Fundamentos de procesamiento de imágenes

Fundamentals of Image Processing

hany.farid@dartmouth.edu http://www.cs.dartmouth.edu/~farid

0. Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.1: Vectors 0.2: Matrices 0.3: Vector Spaces 0.4: Basis 0.5: Inner Products and Projections [*] 0.6: Linear Transforms [*] 1. Discrete-Time Signals and Systems . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.1: Discrete-Time Signals 1.2: Discrete-Time Systems 1.3: Linear Time-Invariant Systems 2. Linear Time-Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1: Space: Convolution Sum 2.2: Frequency: Fourier Transform 3. Sampling: Continuous to Discrete (and back) . . . . . . . . . . . . . . .. . . . . . . . . . 29 3.1: Continuous to Discrete: Space 3.2: Continuous to Discrete: Frequency 3.3: Discrete to Continuous 4. Digital Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.1: Choosing a Frequency Response 4.2: Frequency Sampling 4.3: Least-Squares 4.4: Weighted Least-Squares 5. Photons to Pixels . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1: Pinhole Camera 5.2: Lenses 5.3: CCD 6. Point-Wise Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.1: Lookup Table 6.2: Brightness/Contrast 6.3: Gamma Correction 6.4: Quantize/Threshold 6.5: Histogram Equalize 7. Linear Filtering . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.1: Convolution 7.2: Derivative Filters 7.3: Steerable Filters 7.4: Edge Detection 7.5: Wiener Filter 8. Non-Linear Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.1: Median Filter 8.2: Dithering 9. Multi-Scale Transforms [*] . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10. Motion Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 10.1: Differential Motion 10.2: Differential Stereo 11. Useful Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6911.1: Expectation/Maximization 11.2: Principal Component Analysis [*] 11.3: Independent Component Analysis [*] [*] In progress

0. Mathematical Foundations

0.1 Vectors From the preface of Linear Algebra and its Applications:

0.1 Vectors 0.2 Matrices 0.3 Vector Spaces

“Linear algebra is a fantastic subject. On the one hand it is clean and beautiful.” – Gilbert Strang This wonderful branchof mathematics is both beautiful and useful. It is the cornerstone upon which signal and image processing is built. This short chapter can not be a comprehensive survey of linear algebra; it is meant only as a brief introduction and review. The ideas and presentation order are modeled after Strang’s highly recommended Linear Algebra and its Applications. At the heart of linear algebra is machineryfor solving linear equations. In the simplest case, the number of unknowns equals the number of equations. For example, here are a two equations in two unknowns: 2x − y = 1

0.4 Basis 0.5 Inner Products and Projections 0.6 Linear Transforms

y

2x−y=1 (x,y)=(2,3)

x + y = 5.

(1)
x+y=5

x

There are at least two ways in which we can think of solving these equations for x and y. Thefirst is to consider each equation as describing a line, with the solution being at the intersection of the lines: in this case the point (2, 3), Figure 0.1. This solution is termed a “row” solution because the equations are considered in isolation of one another. This is in contrast to a “column” solution in which the equations are rewritten in vector form: 2 1 x+ −1 1 y = 1 5 . (2)

Figure...