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Summary Introduction to Computer Vision, Bachelor Computer Science & Artificial Intelligence UvA

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  • Course
  • Introduction to Computer Vision (5082ITCV6Y)
  • Institution
  • Universiteit Van Amsterdam (UvA)

Complete summary of the contents for the exam for the course Introduction to Computer Science in the 2nd year of the Computer Science & Artificial Intelligence bachelor's degrees at the University of Amsterdam. The summary is in English, like the course. All lectures and lecture notes are in the s...

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Document information

  • August 3, 2023
  • 17
  • 2022/2023
  • Class notes
  • Dimitris tzionas
  • All classes

Subjects

  • informatica
  • computer vision
  • beeldverwerking
  • linear algebra
  • lineaire algebra
  • kunstmatige intelligentie
  • computer science
  • artificial intelligence

Written for

  • Universiteit van Amsterdam (UvA)
  • Informatica
  • Introduction to Computer Vision (5082ITCV6Y)
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Introduction to Computer Vision Summary CHoogteijling


Introduction to Computer Vision
Contents

1 Interpolation 2

2 Point operators 3

3 Histogram based image operations 3

4 Least Squares Estimators 4

5 Geometric operators 6

6 Homogeneous coordinates 6

7 Local operators 8

8 Local structure 10

9 Image stitching using SIFT 13

10 Pinhole camera 14

11 Convolutional neural networks 16

12 Motion 16




1

,Introduction to Computer Vision Summary CHoogteijling


1 Interpolation

Interpolation allows us to find the value of a function in between the points where the image is
sampled.
Nearest neighbor interpolation is, given the samples F (k), the value of the interpolated function
fˆ at coordinate x. The function is not continuous nor differentiable.


1
fˆ(x) = F (⌊x + ⌋)
2

Linear interpolation is, between adjacent sample points k and k + 1, we assume the function is a
linear function. The function is continuous but not differentiable in the sample points. The second
and higher order derivatives are equal to zero


k ≤ x ≤ k + 1 : fˆ(x) = (1 − (x − k))F (k) + (x − k)F (k + 1)

With cubic interpolation we look at two pixels on the left and two on the right. To interpolate
the function value f (x) for x in between x = k and x = k + 1 we fit a cubic polynomial to the
sample points {k − 1, k, k + 1, k + 2}.


k ≤ x ≤ k + 1 : fˆ(x) = a(x − k)3 + b(x − k)2 + c(x − k) + d

There are better interpolation methods, for example using more samples. A disadvantage of higher
order polynomials is overfitting of the original function: higher order polynomials tend to fluctuate
wildly in between the sample points.
For 2D functions we can also use nearest neighbor, cubic and spline interpolation. We first
interpolate in the x-direction and then in the y-direction.
Extrapolation allows us to find the value of a function outside the domain of the image. To find
the value we can:

• Set the value of a point outside the bounds of the grid to a constant value (often zero).

• Pick a point that is within the bounds of the grid and use the (interpolated) value in that
point.

– Closest point. We select the point inside the bounds of the grid that is closest to the
outside point.
– Mirrored point. We mirror the outside point in the vertical line through the last
sample points in horizontal direction of the grid.
– Wrapping. We select the same point from inside the bounds. Imagine a tiled wall and
each tile showing the same image. This is what the discrete Fourier transform implicitly
assumes.



2

, Introduction to Computer Vision Summary CHoogteijling


1.1 Image histograms

A histogram of all possible scalar pixel values in an image provides a summary of the distribution
of the values over all possible values. There is no science behind choosing an appropriate bin size.
One rule of thumb is Sturges’ rule k = ⌈log2 n⌉ + 1.
The function for an univariate histogram is:
X
hf [i] = [ei ≤ f (x) < ei+1 ]
x∈E

To capture a histogram of data that is multi-dimensional, we can compute a multivariate his-
togram.

X
hf [i, j, k] = [e1,i ≤ f1 (x) < e1,i+1 ][e2,j ≤ f2 (x) < e2,j+1 ][e3,k ≤ f3 (x) < e3,k+1 ]
x∈E



2 Point operators

A point operator γ is a function that constructed by pointwise lifting a value operator to the
image domain. For two images f : D → R and g : D → R′ . Let γ : R × R′ → R′′ be an operator.
The operator γ can be lifted to work on images.


∀x ∈ D : γ(f, g)(x) = γ(f (x), g(x))

α-blending takes the weighted average of two images. Let f and g be two colour images defined
on the same spatial domain. A sequence of images that shows smooth transition from f to g can
be obtained by the following equation for α-values increasing from 0 to 1.


hα = (1 − α)f + αg

Unsharp masking uses alpha-blending to sharpen an image. Let f be an image and g an unsharp
version of the image. The result is obtained by adding β times the difference of f − g to the original
image.


h = f + β(f − g)

Image thresholding uses a relational operator. Let f be a scalar image, then [f > t], for constant
t, results in a binary image.


3 Histogram based image operations

A Monadic point operator is an operator that changes the pixel value f (x) independent of the
position x and independent of all other pixel values in the neighbourhood.

3

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