1. … Sampling … … Filtering … … Reconstruction …
Leonid I. Dimitrov & Miloš Šrámek
Commission for Scientific Visualization
Austrian Academy of Sciences

2. Motivation example (1)

3. Motivation example (2)

4. Motivation example (3)

5. Motivation example (4)
Nice picture
Poor picture

6. Overview Sampling of continuous signals Filtering of discrete signals
Reconstruction of continuous signals
Sampling aspects of volume rendering
Voxelization of geometric objects

7. Why Sampling and Reconstruction?
Real-world signals are continuous 3D
object
Digitalization
and
quantization 3D
reconstruction
and
sampling 2D
reconstruction
and
sampling 3D
scene 2D
image
Measurement
Visualization
Display
Computer representations are discrete

8. Signal and Spectrum Spectrum Signal s 
Fourier
transformation
Spectrum
Signal s 
Frequency/spectral domain
Spatial / time domain
FT: decomposition of a signal into a sum of sinusoids, determined by frequency, amplitude and phase
f(x) - representation in the spatial domain
F() - representation in the frequency domain

9. Fourier Transform
Direct FT:
Inverse FT:
ω=2π𝑓

10. Examples sin(x)+sin(2x) Spectrum of sin(x)+sin(2x) Spectrum music
Spectrum white noise

11. Typically Drawn Spectrum
Symmetric
F is complex, we draw |F|
Falling towards high frequencies
Band limited: has maximum frequency max
|F| f

12. Convolution (1) A weighted sum of f with weight function h placed at s
Applications:
Smoothing, noise removal, edge detection, ...
Other names: filter (e.g. lowpass)

13. Convolution (2) Continuous case: Discrete case:
f: signal, h: convolution kernel

14. Convolution Theorem
Convolution in spatial (frequency) domain corresponds to multiplication in frequency (spatial) domain

15. Dirac's δ-function Can be used to describe the sampling process
Properties: x x0
δ 𝑥− 𝑥 0
−∞ ∞ δ 𝑥 =1
𝑓 𝑥 δ 𝑥− 𝑥 0 =𝑓 𝑥 0 δ 𝑥− 𝑥 0
−∞ ∞ 𝑓 𝑥 δ 𝑥− 𝑥 0 =𝑓 𝑥 0

16. Sampling
Input signal:
Ideal sampling function:
Sampled signal:

17. Sampling in the Frequency Domain
Spectrum of the sampled signal:
S (spectrum of the sampling function):
and

18. Sampling in the Frequency Domain
Sampling a signal causes replication of its spectrum in the frequency domain
Nyquist criterion:

19. Sampling of a Signal
In the spatial domain
In the frequency domain

20. Real-world Sampling Process
The sampling function has a volume
Can be approximated by a Gaussian:
A Gaussian is a low-pass filter, i.e. it causes blurred image, blunt edges

21. Reconstruction Direct reconstruction of a continuous
signal from discrete samples:
The spectrum replicas have to be suppressed by a reconstruction filter
The ideal reconstruction filter is the box function:

22. Reconstruction

23. Ideal Reconstruction Filter

24. Ideal Reconstruction Filter
Bounded support in the frequency domain, but unbounded in the spatial domain
Can’t be realized practically
It has to be approximated (bounded):
The approximations have unbounded support in the frequency domain

25. Problems with the Reconstruction (aliasing)
If the Nyquist criterion is not fulfilled
If the reconstruction filter is too big or too small

26. Prealiasing
Wrong frequencies appear if the Nyquist criterion is not fulfilled:

27. Postaliasing
Wrong frequencies appear also if the reconstruction filter support is too broad

28. Classification of 3D reconstruction filters
Separable:
Sequential application along the axes
Computational complexity: 3n
Spherically symmetrical:
Computational complexity: n3

29. Separable Filters Order 0: nearest neighbor
Order 1: linear interpolation
Order 3: cubic filters:
Cubic B-spline
Catmull-Rom spline

30. Nearest Neighbor
Filter
Spectrum

31. Nearest Neighbor Filtering

32. Nearest Neighbor Filtering

33. Linear Interpolation
Filter
Spectrum

34. Linear Filtering

35. Linear Filtering

36. Trilinear Reconstruction Filter

37. Truncated sinc Filter
Filter
Spektrum

38. Truncated sinc Filtering

39. Truncated sinc Filtering

40. Catmull-Rom Spline
Piecewise cubic, C1-smooth

41. Catmul-Rom Spline
Filter
Spectrum

42. Catmull-Rom Spline Filtering

43. Catmull-Rom Spline Filtering

44. Other Separable Filters
Gaussian filter
Windowed sinc filter

45. An Example
FN = 10, sampling 40 x 40 x 40

46. Reconstruction by the Trilinear Filter

47. Reconstruction by the Cubic B-spline Filter

48. Reconstruction by the Catmull-Rom Spline Filter

49. Reconstruction by the Windowed sinc Filter