JuliaImages / JuliaImages/ImageFiltering.jl
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Description
Please see below how the help for imfilter is displayed, I don't think the formatting characters are supposed to show as such?
It might be related to Julia rather than ImageFiltering?
julia> versioninfo()
Julia Version 1.4.2
Commit 44fa15b150* (2020-05-23 18:35 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: Intel(R) Xeon(R) E-2186M CPU @ 2.90GHz
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-8.0.1 (ORCJIT, skylake)
Environment:
JULIA_NUM_THREADS = 6
julia> using ImageFiltering
help?> imfilter
search: imfilter imfilter! ImageFiltering
imfilter([T], img, kernel, [border="replicate"], [alg]) --> imgfilt
imfilter([r], img, kernel, [border="replicate"], [alg]) --> imgfilt
imfilter(r, T, img, kernel, [border="replicate"], [alg]) --> imgfilt
Filter a one, two or multidimensional array img with a kernel by computing
their correlation.
Details
≡≡≡≡≡≡≡≡≡
The term filtering emerges in the context of a Fourier transformation of an
image, which maps an image from its canonical spatial domain to its
concomitant frequency domain. Manipulating an image in the frequency domain
amounts to retaining or discarding particular frequency components—a process
analogous to sifting or filtering [1]. Because the Fourier transform
establishes a link between the spatial and frequency representation of an
image, one can interpret various image manipulations in the spatial domain
as filtering operations which accept or reject specific frequencies.
The phrase spatial filtering is often used to emphasise that an operation
is, at least conceptually, devised in the context of the spatial domain of
an image. One further distinguishes between linear and non-linear spatial
filtering. A filter is called linear if the operation performed on the
pixels is linear, and is labeled non-linear otherwise.
An image filter can be represented by a function
w: \{s\in \mathbb{Z} \mid -k_1 \le s \le k_1 \} \times \{t \in \mathbb{Z} \mid -k_2 \le t \le k_2 \} \rightarrow \mathbb{R},
where k_i \in \mathbb{N} (i = 1,2). It is common to define k_1 = 2a+1 and
k_2 = 2b + 1, where a and b are integers, which ensures that the filter
dimensions are of odd size. Typically, k_1 equals k_2 and so, dropping the
subscripts, one speaks of a k \times k filter. Since the domain of the
filter represents a grid of spatial coordinates, the filter is often called
a mask and is visualized as a grid. For example, a 3 \times 3 mask can be
potrayed as follows:
\scriptsize
\begin{matrix}
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(-1,-1) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
&
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(-1,0) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
&
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(-1,1) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
\\
\\
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(0,-1) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
&
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(0,0) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
&
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(0,1) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
\\
\\
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(1,-1) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
&
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(1,0) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
&
\boxed{
\begin{matrix}
\phantom{w(-9,-9)} \\
w(1,1) \\
\phantom{w(-9,-9)} \\
\end{matrix}
}
\end{matrix}.
The values of w(s,t) are referred to as filter coefficients.
Discrete convolution versus correlation
=========================================
There are two fundamental and closely related operations that one regularly
performs on an image with a filter. The operations are called discrete
correlation and convolution.
The correlation operation, denoted by the symbol \star, is given in two
dimensions by the expression
\begin{aligned}
g(x,y) = w(x,y) \star f(x,y) = \sum_{s = -a}^{a} \sum_{t=-b}^{b} w(s,t) f(x+s, y+t),
\end{aligned}
whereas the comparable convolution operation, denoted by the symbol \ast, is
given in two dimensions by
\begin{aligned}
h(x,y) = w(x,y) \ast f(x,y) = \sum_{s = -a}^{a} \sum_{t=-b}^{b} w(s,t) f(x-s, y-t).
\end{aligned}
Since a digital image is of finite extent, both of these operations are
undefined at the borders of the image. In particular, for an image of size M
\times N, the function f(x \pm s, y \pm t) is only defined for 1 \le x \pm s
\le N and 1 \le y \pm t \le M. In practice one addresses this problem by
artificially expanding the domain of the image. For example, one can pad the
image with zeros. Other padding strategies are possible, and they are
discussed in more detail in the Options section of this documentation.
One-dimensional illustration
==============================
The difference between correlation and convolution is best understood with
recourse to a one-dimensional example adapted from [1]. Suppose that a
filter w:\{-1,0,1\}\rightarrow \mathbb{R} has coefficients
\begin{matrix}
\boxed{1} & \boxed{2} & \boxed{3}
\end{matrix}.
Consider a discrete unit impulse function f: \{x \in \mathbb{Z} \mid 1 \le x
\le 7 \} \rightarrow \{0,1\} that has been padded with zeros. The function
can be visualised as an image
\boxed{
\begin{matrix}
0 & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{1} & \boxed{0} & \boxed{0} & \boxed{0} & 0
\end{matrix}}.
The correlation operation can be interpreted as sliding w along the image
and computing the sum of products at each location. For example,
\begin{matrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
1 & 2 & 3 & & & & & & \\
& 1 & 2 & 3 & & & & & \\
& & 1 & 2 & 3 & & & & \\
& & & 1 & 2 & 3 & & & \\
& & & & 1 & 2 & 3 & & \\
& & & & & 1 & 2 & 3 & \\
& & & & & & 1 & 2 & 3,
\end{matrix}
yields the output g: \{x \in \mathbb{Z} \mid 1 \le x \le 7 \} \rightarrow
\mathbb{R}, which when visualized as a digital image, is equal to
\boxed{
\begin{matrix}
\boxed{0} & \boxed{0} & \boxed{3} & \boxed{2} & \boxed{1} & \boxed{0} & \boxed{0}
\end{matrix}}.
The interpretation of the convolution operation is analogous to correlation,
except that the filter w has been rotated by 180 degrees. In particular,
\begin{matrix}
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
3 & 2 & 1 & & & & & & \\
& 3 & 2 & 1 & & & & & \\
& & 3 & 2 & 1 & & & & \\
& & & 3 & 2 & 1 & & & \\
& & & & 3 & 2 & 1 & & \\
& & & & & 3 & 2 & 1 & \\
& & & & & & 3 & 2 & 1,
\end{matrix}
yields the output h: \{x \in \mathbb{Z} \mid 1 \le x \le 7 \} \rightarrow
\mathbb{R} equal to
\boxed{
\begin{matrix}
\boxed{0} & \boxed{0} & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{0} & \boxed{0}
\end{matrix}}.
Instead of rotating the filter mask, one could instead rotate f and still
obtained the same convolution result. In fact, the conventional notation for
convolution indicates that f is flipped and not w. If w is symmetric, then
convolution and correlation give the same outcome.
Two-dimensional illustration
––––––––––––––––––––––––––––––
For a two-dimensional example, suppose the filter w:\{-1, 0 ,1\} \times
\{-1,0,1\} \rightarrow \mathbb{R} has coefficients
\begin{matrix}
\boxed{1} & \boxed{2} & \boxed{3} \\ \\
\boxed{4} & \boxed{5} & \boxed{6} \\ \\
\boxed{7} & \boxed{8} & \boxed{9}
\end{matrix},
and consider a two-dimensional discrete unit impulse function
f:\{x \in \mathbb{Z} \mid 1 \le x \le 7 \} \times \{y \in \mathbb{Z} \mid 1 \le y \le 7 \}\rightarrow \{ 0,1\}
that has been padded with zeros:
\boxed{
\begin{matrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \\
0 & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & 0 \\ \\
0 & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & 0 \\ \\
0 & \boxed{0} & \boxed{0} & \boxed{1} & \boxed{0} & \boxed{0} & 0 \\ \\
0 & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & 0 \\ \\
0 & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & 0 \\ \\
0 & 0 & 0 & 0 & 0 & 0 & 0
\end{matrix}}.
The correlation operation w(x,y) \star f(x,y) yields the output
\boxed{
\begin{matrix}
\boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} \\ \\
\boxed{0} & \boxed{9} & \boxed{8} & \boxed{7} & \boxed{0} \\ \\
\boxed{0} & \boxed{6} & \boxed{5} & \boxed{4} & \boxed{0} \\ \\
\boxed{0} & \boxed{3} & \boxed{2} & \boxed{1} & \boxed{0} \\ \\
\boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0}
\end{matrix}},
whereas the convolution operation w(x,y) \ast f(x,y) produces
\boxed{
\begin{matrix}
\boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} \\ \\
\boxed{0} & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{0}\\ \\
\boxed{0} & \boxed{4} & \boxed{5} & \boxed{6} & \boxed{0} \\ \\
\boxed{0} & \boxed{7} & \boxed{8} & \boxed{9} & \boxed{0} \\ \\
\boxed{0} & \boxed{0} & \boxed{0} & \boxed{0} & \boxed{0}
\end{matrix}}.
Discrete convolution and correlation as matrix multiplication
===============================================================
Discrete convolution and correlation operations can also be formulated as a
matrix multiplication, where one of the inputs is converted to a Toeplitz
(https://en.wikipedia.org/wiki/Toeplitz_matrix) matrix, and the other is
represented as a column vector. For example, consider a function f:\{x \in
\mathbb{N} \mid 1 \le x \le M \} \rightarrow \mathbb{R} and a filter w: \{s
\in \mathbb{N} \mid -k_1 \le s \le k_1 \} \rightarrow \mathbb{R}. Then the
matrix multiplication
\begin{bmatrix}
w(-k_1) & 0 & \ldots & 0 & 0 \\
\vdots & w(-k_1) & \ldots & \vdots & 0 \\
w(k_1) & \vdots & \ldots & 0 & \vdots \\
0 & w(k_1) & \ldots & w(-k_1) & 0 \\
0 & 0 & \ldots & \vdots & w(-k_1) \\
\vdots & \vdots & \ldots & w(k_1) & \vdots \\
0 & 0 & 0 & 0 & w(k_1)
\end{bmatrix}
\begin{bmatrix}
f(1) \\
f(2) \\
f(3) \\
\vdots \\
f(M)
\end{bmatrix}
is equivalent to the convolution w(s) \ast f(x) assuming that the border of
f(x) has been padded with zeros.
To represent multidimensional convolution as matrix multiplication one
reshapes the multidimensional arrays into column vectors and proceeds in an
analogous manner. Naturally, the result of the matrix multiplication will
need to be reshaped into an appropriate multidimensional array.
Options
≡≡≡≡≡≡≡≡≡
The following subsections describe valid options for the function arguments
in more detail.
Choices for r
===============
You can dispatch to different implementations by passing in a resource r as
defined by the ComputationalResources
(https://github.com/timholy/ComputationalResources.jl) package. For example,
imfilter(ArrayFireLibs(), img, kernel)
would request that the computation be performed on the GPU using the
ArrayFire libraries.
Choices for T
===============
Optionally, you can control the element type of the output image by passing
in a type T as the first argument.
Choices for img
=================
You can specify a one, two or multidimensional array defining your image.
Choices for kernel
====================
The kernel[0,0,..] parameter corresponds to the origin (zero displacement)
of the kernel; you can use centered to place the origin at the array center,
or use the OffsetArrays package to set kernel's indices manually. For
example, to filter with a random centered 3x3 kernel, you could use either
of the following:
kernel = centered(rand(3,3))
kernel = OffsetArray(rand(3,3), -1:1, -1:1)
The kernel parameter can be specified as an array or as a "factored kernel",
a tuple (filt1, filt2, ...) of filters to apply along each axis of the
image. In cases where you know your kernel is separable, this format can
speed processing. Each of these should have the same dimensionality as the
image itself, and be shaped in a manner that indicates the filtering axis,
e.g., a 3x1 filter for filtering the first dimension and a 1x3 filter for
filtering the second dimension. In two dimensions, any kernel passed as a
single matrix is checked for separability; if you want to eliminate that
check, pass the kernel as a single-element tuple, (kernel,).
Choices for border
====================
At the image edge, border is used to specify the padding which will be used
to extrapolate the image beyond its original bounds. As an indicative
example of each option the results of the padding are illustrated on an
image consisting of a row of six pixels which are specified alphabetically:
\boxed{a \, b \, c \, d \, e \, f}. We show the effects of padding only on
the left and right border, but analogous consequences hold for the top and
bottom border.
"replicate" (default)
–––––––––––––––––––––––
The border pixels extend beyond the image boundaries.
\boxed{
\begin{array}{l|c|r}
a\, a\, a\, a & a \, b \, c \, d \, e \, f & f \, f \, f \, f
\end{array}
}
See also: Pad, padarray, Inner, NA and NoPad
"circular"
––––––––––––
The border pixels wrap around. For instance, indexing beyond the left border
returns values starting from the right border.
\boxed{
\begin{array}{l|c|r}
c\, d\, e\, f & a \, b \, c \, d \, e \, f & a \, b \, c \, d
\end{array}
}
See also: Pad, padarray, Inner, NA and NoPad
"reflect"
–––––––––––
The border pixels reflect relative to a position between pixels. That is,
the border pixel is omitted when mirroring.
\boxed{
\begin{array}{l|c|r}
e\, d\, c\, b & a \, b \, c \, d \, e \, f & e \, d \, c \, b
\end{array}
}
See also: Pad, padarray, Inner, NA and NoPad
"symmetric"
–––––––––––––
The border pixels reflect relative to the edge itself.
\boxed{
\begin{array}{l|c|r}
d\, c\, b\, a & a \, b \, c \, d \, e \, f & f \, e \, d \, c
\end{array}
}
See also: Pad, padarray, Inner, NA and NoPad
Fill(m)
–––––––––
The border pixels are filled with a specified value m.
\boxed{
\begin{array}{l|c|r}
m\, m\, m\, m & a \, b \, c \, d \, e \, f & m \, m \, m \, m
\end{array}
}
See also: Pad, padarray, Inner, NA and NoPad
Inner()
–––––––––
Indicate that edges are to be discarded in filtering, only the interior of
the result is to be returned.
See also: Pad, padarray, Inner, NA and NoPad
NA()
––––––
Choose filtering using "NA" (Not Available) boundary conditions. This is
most appropriate for filters that have only positive weights, such as
blurring filters.
See also: Pad, padarray, Inner, NA and NoPad
Choices for alg
=================
The alg parameter allows you to choose the particular algorithm: FIR()
(finite impulse response, aka traditional digital filtering) or FFT()
(Fourier-based filtering). If no choice is specified, one will be chosen
based on the size of the image and kernel in a way that strives to deliver
good performance. Alternatively you can use a custom filter type, like
KernelFactors.IIRGaussian.
Examples
≡≡≡≡≡≡≡≡≡≡
The following subsections highlight some common use cases.
Convolution versus correlation
================================
# Create a two-dimensional discrete unit impulse function.
f = fill(0,(9,9));
f[5,5] = 1;
# Specify a filter coefficient mask and set the center of the mask as the origin.
w = centered([1 2 3; 4 5 6 ; 7 8 9]);
#=
The default operation of `imfilter` is correlation. By reflecting `w` we
compute the convolution of `f` and `w`. `Fill(0,w)` indicates that we wish to
pad the border of `f` with zeros. The amount of padding is automatically
determined by considering the length of w.
=#
correlation = imfilter(f,w,Fill(0,w))
convolution = imfilter(f,reflect(w),Fill(0,w))
Miscellaneous border padding options
======================================
# Example function values f, and filter coefficients w.
f = reshape(1.0:81.0,9,9)
w = centered(reshape(1.0:9.0,3,3))
# You can designate the type of padding by specifying an appropriate string.
imfilter(f,w,"replicate")
imfilter(f,w,"circular")
imfilter(f,w,"symmetric")
imfilter(f,w,"reflect")
# Alternatively, you can explicitly use the Pad type to designate the padding style.
imfilter(f,w,Pad(:replicate))
imfilter(f,w,Pad(:circular))
imfilter(f,w,Pad(:symmetric))
imfilter(f,w,Pad(:reflect))
# If you want to pad with a specific value then use the Fill type.
imfilter(f,w,Fill(0,w))
imfilter(f,w,Fill(1,w))
imfilter(f,w,Fill(-1,w))
#=
Specify 'Inner()' if you want to retrieve the interior sub-array of f for which
the filtering operation is defined without padding.
=#
imfilter(f,w,Inner())
References
≡≡≡≡≡≡≡≡≡≡≡≡
1. R. C. Gonzalez and R. E. Woods. Digital Image Processing (3rd
Edition). Upper Saddle River, NJ, USA: Prentice-Hall, 2006.
See also: imfilter!, centered, padarray, Pad, Fill, Inner,
KernelFactors.IIRGaussian.
julia>
Contributor guide
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First steps
- Read the whole issue, then the project's contributing guide.
- Comment on the issue to say you are picking it up — it saves two people doing the same work.
- Fork the repository and make your change on a branch.
- Open a pull request that references the issue number.
Research direction
Reproduce the imfilter help output using the Julia 1.4.2 environment details in the report. Start with the ImageFiltering imfilter help entry and compare the displayed formatting characters with the source documentation. Done means the help text renders its formatting correctly rather than showing raw markup.
Written by the indexing model from the issue text.
Assessment
- Tech stack
- julia
- Domain
- computer-vision
- Issue type
- Bug
- Difficulty
- 3/5
- Estimated time
- 1-2 days
- Activity status
- Stale
- Clarity
- Mostly clear
- Newbie friendliness
- 35/100