spectrum.Phase φ(u) : Phase Spectrum.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
A periodic signal and its spectrumA periodic signal and its spectrum
From“Digital Image Warping”byGeorge Wolberg.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
An aperiodic signal and its An aperiodic signal and its spectrumspectrum
From“Digital Image Warping”byGeorge Wolberg.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Fourier Transformation & ConvolutionFourier Transformation & Convolution
)()()(
))()(()(
)()(
)()()(
*
2
2
uHuFuG
dxedxhfuG
dxexguG
dxhfxg
hfg
uxi
uxi
Convolution
Fourier Trans.Using y=x-ξ
Convolution in Spatial Domain ===Multiplication in Frequency Domain.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Fourier Transform and ConvolutionFourier Transform and ConvolutionSpatial Domain (x)g=f * h
g=f x h
Frequency Domain (u)G=F x H
G=F * HAlternative Method of finding g(x)
g = f * h
G = F x H
F.T F.TIFT
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Example: SmoothingExample: Smoothingf(x)
x
NOISYSIGNAL
We want: g(x) = f(x) * h(x) (SMOOTHED)
22
2
2
)2()(
21)(
21
21
ueuH
exhx
Let:
Then:
Example: SmoothingExample: Smoothingh(x)
xσ
H(u)
u
1/(2πσ)
We know: G(u)=F(u) H(u)
H(u) ATTENUATES high frequencies in F(u)(LOW-PASS FILTER)
Sampling TheoremSampling Theoremf(x)
x
CONTINUOUSSIGNAL
S(x)
x
……SHAH FUNCTION
x0
n
nxxxS )()( 0
)(*)()(
)().()().()( 0
uSuFuF
nxxxfxsxfxf
s
ns
Sampled Function:
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Sampling TheoremSampling Theorem
n
s xnu
xuFuF )(1*)()(
00
……1/x0
F(u)Let:
umax
S(u)
uA
Sampling TheoremSampling Theorem
n
s xnu
xuFuF )(1*)()(
00
u……
1/x0
S(u)
F(u)Let:
umax
Fs(u)……
Here: umax <= 1/(2*x0)
u
u
A
A/x0
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Sampling TheoremSampling Theorem
Fs(u)……u
What if umax > 1/(2*x0) ?
A/x0
1/x0
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Sampling TheoremSampling Theorem
Fs(u)……u
What if umax > 1/(2*x0) ?
1/x0
A/x0ALIASING
Can we recover F(u) from Fs(u)?
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Sampling TheoremSampling Theorem
Fs(u)……u
What if umax > 1/(2*x0) ?
1/x0
A/x0
Can we recover F(u) from Fs(u)?
Only if umax <= 1 /(2*x0) (NYQUIST FREQUENCY).
ALIASING
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
FromShreeNayar’snotes.
Figure 8.11. Left: At the top is a 256x256 pixel image showing a grid obtained by multiplying two sinusoids with linearly increasing frequency . one in x and one in y. The other images in the series are obtained by resampling by factors of two, without smoothing (i.e. the next is a 128x128, then a 64x64, etc., all scaled to the same size). Note the substantial aliasing; high spatial frequencies alias down to low spatial frequencies, and the smallest image is an extremely poor representation of the large image. Right: The magnitude of the Fourier transform of each image . displayed as a log, to compress the intensity scale. The constant component is at the center. Notice that the Fourier transform of a resampled image is obtained by scaling the Fourier transform of the original image and then tiling the plane. Interference between copies of the original Fourier transform means that we cannot recover its value at some points . this is the mechanism of aliasing.
Original Image256x256
Resampled128x128
Resampled64x64
CorrespondingFourier Transforms
ALIASING
2-D Domain - Images2-D Domain - ImagesSpatial Domain(x,y) => Frequency Domain Spatial Domain(x,y) => Frequency Domain
(u,v)(u,v)
dudvevuFyxf
dxdyeyxfvuF
vyuxi
vyuxi
)(2
)(2
),(),(
),(),(
f(x,y) g(x,y)h(x,y)LSIS:
δ(x,y) h(x,y)h(x,y)
Point Spread Function
From Forsyth & Ponce
Table 8.1. A variety of functions of two dimensions, and their Fourier transforms. This table can be used in two directions (with appropriate substitutions for u, v and (x, y), because the Fourier transform of the Fourier transform of a function is the function. Observant readers may suspect that the results on infite sums of δ functions contradict the linearity of Fourier transforms; by careful inspection of limits, it is possible to show that they do not.
2πi
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Discrete 2-D Fourier Discrete 2-D Fourier TransformTransform
Fast Fourier Transform (FFT)!
M
k
N
l
NMkmi
M
k
N
l
NMkmi
enmFlkf
elkfMNnmF
1 1
)ln//(2
1 1
)ln//(2
),(),(
),()/1(),(
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
FromShreeNayar’snotes.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
From Forsyth & Ponce.
Image 1
Image 2
Log of Fouriermagnitude Phase Spectrum
Discussion
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
From Forsyth & Ponce.
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Figure 8.11. Left: At the top is a 256x256 pixel image showing a grid obtained by multiplying two sinusoids with linearly increasing frequency . one in x and one in y. The other images in the series are obtained by resampling by factors of two, without smoothing (i.e. the next is a 128x128, then a 64x64, etc., all scaled to the same size). Note the substantial aliasing; high spatial frequencies alias down to low spatial frequencies, and the smallest image is an extremely poor representation of the large image. Right: The magnitude of the Fourier transform of each image . displayed as a log, to compress the intensity scale. The constant component is at the center. Notice that the Fourier transform of a resampled image is obtained by scaling the Fourier transform of the original image and then tiling the plane. Interference between copies of the original Fourier transform means that we cannot recover its value at some points . this is the mechanism of aliasing.
Original Image256x256
Resampled128x128
Resampled64x64
CorrespondingFourier Transforms
ALIASING
Figure 8.11. Left: At the top is a 256x256 pixel image showing a grid obtained by multiplying two sinusoids with linearly increasing frequency . one in x and one in y. The other images in the series are obtained by resampling by factors of two, without smoothing (i.e. the next is a 128x128, then a 64x64, etc., all scaled to the same size). Note the substantial aliasing; high spatial frequencies alias down to low spatial frequencies, and the smallest image is an extremely poor representation of the large image. Right: The magnitude of the Fourier transform of each image . displayed as a log, to compress the intensity scale. The constant component is at the center. Notice that the Fourier transform of a resampled image is obtained by scaling the Fourier transform of the original image and then tiling the plane. Interference between copies of the original Fourier transform means that we cannot recover its value at some points . this is the mechanism of aliasing.
Resampled32x32
Resampled16x16
CorrespondingFourier Transforms
ALIASINGFrom Forsyth & Ponce
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Figure 8.12. Left: Resampled versions of the image of figure 8.11, again by factors of two, but this time each image is smoothed with a Gaussian of σ one pixel before resampling.This filter is a low-pass filter, and so suppresses high spatial frequency components, reducing aliasing. Right: The effect of the low-pass filter is easily seen in these logmagnitude images; the low pass filter suppresses the high spatial frequency components so that components interfere less, to reduce aliasing.
Original Image256x256
CorrespondingFourier Transforms
LOW PASS FILTERING
σ=1 pixel
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Figure 8.12. Left: Resampled versions of the image of figure 8.11, again by factors of two, but this time each image is smoothed with a Gaussian of σ one pixel before resampling.This filter is a low-pass filter, and so suppresses high spatial frequency components, reducing aliasing. Right: The effect of the low-pass filter is easily seen in these logmagnitude images; the low pass filter suppresses the high spatial frequency components so that components interfere less, to reduce aliasing.
CorrespondingFourier Transforms
LOW PASS FILTERING
σ=1 pixel
FromForsyth & Ponce
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Figure 8.12. Left: Resampled versions of the image of figure 8.11, again by factors of two, but this time each image is smoothed with a Gaussian of σ one pixel before resampling.This filter is a low-pass filter, and so suppresses high spatial frequency components, reducing aliasing. Right: The effect of the low-pass filter is easily seen in these logmagnitude images; the low pass filter suppresses the high spatial frequency components so that components interfere less, to reduce aliasing.
Original Image256x256
CorrespondingFourier Transforms
LOW PASS FILTERING
Gaussian σ=2 pixels
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Figure 8.12. Left: Resampled versions of the image of figure 8.11, again by factors of two, but this time each image is smoothed with a Gaussian of σ one pixel before resampling.This filter is a low-pass filter, and so suppresses high spatial frequency components, reducing aliasing. Right: The effect of the low-pass filter is easily seen in these logmagnitude images; the low pass filter suppresses the high spatial frequency components so that components interfere less, to reduce aliasing.
CorrespondingFourier Transforms
LOW PASS FILTERING
σ=2 pixels
FromForsyth & Ponce
Gaussian Smoothing versus Averaging
Filter mask (averaging) Filter mask (gaussian)
Original image (grass)
Result of averaging
Result of Gaussian smoothing
From Forsyth & Ponce
CSc 83020 3-D Computer Vision / Ioannis StamosCSc 83020 3-D Computer Vision / Ioannis Stamos
Figure 8.1. Although a uniform local average may seem to give a good blurring model, it generates effects that are not usually seen in defocussing a lens. The images above compare the effects of a uniform local average with weighted average. The image at the top shows a view of grass. On the left in the second row, the result of blurring this image using a uniform local model and on the right, the result of blurring this image using a set of Gaussian weights. The degree of blurring in each case is about the same, but the uniform average produces a set of narrow vertical and horizontal bars, an effect often known as ringing. The bottom row shows the weights used to blur the image, themselves rendered as an image; bright points represent large values and dark points represent small values (in this example the smallest values are zero).
FromShreeNayar’snotes.
FromShreeNayar’snotes.
FromShreeNayar’snotes.