1 / 41

3-D Computational Vision CSc 83020

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.

dragon
Télécharger la présentation

3-D Computational Vision CSc 83020

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 3-D Computational VisionCSc 83020 Image Processing II - Fourier Transform

  2. 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 CSc 83020 3-D Computer Vision / Ioannis Stamos

  3. 1-D Continuous Fourier Transform Spatial Domain(x) => Frequency Domain (u) Note that F(u) is generally COMPLEX. CSc 83020 3-D Computer Vision / Ioannis Stamos

  4. 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 Stamos

  5. A periodic signal and its spectrum From “Digital Image Warping” by George Wolberg. CSc 83020 3-D Computer Vision / Ioannis Stamos

  6. An aperiodic signal and its spectrum From “Digital Image Warping” by George Wolberg. CSc 83020 3-D Computer Vision / Ioannis Stamos

  7. CSc 83020 3-D Computer Vision / Ioannis Stamos

  8. CSc 83020 3-D Computer Vision / Ioannis Stamos

  9. CSc 83020 3-D Computer Vision / Ioannis Stamos

  10. Fourier Transformation & Convolution Convolution Using y=x-ξ Fourier Trans. Convolution in Spatial Domain === Multiplication in Frequency Domain.

  11. Fourier Transform and Convolution Spatial Domain (x) g=f * h g=f x h Frequency Domain (u) G=F x H G=F * H Alternative Method of finding g(x) g = f * h F.T F.T IFT G = F x H CSc 83020 3-D Computer Vision / Ioannis Stamos

  12. Example: Smoothing f(x) NOISY SIGNAL x We want: g(x) = f(x) * h(x) (SMOOTHED) Let: Then: CSc 83020 3-D Computer Vision / Ioannis Stamos

  13. Example: Smoothing h(x) σ x H(u) 1/(2πσ) u We know: G(u)=F(u) H(u) H(u) ATTENUATES high frequencies in F(u) (LOW-PASS FILTER)

  14. Sampling Theorem f(x) CONTINUOUS SIGNAL x S(x) SHAH FUNCTION … … x0 x Sampled Function:

  15. Sampling Theorem S(u) … … u 1/x0 A Let: F(u) umax CSc 83020 3-D Computer Vision / Ioannis Stamos

  16. Sampling Theorem S(u) … … u 1/x0 A Let: F(u) umax u Fs(u) A/x0 … … u Here: umax <= 1/(2*x0)

  17. Sampling Theorem What if umax > 1/(2*x0)? Fs(u) A/x0 … … u 1/x0 CSc 83020 3-D Computer Vision / Ioannis Stamos

  18. Sampling Theorem What if umax > 1/(2*x0)? ALIASING Fs(u) A/x0 … … u 1/x0 Can we recover F(u) from Fs(u)? CSc 83020 3-D Computer Vision / Ioannis Stamos

  19. Sampling Theorem What if umax > 1/(2*x0)? ALIASING Fs(u) A/x0 … … u 1/x0 Can we recover F(u) from Fs(u)? Only if umax <= 1 /(2*x0)(NYQUIST FREQUENCY). CSc 83020 3-D Computer Vision / Ioannis Stamos

  20. From Shree Nayar’s notes. CSc 83020 3-D Computer Vision / Ioannis Stamos

  21. Original Image 256x256 ALIASING Corresponding Fourier Transforms Resampled 128x128 Resampled 64x64 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.

  22. 2-D Domain - Images Spatial Domain(x,y) => Frequency Domain (u,v) LSIS: f(x,y) g(x,y) h(x,y) Point Spread Function δ(x,y) h(x,y) h(x,y)

  23. From Forsyth & Ponce 2πi 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.

  24. Discrete 2-D Fourier Transform Fast Fourier Transform (FFT)! CSc 83020 3-D Computer Vision / Ioannis Stamos

  25. From Shree Nayar’s notes. CSc 83020 3-D Computer Vision / Ioannis Stamos

  26. From Forsyth & Ponce. Log of Fourier magnitude Phase Spectrum Image 1 Image 2 Discussion CSc 83020 3-D Computer Vision / Ioannis Stamos

  27. From Forsyth & Ponce. CSc 83020 3-D Computer Vision / Ioannis Stamos

  28. CSc 83020 3-D Computer Vision / Ioannis Stamos

  29. Original Image 256x256 ALIASING Corresponding Fourier Transforms Resampled 128x128 Resampled 64x64 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.

  30. ALIASING From Forsyth & Ponce Corresponding Fourier Transforms Resampled 32x32 Resampled 16x16 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.

  31. Original Image 256x256 LOW PASS FILTERING Corresponding Fourier Transforms σ=1 pixel 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. CSc 83020 3-D Computer Vision / Ioannis Stamos

  32. LOW PASS FILTERING From Forsyth & Ponce Corresponding Fourier Transforms σ=1 pixel 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. CSc 83020 3-D Computer Vision / Ioannis Stamos

  33. Original Image 256x256 LOW PASS FILTERING Corresponding Fourier Transforms Gaussian σ=2 pixels 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. CSc 83020 3-D Computer Vision / Ioannis Stamos

  34. LOW PASS FILTERING From Forsyth & Ponce Corresponding Fourier Transforms σ=2 pixels 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. CSc 83020 3-D Computer Vision / Ioannis Stamos

  35. Gaussian Smoothing versus Averaging Original image (grass) From Forsyth & Ponce Result of Gaussian smoothing Result of averaging Filter mask (gaussian) Filter mask (averaging)

  36. 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). CSc 83020 3-D Computer Vision / Ioannis Stamos

  37. From Shree Nayar’s notes.

  38. From Shree Nayar’s notes.

  39. From Shree Nayar’s notes.

More Related