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Image Processing
Fourier Transform
Slide 1
Image Processing: Fourier Transform
Function Spaces
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Images are not vectors. Images are mappings: Moreover, images are functions (continuous domain): However, functions can be seen as vectors as well: → Images are not vectors, they are more, but they are vectors too
Vector
Function
Domain
Mapping
Space
Scalar product
Length
Image Processing: Fourier Transform
Base in vector spaces
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Task: decompose a vector into its “components” in a base with the base vectors and coefficients Properties of base vectors: 1. Vectors should span the space → decomposition exists for all 2. Vectors should be independent (no vector can be represented
as a linear combination of ) → decomposition is unique
Special case – orthonormal base: • All are orthogonal to each other, i.e. for all • All have the same length (=1), i.e. →
Image Processing: Fourier Transform
Base in function spaces
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The space has infinite dimension → • Infinite number of base functions , i.e. , replaces • A continuous function The task is to decompose a given function into the base ones: Orthonormal base means: • orthogonal • normalized
Then
Image Processing: Fourier Transform
Fourier Series
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Space: all periodic functions with the period , i.e. Base functions: and Properties: 1. Orthonormal
2. Span the function space (Jean Baptiste Joseph Fourier, 1822)
Image Processing: Fourier Transform
Fourier Series
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Decomposition: with
Image Processing: Fourier Transform
Fourier Series
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Image Processing: Fourier Transform
Complex numbers
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Euler’s Formula: Decomposition: Coefficients:
Image Processing: Fourier Transform
General functions
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Arbitrary periodic functions – transition Arbitrary non-periodic functions – limit 1. Coefficients become continuous 2. The sequence becomes a complex function of a real-
valued argument
The summands are “not interesting” by themselves, but rather: • amplitude-spectrum and
• phase-spectrum
Image Processing: Fourier Transform
2D Discrete Fourier Transform
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Two primary arguments: and Two frequencies: horizontal and vertical Transform: Inverse:
Image Processing: Fourier Transform
2D Discrete Fourier Transform
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Convolution masks for different frequencies
Image Processing: Fourier Transform
Amplitude-spectrums
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Images Fourier Transforms
Image Processing: Fourier Transform
Amplitude vs. Phase
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Image Processing: Fourier Transform
Example – Directions
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Image Processing: Fourier Transform
Example – Directions
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Image Processing: Fourier Transform
: operator (Fourier Transform) : the image of a function in the frequency space Proof: … analogously
Convolution Theorem
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Image Processing: Fourier Transform
Convolution Theorem
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Corollary 1: a convolution can be performed in the frequency space by Time complexity: Fourier Transform can be done with Component-by-component multiplication: → all together instead of by the direct implementation Corollary 2: each filter has its spectral characteristics in the frequency space. 1. It is possible to analyze filter characteristics 2. It is possible to design filters with the necessary properties
Image Processing: Fourier Transform
Convolution Theorem
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Some filters and their spectrums
Image Processing: Fourier Transform
Further themes:
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Image say “where” but not “what”. Spectrums say “what” but not “where”. Windowed Fourier Transform – spectrums for (small) windows at each position. Cosine Transform (1D, discrete, DCT-II – JPEG): Wavelet Transform:
− a “mother function”, e.g. Complex Mexican hat wavelet
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