1. Fundamentals
Data

2. Outline Visualization Discretization Sampling Quantization
Representation
Continuous
Discrete
Noise

3. Data Data : Function dependent on one or more variables. Example
Audio (1D) - depends on time t - 𝐴(𝑡)
Image (2D) - depends on spatial coordinates x and y - 𝐼 𝑥,𝑦
Video (3D) - depends on spatial coordinate (x,y) and time t 𝑉(𝑥,𝑦,𝑡)

4. Data (1D and 2D)
Plot of dependent variable with respect to independent ones
2D plot is a height field

5. Two image representations

6. Color Image Representation
Three color channels: 𝑅 𝑥,𝑦 , 𝐺 𝑥,𝑦 𝑎𝑛𝑑 𝐵(𝑥,𝑦)

7. An Alternative Representation (1D)
frequency domain representation
A signal is a linear combination of sine or cosine waves
𝑐 𝑡 = 𝑖=1 ∞ 𝑎 𝑖 𝐶𝑜𝑠( 𝑓 𝑖 + 𝑝 𝑖 )
Signal can be represented by the coefficients of these sine or cosine waves

8. Frequency Domain Representation
Amplitude and Phase
What about 2D?
Frequency and Orientation
Polar Representation

9. Discretization Data exists in nature as a continuous function.
Convert to discrete function for digital representation
Discretization
Two concepts
Sampling
Quantization

10. Sampling Set of values of continuous 𝑓 𝑡 at specific values of t.
Reduces continuous function 𝑓 𝑡 to discrete form
𝐴(𝑡) t
Sampling

11. Uniform vs Non-uniform sampling

12. Reconstruction Get the continuous function from the discrete function
Correct Reconstruction
Sampling

13. Reconstruction Accurate reconstruction needs adequate samples Sampling
Incorrect Reconstruction

14. Aliasing Incorrect representation of some entity Zero frequency
A much lower frequency
Zero frequency

15. Correct Reconstruction
Nyquist Sampling Rate
By sampling at least twice the frequency (2 samples per cycle), signal can be reconstructed correctly. t
Sampling
Correct Reconstruction

16. Quantization ½ step size
A analog signal can have any value of infinite precision
Digital signal can only have a limited set of value
Step Size
Max error
½ step size

17. Quantization Error

18. Quantization Depends on number of bits 12 bits = 4095 levels
0.0 ≤ Voltage ≤4.096
2.56 and TO 2560
Each level (LSB) = 0.001
Error ≤ ±½ LSB
Called Quantization Error

19. Noise Addition of random values at random locations in the data.
Random noise

20. Noise Outlier Noise An example of such noise is salt and pepper noise
Can be solved by Median filter

21. Noise
Some noise look random in spatial domain but can be isolated to a few frequencies in spectral domain.
Can be removed by Notch filter.

22. Technqiues Interpolation Linear Interpolation Bilinear Interpolation
Geometric Intersections

23. Interpolation
Estimate function for values which it has not been measured.
Linear interpolation:
Assuming a line between samples.
Change abruptly at sample points
No gradient continuity

24. Interpolation Non-linear interpolation:
A smooth curve passes through samples
First derivative continuous
Second derivative continuous

25. Linear Interpolation Point V on the line segment 𝑉 1 𝑉 2 is given by
𝑉=𝛼 𝑉 1 + 1−𝛼 𝑉 2 , ≤𝛼≤1
Example: Linear interpolation of color at point V
𝐶 𝑉 =C 𝛼 𝑉 1 + 1−𝛼 𝑉 2 =𝛼𝐶( 𝑉 1 )+ 1−𝛼 𝐶(𝑉 2 )

26. Bilinear Interpolation
2D Data
Interpolating in one direction followed by interpolating in the second direction.