🌀 I Tried to Explain a Picture That Contains Its Own World in
German

I thought describing an image in German would be easy. I had described
photographs before. There is a person. There is a building. Something is in
the foreground. Something else is in the background. If necessary, I can
become extremely adventurous and mention that an object is „links
im Bild“.

Then a colleague showed me an image that seemed to contain another
version of its own scene.

That immediately destroyed my comfortable little vocabulary plan.

The picture was not simply showing an object. Its geometry appeared to
bend into itself. A region of the scene seemed connected to another
representation of the same visual world, creating the impression that the
image was somehow referring back to itself.

I stared at it for a while and produced my first highly technical German
analysis: „Das Bild ist sehr seltsam.“

It was not wrong.

It was also not going to carry me very far through a conversation about
recursive geometry and generative models.

I started with a more useful expression: „Das Bild verweist auf
sich selbst.“ The image refers to itself.

That gave me a simple way to describe self-reference without immediately
trying to explain the mathematics behind it. At an early language level,
that was enough. I could say that the image looked unusual, that one part
appeared inside another part, and that the scene seemed to repeat
itself.

But the more closely I looked, the less accurate the word “repeat”
became.

This was not ordinary repetition like copying the same photograph three
times. The structure had been transformed. Positions changed. Scale
changed. Straight relationships could become curved ones. The scene was
recognizable, but its geometry followed a different arrangement.

I learned to say, „Die Geometrie des Bildes wurde
transformiert.“

That sentence was already much better than “very strange.” It told my
colleague that I was thinking about the spatial structure of the image
rather than merely its appearance.

Then the mathematics arrived.

The discussion involved a mapping between an ordinary source
representation and a distorted representation. In a simpler picture-editing
workflow, I might create an image first and transform it afterward. But
that can produce a problem: structures that should meet naturally in the
transformed geometry may no longer connect convincingly.

I needed a German sentence for that too: „Eine nachträgliche
Transformation kann die Verbindungen im Bild beschädigen.“

That idea became surprisingly important.

If I distort a finished scene only after generation, I am asking a
geometric operation to reorganize content that was never created with that
final geometry in mind. Lines can stop meeting properly. Objects can
stretch in awkward ways. Boundaries can become visually inconsistent.

My first instinct was obvious: why not apply the transformation while
the image is being generated?

Unfortunately, generative models have their own opinions.

A denoising model is trained to turn noisy intermediate representations
into increasingly plausible images. If I impose an unusual geometric
distortion during that process, the denoiser may interpret the distortion
as something that needs to be corrected.

I found the German phrase „Das Modell versucht, die Verzerrung
zu korrigieren.“ extremely useful.

The funny part was that the model could be doing exactly what it had
learned to do well while simultaneously destroying the structure I
wanted.

I was not asking it to repair the geometry.

I was asking it to respect the geometry.

That distinction moved my vocabulary into much more precise territory.
Instead of talking only about a transformation, I needed to talk about a
„geometrische Nebenbedingung“—a geometric constraint.

The generated image should not merely look attractive. It should remain
compatible with a prescribed spatial structure.

At an intermediate level, I could explain the problem like this:
„Das Modell soll ein plausibles Bild erzeugen, ohne die
vorgeschriebene Geometrie zu verlassen.“

That sentence captured the tension surprisingly well. There were really
two objectives. The image needed to look visually coherent, but it also
needed to remain geometrically admissible.

Then my colleague introduced another complication: the transformation
was not necessarily invertible.

I knew the word „umkehrbar“, but suddenly I needed its
more technical counterpart: „invertierbar“.

If a transformation is invertible, I can conceptually move from one
representation to another and then recover the original. A non-invertible
transformation is more difficult because some information can be merged,
constrained, or lost.

So simply saying “apply the inverse” was no longer enough.

We needed something more general.

That was how „verallgemeinerte Inverse“ entered my
German vocabulary.

I practiced the sentence „Für die nicht invertierbare
Transformation benötigen wir eine verallgemeinerte Inverse.“

This was the moment when I realized my German study session had moved a
considerable distance from ordering coffee.

The generalized inverse was useful because it provided a way to move
information back toward the source representation even when an ordinary
inverse was unavailable. More importantly, it could be designed around the
recursive constraint of the transformation rather than treated as a generic
image-editing trick.

At B2, I wanted to explain why that mattered rather than merely name the
operation.

I could say, „Die verallgemeinerte Inverse rekonstruiert eine
geeignete Darstellung im Quellraum.“

The word „Quellraum“ became central to the entire
discussion.

I began thinking of the process as movement between two related spaces.
In the source space, the scene could develop in a comparatively ordinary,
untwisted representation. In the transformed space, the image could be
evaluated and refined according to the unusual final geometry.

This was much easier for me to understand than imagining one model
desperately trying to generate a perfect recursive image in a single
representation.

The process could alternate.

A few steps could improve the source representation. Then the
transformation could carry the current result into the target geometry.
Additional denoising could improve appearance and local connections there.
A generalized inverse could then return useful information toward the
source representation.

I learned the sentence „Die Verarbeitung wechselt zwischen
Quellraum und transformiertem Raum.“

That one sentence contained the central intuition I had been missing.

The source and transformed representations were not competing versions
of the image. They had different responsibilities.

The source space made it easier to develop recognizable content. The
transformed space made it possible to refine the image where the recursive
geometry actually mattered.

By C1, I could describe this as an interleaved process rather than a
simple sequence.

„Die Entrauschungsschritte werden mit der Transformation und
ihrer verallgemeinerten Inversen verschränkt.“

I liked the word „verschränkt“ here because it
communicated that the operations were deliberately interwoven. We were not
finishing one entire process and then beginning another. Scene formation
and geometric enforcement developed together.

That distinction also changed how I thought about image generation.

A conventional post-processing workflow says: first create the picture,
then distort it.

The more interesting approach says: allow the picture and the distortion
to influence one another while the picture is still forming.

That sounds like a small procedural difference until you consider what
happens to structures crossing transformed regions. If two parts of the
image are supposed to meet under a recursive mapping, waiting until the end
may be too late. Their visual relationship needs to develop under the
constraint.

I could now explain this in German: „Die Szene und ihre
geometrische Verzerrung entwickeln sich gemeinsam.“

Then came the word „Projektion“.

In ordinary conversation, projection might make me think of a
presentation screen. Here it meant something much more mathematical: an
operation that maps a candidate representation back toward the set of
images satisfying the geometric constraint.

An especially useful property was idempotence.

I had definitely not expected „idempotent“ to become
part of my German vocabulary that day.

An idempotent projection has the useful property that applying the
projection again does not continue changing an image that is already in the
projected set. Informally, once the representation satisfies the relevant
constraint, projecting it again should leave it there.

I practiced: „Die Projektion ist idempotent.“

Short sentence. Considerably less short explanation.

But even projection did not solve everything.

A geometrically admissible image is not automatically a natural image
from the perspective of a diffusion model. If I force the denoiser to
operate only on heavily transformed representations, I may be asking it to
process inputs that differ substantially from the distribution it
encountered during training.

That introduced another wonderfully compact technical expression:
„außerhalb der Trainingsverteilung“.

I could say, „Die transformierte Darstellung kann außerhalb der
Trainingsverteilung des Modells liegen.“

This helped explain why enforcing the geometry and obtaining good
denoising were not the same problem.

A projection could enforce a mathematical constraint while the denoiser
still struggled with an unfamiliar representation.

That was why alternating between spaces was so appealing. The model
could spend part of the process working in a representation closer to
ordinary images, while other steps maintained and refined the unusual
target geometry.

At C2, the conversation became less about translating individual
technical words and more about expressing relationships between them
precisely.

I wanted to distinguish an inverse from a generalized inverse,
plausibility from admissibility, transformation from projection, and
ordinary denoising from denoising under a recursive constraint.

I could say, „Die geometrische Zulässigkeit allein gewährleistet
noch keine verteilungsnahe Eingabe für den Entrauscher.“

That sentence would have been completely inaccessible to me at A1. But
the underlying idea was not mysterious anymore.

Something can satisfy the geometry and still look statistically
unfamiliar to the model.

I could also explain the role of alternating representations more
precisely: „Quellraum-Schritte stabilisieren die untransformierte
Szenenstruktur, während Schritte im Zielraum die Kohärenz innerhalb der
vorgeschriebenen Geometrie verfeinern.“

At that point, I was no longer merely describing a strange picture. I
was describing a computational strategy.

That was the most interesting part of the entire language exercise.

At A1, I needed words for picture, shape, strange, inside, and
repeat.

At A2, I could describe a transformation.

At B1, I could explain that a model might unintentionally correct an
intended distortion.

At B2, I could discuss non-invertibility, generalized inverses, and
movement between representations.

At C1, I could explain projection, denoising, constraints, and
interleaved processing.

At C2, I could distinguish mathematical admissibility from statistical
familiarity and explain why the source scene and transformed geometry might
need to evolve together.

The image had not become less peculiar.

I had simply acquired more precise ways to explain why it was
peculiar.

And somewhere between „Das Bild ist seltsam“ and
„geometrisch zulässige Projektion“, my German vocabulary
apparently wandered into a machine-learning laboratory and decided to stay
there.