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Activating new project at `/tmp/jl_NBHhej` Updating registry at `~/.julia/registries/General.toml` Resolving package versions... 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Singular Value Decomposition
Lorem ipsum SVD est.
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Step 1: upload your favorite image:
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Enable 📸
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missingraw_camera_data
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MethodError: no method matching getindex(::Missing, ::String)
Here is what happened, the most recent locations are first:
- process_raw_camera_data
(raw_camera_data::Missing) from Other cell: line 12# So to get the red values for each pixel, we take every 4th value, starting at# the 1st:reds_flat = UInt8.(raw_camera_data["data"][1:4:end])greens_flat = UInt8.(raw_camera_data["data"][2:4:end])blues_flat = UInt8.(raw_camera_data["data"][3:4:end]) - Show more...
C'est la vie !
img = process_raw_camera_data(raw_camera_data)
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Another cell defining img contains errors.
color .- img
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The camera image can be used as a variable inside Julia! Let's have a look at its type:
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Everything is going to be okay!
typeof(img)
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md"It's a 2D array, and its elements are of the type _$(eltype(img))_"
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Pixels
To get a single pixel from the image, we just need to get a value from the 2D array. This is done with 2D indexing:
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firstpixel = img[1,1]
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typeof(firstpixel)
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To get its values, we use the functions red, green, blue. These are part of the Images.jl package.
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red(firstpixel), green(firstpixel), blue(firstpixel)
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red.(img)
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You can use img_data like any other Julia 2D array! For example, here is the top left corner of your image:
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topleft = let
# the first coordinate is vertical, the second is horizontal (it's a matrix!)
half_height = size(img)[1] ÷ 2
half_width = size(img)[2] ÷ 2
img[1:half_height, 1:half_width]
end
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Step 2: running the SVD
The Julia standard library package LinearAlgebra contains a method to compute the SVD.
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using LinearAlgebra
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bw = Gray.(img)
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📚 = svd(bw);
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Let's look at the result.
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📚
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Let's verify the identity
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bw_reconstructed = 📚.U * Diagonal(📚.S) * 📚.V'
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Are they equal?
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bw == bw_reconstructed
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It looks like they are not equal - how come?
Since we are using a computer, the decomposition and multiplication both introduce some numerical errors. So instead of checking whether the reconstructed matrix is equal to the original, we can check how close they are to each other.
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One way to quantify the distance between two matrices is to look at the point-wise difference. If the sum of all differences is close to 0, the matrices are almost equal.
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UndefVarError: img_data_reconstructed not defined
Here is what happened, the most recent locations are first:
Probably not your fault!
p1_dist = sum(abs.(img_data_reconstructed - img_data))
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There are other ways to compare two matrices, such methods are called matrix norms.
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The 👀-norm
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Another popular matrix norm is the 👀-norm: you turn both matrices into a picture, and use your 👀 to see how close they are:
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UndefVarError: BWImage not defined
Here is what happened, the most recent locations are first:
[BWImage(img_data), BWImage(img_data_reconstructed)]
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How similar are these images?
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missing👀_dist
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In some applications, like image compression, this is the most imporant norm.
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Step 3: compression
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Oh no! 🙀
@bind keep HTML("<input type='range' max='$(length(📚.S))' value='10'>")
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md"Showing the **first $(keep) singular pairs**."
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Gray.(
📚.U[:,1:keep] *
Diagonal(📚.S[1:keep]) *
📚.V'[1:keep,:]
)
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[Gray.(
📚.U[:,1:keep] *
Diagonal(📚.S[1:keep]) *
📚.V'[1:keep,:]
) for keep in 0:20]
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Store fewer bytes
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#compressed_size(keep), uncompressed_size()
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uncompressed_size (generic function with 1 method)function uncompressed_size()
num_el = length(img_data)
return num_el * 8 ÷ 8
end
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compressed_size (generic function with 1 method)function compressed_size(keep)
num_el = (
length(📚.U[:,1:keep]) +
length(📚.S[1:keep]) +
length(📚.V'[1:keep,:])
)
return num_el * 16 ÷ 8
end
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#BWImage(Float16.(F.U)[:,1:keep] * Diagonal(Float16.(F.S[1:keep])) * Float16.(F.V)'[1:keep,:])
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JPEG works in a similar way
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Individual pairs
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@bind pair_index HTML("<input type='range' min='1' max='$(length(📚.S))' value='10'>")
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You got this!
Gray.(normalize_mat((
📚.U[:,pair_index:pair_index] *
Diagonal(📚.S[pair_index:pair_index]) *
📚.V'[pair_index:pair_index,:]
), Inf))
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normalize_mat (generic function with 2 methods)normalize_mat(A, p=2) = A ./ norm(A, p)
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Going further
More stuff to learn about SVD
To keep things simple (and dependency-free), this notebook only works with downscaled black-and-white images that you pick using the button. For color, larger images, or images from your disk, you should look into the Images.jl package!
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Appendix
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Enable 📸
camera_input()
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camera_input (generic function with 1 method)function camera_input(;maxsize=200, default_url="https://i.imgur.com/VGPeJ6s.jpg")
"""
<span class="pl-image">
<style>
.pl-image video {
max-width: 250px;
}
.pl-image prompt {
max-width: 250px;
}
.pl-image video.takepicture {
animation: pictureflash 200ms linear;
}
@keyframes pictureflash {
0% {
filter: grayscale(1.0) contrast(2.0);
}
100% {
filter: grayscale(0.0) contrast(1.0);
}
}
</style>
<div id="video-container" title="Click to take a picture">
<video playsinline autoplay></video>
<div id="prompt">Enable 📸</div>
</div>
<script>
// based on https://github.com/fonsp/printi-static (by the same author)
const span = this.currentScript.parentElement
const video = span.querySelector("video")
const maxsize = $(maxsize)
const send_source = (source, src_width, src_height) => {
const scale = Math.min(1.0, maxsize / src_width, maxsize / src_height)
const width = Math.floor(src_width * scale)
const height = Math.floor(src_height * scale)
const canvas = html`<canvas width=\${width} height=\${height}>`
const ctx = canvas.getContext("2d")
ctx.drawImage(source, 0, 0, width, height)
span.value = {
width: width,
height: height,
data: ctx.getImageData(0, 0, width, height).data,
}
span.dispatchEvent(new CustomEvent("input"))
}
navigator.mediaDevices.getUserMedia({
audio: false,
video: {
facingMode: "environment",
},
}).then(function(stream) {
window.stream = stream
video.srcObject = stream
window.cameraConnected = true
video.controls = false
video.play()
video.controls = false
}).catch(function(error) {
console.log(error)
});
span.querySelector("#video-container").onclick = function() {
const cl = video.classList
cl.remove("takepicture")
void video.offsetHeight
cl.add("takepicture")
video.play()
video.controls = false
console.log(video)
send_source(video, video.videoWidth, video.videoHeight)
};
const img = html`<img crossOrigin="anonymous">`
img.onload = () => {
send_source(img, img.width, img.height)
}
img.src = "$(default_url)"
</script>
</span>
""" |> HTML
end
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process_raw_camera_data (generic function with 1 method)function process_raw_camera_data(raw_camera_data)
# the raw image data is a long byte array, we need to transform it into something
# more "Julian" - something with more _structure_.
# The encoding of the raw byte stream is:
# every 4 bytes is a single pixel
# every pixel has 4 values: Red, Green, Blue, Alpha
# (we ignore alpha for this notebook)
# So to get the red values for each pixel, we take every 4th value, starting at
# the 1st:
reds_flat = UInt8.(raw_camera_data["data"][1:4:end])
greens_flat = UInt8.(raw_camera_data["data"][2:4:end])
blues_flat = UInt8.(raw_camera_data["data"][3:4:end])
# but these are still 1-dimensional arrays, nicknamed 'flat' arrays
# We will 'reshape' this into 2D arrays:
width = raw_camera_data["width"]
height = raw_camera_data["height"]
# shuffle and flip to get it in the right shape
reds = reshape(reds_flat, (width, height))' / 255.0
greens = reshape(greens_flat, (width, height))' / 255.0
blues = reshape(blues_flat, (width, height))' / 255.0
# we have our 2D array for each color
# Let's create a single 2D array, where each value contains the R, G and B value of
# that pixel
RGB.(reds, greens, blues)
end
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