Expanding Dimensions

Bond Reviews the Bond Stories

After knocking on Bond’s door and receiving a slurred permission, I entered and froze on the doorstep. James was deeply absorbed in reading a few pages when I realized, with a jolt, that they were my own stories about him.

“Ah, that’s you, Mr. Conan-Doyle-Malevich?” he said. “Come in, come in! I am just reviewing your opuses.”

Nothing encouraging was written on his face. I felt deeply embarrassed. However, as an experienced reviewer, James began with compliments.

“I appreciate very much that you hid my real name under a pseudonym, James Bond, — a quite ordinary English name that tells people nothing. Although, perhaps, John Smith would have been even better. It is commendable that you used a senseless abbreviation, namely MI-6, to refer to my agency. Its real name is, of course, top secret. But you chose too short an abbreviation. It may eventually coincide with the name of some real company, and then you would be in trouble. You should rather use a generator of random keys and get something like…” — he clicked a few buttons on his device — “for instance, KGBfbiN@S@.”

“I tried to do my best…” I mumbled.

“You did, I see. But why do you always portray me with such an idiotic face?” he winced. “And honestly, I find ambiguous statements and inaccurate formulations on every page.” He looked at me very skeptically. “Taking into account your very moderate educational background, this is probably excusable.”

I uttered not a word.

“However, I cannot tolerate,” Bond continued, “when you distort the very meaning of my words. For instance, look at our conversation in the bar some years ago ( Squeezing-dimensions). Was it not clear that I meant extending the dimensionality, not squeezing it? You ignored that and used my words to promote your own ideas about automatic reduction of dimensionality in PCA…”

Patch-based denoising of 2D Images

As always when receiving reviews on my papers, I experienced a sober mixture of shame and rage. Yet I had to admit that James’s story about finding a way out of a labyrinth ( Squeezing-dimensions) indeed implied the introduction of an extra dimension, not the reduction of existing ones.

I began to guess where and how Bond might have applied such a strategy and soon found an example. Remember how we denoised 3D spectrum images by finding correlations along their third (spectral) dimension? Then PCA removed much of the noise by clipping components in that dimension. Could we design something similar for ordinary 2D images?

James taught that we must create an extra dimension. For instance, suppose there is a certain correlation among neighbouring pixels in an image. Then, from each pixel, we construct a third dimension consisting of its neighbours. The size of this new dimension depends on how large a neighbourhood we consider.

Does this sound like an unnecessary complication? But if the neighbourhood is correlated, PCA is able to capture the correlation and truncate the third dimension down to the strictly necessary size. When we fold this cube back into a 2D shape, a great deal of random noise disappears. This is called patch-based PCA.

I think I now understand better what James meant when he said that he found a way out of the labyrinth by looking from an extra dimension. Make geometry more complex in order to see a simpler solution! This principle is used in many data science methods, for instance, in Support Vector Machines.

The Python codes can be found in the pdf version of this document: Full Text with Codes.

If you have any comments or suggestions, please email pavel@temdm.com ".

Posted January 2, 2026

Steve :

(received via Email, January 7, 2026)

Denoise looks not significant. I can do same by simple median filering.

Pavel: .

You are fully right. Denoising is not as straightforward as that for spectrum-images. However, this example is only an illistration of idea. In reality, patch PCA requires much more efforts like sorting, grouping and optimization of patches which brings us to the domain of dictionary learning. When doing everything correctly, we denoise 2D images without loss of resolution, thus outperform any filtering.