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WIP: rework gabor filter
This commit introduces an lazy array interface for the Gabor filter, benchmark shows that it's 2-3x faster than the previous version. The documentation is heavly rewrote to explain this filter and all its parameters.
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‎docs/Project.toml

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[deps]
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DemoCards = "311a05b2-6137-4a5a-b473-18580a3d38b5"
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Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
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FileIO = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549"
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ImageContrastAdjustment = "f332f351-ec65-5f6a-b3d1-319c6670881a"
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ImageCore = "a09fc81d-aa75-5fe9-8630-4744c3626534"
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ImageDistances = "51556ac3-7006-55f5-8cb3-34580c88182d"

‎docs/demos/filters/gabor_filter.jl

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# ---
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# title: Gabor filter
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# id: demo_gabor_filter
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# cover: assets/gabor.png
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# author: Johnny Chen
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# date: 2021-11-01
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# ---
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# This example shows how one can apply frequency kernesl
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# [Gabor](https://en.wikipedia.org/wiki/Gabor_filter) and [Log
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# Gabor](https://en.wikipedia.org/wiki/Log_Gabor_filter) to extract image features.
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using ImageCore, ImageShow, ImageFiltering # or you could just `using Images`
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using FFTW
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using TestImages
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# ## Definition
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#
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# Mathematically, Gabor kernel is defined in frequency space:
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#
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# ```math
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# g(x, y) = \exp(-\frac{x'^2 + \gamma^2y'^2}{2\sigma^2})\exp(i(2\pi\frac{x'}{\lambda} + \psi))
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# ```
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# where ``i`` is imaginary unit `Complex(0, 1)`, and
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# ```math
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# x' = x\cos\theta + x\sin\theta \\
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# y' = -x\sin\theta + y\cos\theta
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# ```
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#
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# First of all, Gabor kernel is defined in frequency space so it is a complex-valued matrix:
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bandwidth, orientation, wavelength, phase_offset = 0.1, 0, 2, 0
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kern = Kernel.Gabor((10, 10); bandwidth, orientation, wavelength, phase_offset)
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# !!! tip "Lazy array"
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# The `Gabor` type is a lazy array, which means when you build the Gabor kernel, you
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# actually don't need to allocate any memories.
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#
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# ```julia
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# using BenchmarkTools
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# @btime Kernel.Gabor((64, 64); σ=5, θ=0, λ=1); # 1.705 ns (0 allocations: 0 bytes)
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# ```
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# To explain the parameters of Gabor filter, let's introduce one small helper function to
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# display frequency kernels.
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## You can also try display the real part: `@. Gray(log(abs(real(kern)) + 1))`
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show_phase(kern) = @. Gray(log(abs(imag(kern)) + 1))
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show_mag(kern) = @. Gray(log(abs(real(kern)) + 1))
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show_abs(kern) = @. Gray(log(abs(kern) + 1))
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nothing #hide
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# ## Keywords
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#
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# ### `wavelength` (λ)
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# λ specifies the wavelength of the sinusoidal factor.
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bandwidth, orientation, phase_offset, aspect_ratio = 1, 0, 0, 0.5
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f(wavelength) = show_abs(Kernel.Gabor((100, 100); wavelength, bandwidth, orientation, aspect_ratio, phase_offset))
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mosaic(f.((5, 10, 15)), nrow=1)
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# ### `orientation` (θ)
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# θ specifies the orientation of the normal to the parallel stripes of a Gabor function.
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wavelength, bandwidth, phase_offset, aspect_ratio = 10, 1, 0, 0.5
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f(orientation) = show_abs(Kernel.Gabor((100, 100); wavelength, bandwidth, orientation, aspect_ratio, phase_offset))
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mosaic(f.((0, π/4, π/2)), nrow=1)
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# ### `phase_offset` (ψ)
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wavelength, bandwidth, orientation, aspect_ratio = 10, 1, 0, 0.5
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f(phase_offset) = show_phase(Kernel.Gabor((100, 100); wavelength, bandwidth, orientation, aspect_ratio, phase_offset))
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mosaic(f.((-π/2, 0, π/2, π)), nrow=1)
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# ### `aspect_ratio` (γ)
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# γ specifies the ellipticity of the support of the Gabor function.
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wavelength, bandwidth, orientation, phase_offset = 10, 1, 0, 0
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f(aspect_ratio) = show_abs(Kernel.Gabor((100, 100); wavelength, bandwidth, orientation, aspect_ratio, phase_offset))
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mosaic(f.((0.5, 1, 2)), nrow=1)
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# ### `bandwidth` (b)
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# The half-response spatial frequency bandwidth (b) of a Gabor filter is related to the
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# ratio σ / λ, where σ and λ are the standard deviation of the Gaussian factor of the Gabor
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# function and the preferred wavelength, respectively, as follows:
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#
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# ```math
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# b = \log_2\frac{\frac{\sigma}{\lambda}\pi + \sqrt{\frac{\ln 2}{2}}}{\frac{\sigma}{\lambda}\pi - \sqrt{\frac{\ln 2}{2}}}
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# ```
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wavelength, orientation, phase_offset, aspect_ratio = 10, 0, 0, 0.5
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f(bandwidth) = show_abs(Kernel.Gabor((100, 100); wavelength, bandwidth, orientation, aspect_ratio, phase_offset))
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mosaic(f.((0.5, 1, 2)), nrow=1)
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# ## Examples
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#
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# To apply the filter, we need to use [the convolution
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# theorem](https://en.wikipedia.org/wiki/Convolution_theorem):
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#
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# ```math
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# \mathcal{F}(x \circledast k) = \mathcal{F}(x) \odot \mathcal{F}(k)
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# ```
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# where ``\circledast`` is convolution, ``\odot`` is pointwise-multiplication, and
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# ``\mathcal{F}`` is the fourier transformation.
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#
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# Because Gabor kernel is defined around center point (0, 0), we have to apply `fftshift`
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# first before we do pointwise-multiplication. Because `fftshift(fftshift(x)) == x`, this
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# can be applied to either `fftshift(fft(x))` or `fftshift(kern)`.
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img = TestImages.shepp_logan(127)
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kern = Kernel.Gabor(size(img); orientation=π/4, wavelength=20, bandwidth=2, phase_offset=0)
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out = ifft(centered(fft(channelview(img))) .* fftshift(kern))
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mosaic(img, show_abs(kern), show_abs(out); nrow=1)
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# A filter bank is just a list of filter kernels, applying the filter bank generates
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# multiple outputs:
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filters = [Kernel.Gabor(size(img);
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orientation,
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wavelength=20,
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bandwidth=50,
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phase_offset=0,
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)
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for orientation in -1.3:π/4:1.3
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];
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f(X, kern) = ifft(centered(fft(channelview(X))) .* fftshift(kern))
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mosaic(
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map(show_abs, filters)...,
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map(kern->show_abs(f(img, kern)), filters)...;
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nrow=2, rowmajor=true
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)
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## save covers #src
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using FileIO #src
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mkpath("assets") #src
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filters = [Kernel.Gabor((32, 32); #src
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orientation, #src
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wavelength=5, #src
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bandwidth=2, #src
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phase_offset=0, #src
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) #src
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for orientation in range(-π/2, stop=π/2, length=9) #src
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]; #src
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save("assets/gabor.png", mosaic(map(show_mag, filters); nrow=3, npad=2, fillvalue=Gray(1)); fps=2) #src

‎docs/src/function_reference.md

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Kernel.DoG
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Kernel.LoG
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Kernel.Laplacian
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Kernel.gabor
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Kernel.Gabor
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Kernel.moffat
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```
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‎src/gaborkernels.jl

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module GaborKernels
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export Gabor
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"""
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Gabor(kernel_size; kwargs...)
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Gabor(kernel_axes; kwargs...)
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Generates the 2-D Gabor kernel in frequency space.
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# Arguments
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- `kernel_size::Dims{2}`: the Gabor kernel size. The axes at each dimension will be `-r:r`
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if the size is odd.
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- `kernel_axes::NTuple{2, <:AbstractUnitRange}`: the axes of the Gabor kernel.
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# Keywords
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## `wavelength::Real`(λ>=2)
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This is the wavelength of the cosine factor of the Gabor filter kernel and herewith the
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preferred wavelength of this filter. Its value is specified in pixels.
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The value `λ=2` should not be used in combination with phase offset `ψ=-π/2` or `ψ=π/2`
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because in these cases the Gabor function is sampled in its zero crossings.
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In order to prevent the occurence of undesired effects at the image borders, the wavelength
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value should be smaller than one fifth of the input image size.
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## `orientation::Real`(θ∈[0, 2pi]):
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This parameter specifies the orientation of the normal to the parallel stripes of a Gabor
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function [3].
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## `bandwidth::Real` (b>0)
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The half-response spatial frequency bandwidth b (in octaves) of a Gabor filter is related to
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the ratio σ / λ, where σ and λ are the standard deviation of the Gaussian factor of the
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Gabor function and the preferred wavelength, respectively, as follows:
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```math
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b = \\log_2\\frac{\\frac{\\sigma}{\\lambda}\\pi + \\sqrt{\\frac{\\ln 2}{2}}}{\\frac{\\sigma}{\\lambda}\\pi - \\sqrt{\\frac{\\ln 2}{2}}}
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```
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## `aspect_ratio`(γ=0.5)
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This parameter, called more precisely the spatial aspect ratio, specifies the ellipticity of
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the support of the Gabor function [3]. For `γ = 1`, the support is circular. For γ < 1 the
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support is elongated in orientation of the parallel stripes of the function.
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## `phase_offset` (ψ∈[-π, π])
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The values `0` and `π` correspond to center-symmetric 'center-on' and 'center-off'
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functions, respectively, while -π/2 and π/2 correspond to anti-symmetric functions. All
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other cases correspond to asymmetric functions.
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# Examples
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To apply gabor filter `g` on image `X`, one need to use convolution theorem, i.e., `conv(A,
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K) == ifft(fft(A) .* fft(K))`. Because this `Gabor` function generates Gabor kernel directly
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in frequency space, we don't need to apply `fft(K)` here:
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```jldoctest
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julia> using ImageFiltering, FFTW, TestImages, ImageCore
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julia> img = TestImages.shepp_logan(256);
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julia> kern = Kernel.Gabor(size(img); wavelength=30, orientation=0, bandwidth=2, phase_offset=0, aspect_ratio=0.5);
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julia> g_img = ifft(centered(fft(channelview(img))) .* fftshift(kern)); # apply convolution theorem
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julia> @. Gray(abs(g_img));
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```
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See the [gabor filter demo](@ref demo_gabor_filter) for more examples.
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# Extended help
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Mathematically, gabor filter is defined in frequency space:
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```math
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g(x, y) = \\exp(-\\frac{x'^2 + \\gamma^2y'^2}{2\\sigma^2})\\exp(i(2\\pi\\frac{x'}{\\lambda} + \\psi))
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```
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where ``x' = x\\cos\\theta + y\\sin\\theta`` and ``y' = -x\\sin\\theta + y\\cos\\theta``.
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# References
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- [1] [Wikipedia: Gabor filter](https://en.wikipedia.org/wiki/Gabor_filter)
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- [2] [Wikipedia: Gabor transformation](https://en.wikipedia.org/wiki/Gabor_transform)
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- [3] [Wikipedia: Gabor atom](https://en.wikipedia.org/wiki/Gabor_atom)
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- [4] [Gabor filter for image processing and computer vision](http://matlabserver.cs.rug.nl/edgedetectionweb/web/edgedetection_params.html)
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"""
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struct Gabor{T<:Complex, TP<:Real, R<:AbstractUnitRange} <: AbstractMatrix{T}
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ax::Tuple{R,R}
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λ::TP
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θ::TP
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ψ::TP
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σ::TP
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γ::TP
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# cache values
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sc::Tuple{TP, TP} # sincos(θ)
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λ_scaled::TP # 2π/λ
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function Gabor{T,TP,R}(ax::Tuple{R,R}, λ::TP, θ::TP, ψ::TP, σ::TP, γ::TP) where {T,TP,R}
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λ > 0 || throw(ArgumentError("`λ` should be positive: $λ"))
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new{T,TP,R}(ax, λ, θ, ψ, σ, γ, sincos(θ), 2π/λ)
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end
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end
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function Gabor(
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size_or_axes::Tuple;
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wavelength::Real,
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orientation::Real,
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bandwidth::Real,
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phase_offset::Real,
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aspect_ratio::Real=0.5,
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)
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bandwidth > 0 || throw(ArgumentError("bandwidth should be positive: $bandwidth"))
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wavelength >= 2 || @warn "wavelength should be equal to or greater than 2" wavelength
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# we still follow the math symbols in the implementation
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# orientation: Julia follow column-major order, thus here we compensate the orientation by π/2
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λ, θ, γ, ψ = wavelength, mod2pi(orientation+π/2), aspect_ratio, phase_offset
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σ = (λ/π)*sqrt(0.5log(2)) * (2^bandwidth+1)/(2^bandwidth-1)
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params = float.(promote(λ, θ, ψ, σ, γ))
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T = typeof(params[1])
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ax = _to_axes(size_or_axes)
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all(r->λ .<= length(r)./5, ax) || @warn "wavelength `λ=$λ` is too large for image of size $(map(length, ax))"
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Gabor{Complex{T}, T, typeof(first(ax))}(ax, params...)
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end
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@inline Base.size(kern::Gabor) = map(length, kern.ax)
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@inline Base.axes(kern::Gabor) = kern.ax
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@inline function Base.getindex(kern::Gabor, x::Int, y::Int)
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ψ, σ, γ = kern.ψ, kern.σ, kern.γ
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s, c = kern.sc # sincos(θ)
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λ_scaled = kern.λ_scaled # 2π/λ
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xr = x*c + y*s
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yr = -x*s + y*c
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yr = γ * yr
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return exp(-(xr^2 + yr^2)/(2σ^2)) * cis(xr*λ_scaled + ψ)
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end
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# Utils
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function _to_axes(sz::Dims)
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map(sz) do n
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r=n÷2
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isodd(n) ? UnitRange(-r, n-r-1) : UnitRange(-r+1, n-r)
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end
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end
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_to_axes(ax::NTuple{N, T}) where {N, T<:AbstractUnitRange} = ax
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end

‎src/kernel.jl

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- `DoG` (Difference-of-Gaussian)
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- `LoG` (Laplacian-of-Gaussian)
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- `Laplacian`
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- `gabor`
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- `Gabor`
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- `moffat`
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See also: [`KernelFactors`](@ref).
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using ..ImageFiltering.KernelFactors
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import ..ImageFiltering: _reshape, IdentityUnitRange
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include("gaborkernels.jl")
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using .GaborKernels
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# We would like to do `using ..ImageFiltering.imgradients` so that that
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# Documenter.jl (the documentation system) can parse a reference such as `See
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# also: [`ImageFiltering.imgradients`](@ref)`. However, imgradients is not yet

‎test/runtests.jl

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include("mapwindow.jl")
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include("extrema.jl")
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include("basic.jl")
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include("gabor.jl")
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# include("gabor.jl")
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include("models.jl")
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