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During the preparation of a general audience talk on why mathematicians use dimensions higher than three (or four) even for concrete applications, I came up with the following enjoyable observation : Taking the negative of a picture is a central reflexion of the hypercube. I'm pretty sure many people already observed this, so my question is :

Are you aware of other similar transformations for "everyday life high dimensional objects" which have a nice geometric interpretation?

To provide more details, consider a picture of $240\times 240$ pixels, where each pixel has a grey level ranging from $0$ to $255$. Each grey level can be written as a $8$ digit number in base $2$ (and now your computer understands what grey level means). With this identification, this picture is a vertex of the $$ 240\times240\times8=460800\mbox{-dimensional hypercube: }\quad [0,1]^{ 460800}.$$ Now, the negative of a picture is the picture you obtain by reversing the grey scale, meaning mapping a grey level $\alpha$ to $255-\alpha$ for every pixels. In binary this means you switch every zero to one and vice versa. Thus the negative picture, seen as a vertex of the hypercube, is the central reflexion of your initial picture with respect to the center of the hypercube.

I would like to see if one can find other examples of nice geometric interpretations for (colored or not) image transformations, sound effects (i.e. Fourier series of reasonable functions), etc.

enter image description here

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    $\begingroup$ Sorry, but I don't see what is illuminating about this view of "reflection in the hypercube". The simple "255 - image" is perfectly nice for me and the reflection in the hypercube makes things much more complicated without giving any insight (well, these mathematicians surely like it complicated…). If $240^2$ is not high enough for your dimension, just talk about pictures that your smartphone takes which have about a few million pixels or movies which may have a billion voxels… $\endgroup$ – Dirk Jul 3 '17 at 17:44
  • $\begingroup$ @Dirk Identifying numerical data with high dimensional objects is basic science indeed and this is not what my question is about: It is about specific transformations of numerical data which still have a nice geometric interpretation after this identification; notice I didn't say "more simple" but "nice". For instance, It was a surprise to me that reversing an image has again a simple geometric interpretation after this identification. $\endgroup$ – Adrien Hardy Jul 3 '17 at 20:21
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The wall that separates me from my neighbor acts as a low-pass filter. That means I get to listen to the projection of Justin Bieber onto the subspace of low-frequency audio signals.

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  • Changing the tune/pitch of a sound-piece is a scaling/dilation of the Fourier transform.

  • Similarly, zooming of an image is also a scaling/dilation of the (now 2D) Fourier transform.

  • JPEG compression is quantization after a "rotation" of images patches (more precisely: a DCT of 8x8 pixels block) - well plus a number of other tricks that are not that geometrical…

  • Some image denoising techniques rely on the idea of (non-linear) diffusion, e.g. interpreting the gray value as heat and letting the heat diffuse according to some (non-linear, heat- or heat-gradient-dependent) way.

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