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2026-06-27T22:02:55Z

The Angel in the Withywindle

There is a moment, the old Navigators say, when the Warp does not merely carry a soul but considers it — turns it over like a jeweler turns a stone to the light, and decides, on a whim older than stars, to set it somewhere else entirely.

Sanguinius felt that moment as a falling.

One instant he stood upon the embarkation deck of the Red Tear, gold armor catching the hell-glow of the Siege, his great wings half-mantled, his thoughts fixed upon a single black ship hanging in the ruined sky — the Vengeful Spirit, and the brother who waited aboard it. He had spoken the words of leaving. He had stepped toward the teleportarium, toward the death he had already seen in a hundred dreams, walking into it the way a man walks into cold water, unflinching because flinching changes nothing.

Then the deck was gone.

The roar of the bombardment was gone. The screaming of ten thousand vox-channels, the prayers, the engines, the grinding of a galaxy tearing itself apart — all of it folded shut like a book.

And there was birdsong.

He stood to his full towering height in a green and golden world, and for the space of three heartbeats the Angel of Baal did not understand it.

A river ran below him, brown and slow and patient, its banks crowded with willows that leaned out over the water as if to whisper to their own reflections. The light was the long honeyed light of a late afternoon that had clearly never heard of war. Reeds nodded. Somewhere a dragonfly stitched the air. The smell — Throne, the smell — was of cut grass and crushed mint and wet earth, and Sanguinius, who had not drawn a breath untainted by smoke or blood or incense in longer than he could remember, found that he had stopped breathing altogether.

His wings were spread wide, primaries trembling, every warrior instinct mapping the terrain for threat. There was none. There was only the valley, drowsing.

“This is a trick,” he said aloud, and his voice — built to carry over the din of planetary assault — came out small here, swallowed by the soft green hush. The Warp counterfeited paradise as easily as it counterfeited terror. He knew the shape of a lie. He had been lied to by the most beautiful things in creation.

And yet the grass did not melt into worms. The river did not run red. The willows simply went on being willows.

Then he heard the singing.

It came up the valley path with a great deal of confidence and no melody whatsoever that Sanguinius could name — a tumbling, nonsense, rhyming sort of singing, the kind a man makes when he is entirely alone and entirely happy and would not change a single thing about either circumstance:

Hey now! merry-o! down along the meadow, willow-water, weeping-water, river running shallow, who comes a-walking with the sun behind his shoulder? Old Tom knows the small things and the small things know him older!

And around the bend in the path came a man.

He was not a tall man. He was broad and stumping and old in the way that hills are old — which is to say not old at all, merely present, and likely to go on being present after everything taller had fallen down. He wore a blue coat and yellow boots, and a battered hat with a long feather in the band, and his beard was brown and his eyes were a startling, merry blue, and he was carrying a bundle of white waterlilies cradled in one arm as carefully as a man might carry a sleeping child.

He stopped when he saw the Angel. He looked up. And up. And up.

He did not draw a weapon, because he had none. He did not fall to his knees, as primarchs’ subjects fell. He did not scream, as the enemy screamed. He did not even seem especially surprised.

He simply tilted his head, the way a robin considers a worm, and grinned through his beard.

“Well now,” said Tom Bombadil. “There’s a big bird come down in my valley. Big golden bird with a sorrowful face. Whatever’s the matter, my fine feathered fellow? You’ve a look on you like the last leaf in November.”

Sanguinius had been called many things. The Great Angel. The Brightest of the Sons. The Pure One. Demigods had bowed to him. A galaxy had been conquered partly on the strength of the awe his very presence inspired. He could not remember the last being who had addressed him as my fine feathered fellow.

“You can see me,” Sanguinius said slowly. “What I am.”

“I see a great tall fellow with white wings and golden coat and a sword too heavy for the asking,” said Tom cheerfully, settling himself down on a mossy stone and patting the lilies into a neater bundle. “I see you’re far from home and dripping with somewhere-else. The water knew it before you came — went all a-shiver, it did. Things wash up in the Withywindle now and then. Mostly sticks. You’re a fair bit more than a stick.”

“Do you know what I am?”

“Does the river know what a fish is?” Tom shrugged. “It knows it’s wet, and it knows it’s passing through, and that’s enough for the river. You’re wet with battle, fish. Passing through. That much old Tom can see.”

“My name,” said the Angel, “is Sanguinius.”

“Is it!” Tom seemed delighted by this, the way he might be delighted by an interesting beetle. “A grand sloshing red sort of name, all blood and wine in it. Well, that’s your name, and you’re welcome to it, but old Tom won’t be needing it. Names are luggage, fish. You haul them about so other folk know which of them owns you. I’m not owned, so I travel light.” He patted the stone beside him. “Sit, sit. You’re blocking my evening sun, stood up there like a steeple.”

Sanguinius did not sit. Not at first. He stood at the river’s edge and looked down at this small, ridiculous, unafraid old man, and felt a suspicion sharpen in him that had nothing to do with the smell of mint or the patient brown water.

“You are not afraid of me,” he said. “Every living thing is afraid of me, on some level beneath thought. It is built into me. My father made me to be feared, among other things. Beasts flee me. Strong men weep. And you sit on a rock and tell me I am blocking your light.”

“You are, though,” said Tom mildly.

“Why are you not afraid.”

The old man considered this seriously, scratching at his beard. The feather in his hat bobbed. “Reckon it’s because I’m not for sale,” he said at last. “Fear’s a kind of buying and selling, isn’t it? A great big fellow comes along and says be afraid, and the small fellow looks at all he’s got — his life, his little patch, his tomorrow — and reckons up what he stands to lose, and pays it over in fright. But old Tom’s got nothing on the table. My patch was here before the big fellows learned to walk upright, and it’ll be here after. I can’t be made to fear losing what I never grabbed hold of in the first place.” He smiled. “So. Sit down, unsellable to unsellable, and tell me why a creature built for fearing has got such a frightened face of his own.”

And the Angel of Baal, who had refused thrones, sat down on the mossy bank of the Withywindle.

“There is a war,” Sanguinius said.

“There’s always a war, somewhere,” said Tom. “Willows war with the reeds for the riverbank. Badger wars with the fox over the good den. The frost wars with the spring every single year, and loses, every single year, and tries again. War’s just the world disagreeing with itself. What’s yours about, then?”

“It is not like the frost and the spring.” Sanguinius drew a breath. The words came slowly, because he had never had to explain the Heresy — everyone he had ever spoken to already lived inside it. “There was a great work. The greatest work ever attempted. My father set out to unite every world where men live, to throw down the false gods and the tyrants and the things in the dark that feed on us, and to build, in their place, an age of reason. A galaxy lit by mankind alone, beholden to nothing. We were his instruments. His sons. Twenty of us, made in his image, given to him to lead his armies and carry out the work.”

“Twenty sons!” Tom whistled. “That’s a deal of sons. Goldberry and I have got none, and I’ll confess I sleep the better for it. And these twenty — they’re like you? Big golden frightening fellows?”

“None of them is like me,” said Sanguinius, with the faintest ghost of a smile. “We are all of us different. That was the design. One is a hunter, silent as snowfall. One is a scholar who would rather read than rule. One is a brute and proud of it. One is wracked with rage he cannot master. One is the cleverest creature I have ever known, and one the kindest, and one so beautiful and so beloved that we all of us, every brother, would have followed him into any fire.” His voice changed on that last. Tom noticed. Tom noticed everything.

“Ah,” said Tom softly. “And that’s the one, isn’t it. The fire one.”

“His name is Horus.” Sanguinius said it the way a man touches an old wound to see if it still bleeds. It did. “He was the first of us, and the best of us, and our father loved him most, and named him Warmaster — first among the sons, given command of all the rest when our father returned to his great work and left the war to us.” He looked at the river. “And the things in the dark that my father set out to destroy — they are not so easily destroyed. They are old. Older than mankind. They are made of every cruel and hungry and frightened thing that has ever crawled through a living mind, and they have been waiting in the deep currents of the universe since before there were stars to mark the time. We have a name for that deep. We call it the Warp. It is how I came to be here — it threw me out of my own story and into yours, like a wave throwing a fish onto the grass.”

“I know the deep dark you mean,” said Tom, and for the first time the merriness in his eyes went still — not frightened, never that, but grave, the way deep water is grave. “It came knocking here too, fish, long and long ago, in shapes you’d not have a name for. It wanted the same thing it always wants. To be let in. To be agreed with. The dark can’t take a thing that won’t agree to be taken — that’s the whole sorrow and the whole mercy of it. It can only whisper, and wait for the yes.” He looked at the Angel keenly. “Your fire-brother. Your Horus. He said yes.”

Sanguinius closed his eyes. “He said yes.”

The light shifted by a degree. A kingfisher dropped from a willow-branch, took something silver from the river, and was gone in a blue flash. Neither of them watched it.

“Tell me how it was,” said Tom. Not a command. An offering — the way you’d offer to take a heavy basket from someone whose arms were failing.

“They poisoned him,” said Sanguinius. “Wounded him, body and spirit, and while he lay dying the dark offered him a bargain, the way it always does. Power to set the galaxy right as he saw fit. He took it. And then he came for the rest of us, brother by brother, and he asked us each the same question he’d asked himself — will you serve the failing dream of an absent father, or will you take what is owed to you, here, now, with me? And half of us said yes to him. Half. My brothers. Boys I had bled beside for two centuries turned their guns on the worlds we built together, and the galaxy has been burning ever since, and now it has come down to one city, on one world, the world that is my father’s seat — and the black ship hangs over it, and inside the black ship sits the thing that used to be the brother I loved most after my father, and I am going up to it. Alone. To face him.” He opened his eyes. “Because someone must, and because of all the loyal sons I am the one he might still listen to, for the length of a heartbeat, before the end. And because—”

He stopped.

“Because?” said Tom.

“Because I have already seen how it ends,” said Sanguinius, very quietly. “I have the curse of it. Foresight. I dream of things before they come, and they come. And I have dreamed, since I was young, of a golden angel dying in the dark at the hands of his brother. I have dreamed it so many times I know the shape of the wound. I am walking up to that ship to die. I have known it for a hundred years. And I am going anyway, because to turn aside is to let everything and everyone else die in my place, and I would rather it were me.”

For a long while Tom Bombadil said nothing at all, which was, for Tom, a thing as rare as snow in high summer.

Then he reached out, and with one weathered brown hand he plucked a single stem of grass, and he turned it over, and he looked at it as though it were the most interesting thing in three worlds.

“You’ve a great heavy gift, and it’s worked a great cruel trick on you,” he said at last. “Want to know the trick of it, fish?”

“Tell me.”

“You’ve gone and mistaken a dream for a door.” Tom held up the grass-stem. “You dream the ending, and you walk to the ending, and all the long way there you’re already half in the grave, because in your own head you’ve died a hundred deaths before the real one ever comes. You think the dream is a door you’ve got to walk through. But a dream’s not a door, fish. A dream’s a window. You can look through it. It doesn’t make you go.”

“It is fate,” said Sanguinius. “It is certain.”

“Is it!” Tom looked genuinely curious. “And how many of your dreams have come true, then, just exactly as you dreamed?”

“…Most.”

“Most.” Tom pounced on the word with great satisfaction. “Not all. So there’s a crack in your certain fate, fish, a crack the width of most-but-not-all, and you could drive a whole river through a crack like that. I’ve seen it done.” He tossed the grass-stem into the Withywindle, and they both watched it spin away on the brown water. “Here’s a thing old Tom knows that your great father with his great work never learned, for all his cleverness. The dark loves certainty. It loves a fellow who’s sure how the song ends, because a fellow who’s sure how the song ends has already stopped singing. He’s just waiting. And waiting’s the same as agreeing, in the deep dark. Waiting’s a kind of yes.” He fixed the Angel with his bright blue stare. “You’re not walking up to that ship to fight your brother, fish. You stopped fighting a hundred years ago. You’re walking up there to keep an appointment. And that, I’d wager, is exactly what the thing in the dark wants of you.”

Sanguinius was silent. It was the silence of a man who has been struck somewhere he did not know he could be struck.

“You speak as if you know it,” he said finally. “The dark. You speak as if you have stood where I stand.”

“Bless you, no.” Tom laughed, and the gravity broke off him like ice off a thawing branch. “I never stood where you stand. I stood here, which is the whole point of me. The dark came to Tom, in the deep of years — came as a great Rider all in black, came as a will that wanted everything and meant to have it, came carrying a little gold ring that’s bent better and braver folk than you into shadows of themselves.” He waved a hand dismissively. “Terrible thing, that ring. Whole world hangs on it, near enough. They’ll be passing it through my borders before long, I shouldn’t wonder — I can feel that one coming the way you feel rain. And do you know what it’d do to me, that great terrible world-ending ring of power that turns kings into wraiths?”

“What?”

Tom Bombadil grinned his enormous grin. “Nothing. Not a blessed thing. I could put it on my finger this minute and I’d not so much as flicker, because the ring’s whole power is over folk who want. It promises you the thing you’re hungry for. And I’m not hungry, fish. I’ve got my river and my willows and my Goldberry waiting up the hill with the kettle on, and I want for nothing under the sun, so there’s no handle on me for the dark to grab. That’s the secret. That’s the only secret there is. Want nothing it can give you, and it has no power over you at all.”

“I want nothing it can give me,” said Sanguinius. “I have refused it every gift it ever offered.”

“Have you, though?” Tom’s eyes glinted. “You want to be the one who dies so the others live. That’s a noble want, fish, a fine and golden want, the best want I ever heard. But it’s still a want, and the dark’s got hold of it. It’s whispering to you right now, I’d wager, in that grand sorrowful head of yours. Yes, go, die beautifully, die the way you dreamed, it’s already written, it’s already yours. The dark doesn’t always tempt you with the worst of you, fish.” He said it gently, almost tenderly. “Sometimes it tempts you with the best of you. The noble death. The beautiful sacrifice. It’ll take your finest feeling and turn it into a leash, and lead you right up the gangway of that black ship telling you all the way how brave you are.”

Sanguinius found that his hands were trembling. Gold gauntlets, faintly shaking in the grass. “Then what would you have me do?” he said, and there was something raw in it now, something that had not spoken in a very long time. “Refuse to go? Let the city fall? Let my father be torn down and my loyal brothers die unaided so that I might keep myself unleashed? You ask me to want nothing. I am made of want. I want to save them. I cannot put that down.”

“I’d not ask you to put it down.” Tom’s voice was suddenly firm, no nonsense in it at all. “Listen to me now, fish, for old Tom’s about to say a true thing and he doesn’t do it often. I’d not have you turn aside. The dying needs doing, maybe — if a thing’s coming for the folk you love, you stand between, that’s not a leash, that’s just love with its boots on. Go up to your black ship. Face your fire-brother. Do the brave hard thing. I’d not take that from you for all the lilies in the Withywindle.”

He leaned in close, and the whole green valley seemed to lean in with him.

“But go up there wanting to live. You hear me? Not certain you’ll die. Wanting to live. Go up that gangway like a fellow who’s got a kettle on the hill and means to come home to it. Fight him like you’ve got a hundred more suppers in you and you’re cross at him for trying to take them. The dark wants you to walk in already-buried, fish, all calm and golden and certain, because a fellow who’s made his peace with dying is a fellow who’s stopped fighting to live — and that’s the only way that thing ever wins. It can’t beat a creature that wants its next sunrise badly enough. It can only beat one that’s agreed to do without.”

“And if the dream is true?” Sanguinius whispered. “If I want to live with all that I am, and I die anyway?”

“Then you die wanting, and fighting, and alive right up to the last breath,” said Tom Bombadil, “instead of dying like a ghost who quit a hundred years ago and only kept walking out of politeness. Same ending, maybe. Maybe. But oh, fish — what a different song.”

The light was going amber now, sliding toward gold, and the first cool breath of evening came up off the water. Tom hauled himself up off his stone, gathered his lilies, and brushed at his blue coat. He seemed, abruptly, to remember something, and reached into the coat and drew out a heel of brown bread and a small wedge of yellow cheese.

“Here,” he said, holding them up. “You’ll not have eaten in a while, I’ll wager. Big golden fellows always forget to eat. Goldberry’s the same with me — Tom, you old fool, you’ve been out since morning with not a crumb. Eat.”

Sanguinius looked at the bread.

He was a primarch. He did not need it. His body was an engine of war that could run on its own catastrophic chemistry for days. The gesture was, by every measure of strategy and survival he possessed, entirely meaningless.

He took the bread, and he ate it, and it was the best thing he had ever tasted. He told Tom so.

“Course it is,” said Tom comfortably. “Everything is, when you eat it like a thing that wants to go on living. That’s not the bread, fish. That’s you, coming back. Tasting like you mean it. You’ve not done that in a hundred years either, I’d wager.”

They walked a little way along the bank together as the sun lowered, the Angel pacing slow to match the old man’s stump, his great wings folded now and trailing in the long grass.

“I have one more question,” said Sanguinius. “You have answered everything I asked, in your way. Answer this. What are you? In all my father’s galaxy, across a million worlds, I have met gods and daemons and engines that think and men who became more than men. I have never met anything like you. You are immune to a power that bends kings. You have lived longer than the stars. Are you a god? An angel older than I am? Something my father’s truth has no shelf for?”

Tom Bombadil stopped walking. He looked out across his valley — at the willows going dark, at the patient brown water catching the last of the gold, at the first faint star pricking through over the eastern hills. And when he answered, his voice had in it the whole long weight of the years he carried so lightly.

“I’m the eldest, fish,” he said. “That’s the nearest word for it in any tongue I know. I was here before the river found its bed. I knew the first raindrop and the first acorn and the first dark thing that came creeping, and I’ll likely be here pottering about when the last of them’s gone. I’m not a god — gods want, and I’ve told you I don’t. I’m not an angel — angels serve, and I serve nobody. I’m not the dark, and I’m not the light that fights it.” He smiled, and it was the oldest and kindest thing Sanguinius had ever seen on a face. “I’m just the fellow who stayed. Who watched it all go by and didn’t grab at any of it. Master of wood and water and hill — not because I conquered them, mind, but because I love them and want nothing of them but their company. That’s a master worth being, fish. The other kind — the kind your fire-brother’s trying to be, the kind that takes and takes — that kind isn’t master of anything. Not even himself. Specially not himself.”

“My father,” said Sanguinius slowly, “would not have known what to do with you.”

“No,” Tom agreed, with great cheer. “Reckon he wouldn’t. Folk who set out to own everything never do know what to make of a fellow who owns nothing and is happier than they are. Galls them something terrible.” He chuckled. “You tell your father, if you ever see him again — there’s an old fool in a blue coat at the edge of the world who beat the dark without ever raising a hand to it, just by being too content to be tempted. He’ll hate that. Good for him.”

The valley was beginning to fold shut.

Sanguinius felt it before he saw it — the pull at the edges of him, the Warp remembering its appointment, the brown river beginning to run not into the eastern hills but into light, pure and directionless. The willows blurred at their margins. The birdsong thinned. Somewhere very far away, on the other side of a wound in creation, a black ship hung over a burning city, and a thing wearing his brother’s face was waiting with infinite patience for an appointment the Angel had no intention, now, of keeping in the old way.

“It is taking me back,” he said.

“Course it is. You’re a fish, fish. You don’t belong on my grass, however nice it was to have you.” Tom planted his yellow boots and looked up at the towering golden figure already going translucent at the edges, and he did not look sorrowful, and he did not look afraid, because Tom Bombadil had never in his long, long life learned how to do either. “Go on, then. Up to your black ship. Do the brave hard thing.”

“I am still going to die,” said Sanguinius. But it came out differently now. Not as a verdict. As a fact among other facts — and not the largest of them.

“Maybe!” said Tom, with maddening, glorious unconcern. “Everything green does, fish. And comes up green again somewhere, some spring, whether it knows the way or not. So go and do your dying, if it needs doing and you’re the one to do it.” He reached up — up — and patted the Angel’s golden vambrace one last time, the way one might gentle a great horse. “But go up there alive, you hear me? Wanting your next sunrise. Cross as a wet cat that he’s trying to take it from you. Tasting it like the bread. Not as a ghost who quit a hundred years ago. As a fellow who had his supper by the Withywindle and means to have a thousand more. Promise old Tom that much.”

The light was nearly all there was now. Sanguinius was a shape of gold and white dissolving into it, and the last of the green world ran through his fingers like water.

“I promise,” said the Great Angel.

And the very last thing he heard, as the valley let him go and the iron stink of the Siege came roaring back and the deck of the teleportarium reassembled itself cold and hard beneath his golden boots — the last thing, warm and merry and utterly, gloriously without fear, was a singing voice tumbling away down a river that no Imperial chart would ever hold:

Hey! Come merry-o! the sun’s gone under hilltop, Tom’s away to supper now, his singing’ll not stop, Eldest he, and master, and the dark has got no daughter Can put a single shadow on the bright of Tom’s own water!

Aboard the Vengeful Spirit, in the red dark, Sanguinius of the Blood Angels straightened to his full and terrible height. His sword was in his hand. The thing wearing his brother’s face turned to greet him, and began, in his brother’s beloved voice, to speak its lies — the old whisper, dressed in gold this time, the noble death, the beautiful sacrifice, the ending already written, yes, come, it is already yours.

The Angel was smiling.

It was not the smile of a being who did not know how this ended. He knew exactly how it might end. He had always known.

But somewhere on the far side of a wound in the world there was a slow brown river, and an old man in yellow boots who owned nothing and feared nothing and had beaten the dark a thousand times over simply by being too content to be taken — and that old man had given him bread, and told him the secret, and asked him for a promise.

So when the thing in the dark whispered yes, it is already yours, the Great Angel raised his blade, and wanting his next sunrise with every fibre of his golden being, cross as a wet cat that anything should dare try to take it from him, alive right down to the marrow of him for the first time in a hundred years —

— he said no.

And whatever came after, he met it singing.

Constancy of the Speed of Light

2026-06-27T22:02:55Z

The postulate

Special relativity rests on two statements:

The laws of physics take the same form in every inertial frame.
Light propagates in vacuum with the same speed $c$ in every inertial frame, independently of the motion of the source.

The second postulate is the radical one. Galilean intuition says velocities add: a photon emitted from a moving train should travel at $c + v$ . Experiment — beginning with Michelson–Morley in 1887 — says it does not.

Where it comes from

The postulate is not an arbitrary axiom; it is forced by electromagnetism. Maxwell’s equations in vacuum yield the wave equation

\nabla^2 \mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0, \qquad c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}},

with $c$ built out of constants of nature, with no reference to any preferred frame. If the first postulate is to hold for electromagnetism, every inertial observer must measure the same $c$ .

What it costs

Keeping $c$ invariant means abandoning absolute simultaneity. The transformation between inertial frames can no longer be Galilean; it must preserve the quantity

\Delta s^2 = -c^2\,\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2,

the spacetime interval. The linear maps that do so are the Lorentz transformations; for relative velocity $v$ along $x$ ,

t' = \gamma\left(t - \frac{vx}{c^2}\right), \qquad x' = \gamma\,(x - vt), \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.

Time dilation and length contraction are immediate corollaries — not optical illusions but consequences of the geometry of the interval.

The light cone

Because $\Delta s^2 = 0$ characterises light rays in every frame, the light cone through an event is an invariant structure. It partitions spacetime into timelike, null, and spacelike regions, and with it, causality itself becomes frame-independent: no observer can disagree about what may influence what.

Conditional Probability

2026-06-27T22:02:55Z

Talking about probability is talking about assigning a value to some elements of sets, or to some subsets of sets.

A outcome is an element. Outcomes must be mutually exclusive. No two outcomes can happen at the same time. An event is a subset with outcomes as its elements.

Both the value and the sets must fulfill some conditions to allow this assignment to be without contradictions:

The set of all possible outcomes $\Omega$ can be whatever set.
The set of all event $F$ must form a $\sigma$ -algebra.
The probability is a measure that assigns to each event $E\in F$ its probability value $P(E)$ (or $Pr(E)$ )
It is a convention that $P(\Omega)=1$
For general uncountable sets, we need specific $\sigma$ -algebra called the Borel algebra

We assign probabilities to events. A single outcome can be an event.

For all events, their probabilities are non-negative values.

The probability of an event $A$ happening or event $B$ happening (the probability of situation where at least one happens) is the sum of their respective probabilities.

It is important to distinguish between colloquial naming and mathematical definitions.

::: note Example Let $\Omega$ be the set of outcomes:

$A$ and name it: It rains and the grass is wet
$B$ and name it: It rains and the grass is not wet
$C$ and name it: It doesn’t rain and the grass is wet
$D$ and name it: It doesn’t rain the grass is not wet

Let $F$ be the set of events:

It rains: a subset of $A, B$
It doesn’t rain: a subset of $C, D$
The grass is wet: a subset of $A, C$
The grass is not wet: a subset of $B, D$
Plus the four above :::

We are Bayesian statisticians.

We imagine a world where we know it may rain but it doesn’t have to. We also know the grass may get wet but it doesn’t have to.

We think nothing else can happen. Everything except raining and grass getting wet has a zero probability of happening.

Outcomes:

Let $Pr(a)$ be the probability of outcome $a$ called: it rains and the grass is wet.
Let $Pr(b)$ be the probability of outcome $b$ called: it rains and the grass is not wet.
Let $Pr(c)$ be the probability of outcome $c$ called: it doesn’t rain and the grass is wet.
Let $Pr(d)$ be the probability of outcome $d$ called: it doesn’t rains and the grass is not wet.

Events:

Event called: it rains - a subset $E_{\text{rain}} = (a, b)$
Event called: it doesn’t rain - a subset $E_{\text{no rain}} = (c, d)$
Event called: the grass is wet - a subset $E_{\text{grass wet}} = (a, c)$
Event called: the grass is not wet - a subset $E_{\text{grass not wet}} =(b, d)$
Plus the four above

We are tasked with defining this:

“what is the probability that the grass is wet if it is given that it rains”

It would be sensible to ask this question: what is the difference between the ‘probability that the grass is wet and that it rains’ and the ‘probability that the grass is wet given it rains’?

The first case is asking what is the probability of the intersection of events ‘it rains’ and ‘grass is wet’ which is the outcome $A$ with probability $P(a)$ .

The second case is asking what is the probability of an event ‘grass is wet’ when any other outcome can come only from the event ‘it rains’. Meaning, when fixing any event trough the statement: ‘it is given that…’, we create a new space of all possible outcomes - those that can occur only in the event we are taken as given.

In our case, giving ‘it rains’, we limit ourselves only to outcomes in event $E_{rains}$ : $(a, b)$ .

We, however, did not touch the function or parameters of the world that determine the values of probabilities. This means, if no other outcome is possible, we would get $P(a)+P(b)\neq 1$ . The probabilities of $P(a)$ and $P(b)$ must thus be recalculated as those are the only two outcomes in our new universe of outcomes. Let the probability be $P'$ .

The relation ship between the remaining outcomes has not changed. If one was twice as probable as the other one, that ratio must be conserved.

We want to scale the probabilities to achieve the normalization condition.

P'(a) = \alpha P(a) \\ P'(b) = \alpha P(b) \\ \alpha P(a) + \alpha P(a) = 1\\ \alpha = \frac{1}{P(b) + P(b)}

From axiom of probability addition we may call the probability $P(a) + P(b)$ the probability of event ‘it rains’: $P(E_{rains})$ .

The new probabilities are thus $P'(a) = \frac{P(a)}{P(E_{rains})}$ and $P'(b) = \frac{P(b)}{P(E_{rains})}$ .

If we denote $P'(a)\equiv P(a|E_{rains})$ and $P'(b)\equiv P(b|E_{rains})$ we have:

P(a|E_{rains}) = \frac{P(a)}{P(E_{rains})}

P(b|E_{rains}) = \frac{P(b)}{P(E_{rains})}

Which is the definition of Conditional Probability:

P(X \mid Y) = \frac{P(X \cap Y)}{P(Y)}

More straight forward way is this.

Consider a space of possible outcomes $\Omega$ . A probability of a event $A$ occurring is defined (by the way of measure) as

Pr(A) = \frac{|A|}{|\Omega|}

Now, assigning a probability that event (or some event from events) $A$ happened given that event (or some event from events) $B$ happened is saying that some event $C$ happened and $C$ must be a possible event in $A$ and $B$ .

Functional Analysis

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Scope

Notes following the functional analysis lecture course. Core results covered:

Hahn–Banach: every bounded functional on a subspace extends to the whole space without increasing its norm.
Open mapping: a surjective bounded operator between Banach spaces is open.
Uniform boundedness: pointwise-bounded families of operators are norm-bounded.

A first taste

A normed space $X$ is Banach if it is complete. The dual $X^* = B(X, \mathbb{R})$ is always Banach, even when $X$ is not, since

\|f\| = \sup_{\|x\| \le 1} |f(x)|

inherits completeness from $\mathbb{R}$ .

(Lecture-by-lecture notes to be added as separate pages in this section.)

Algebra 1

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Scope

Notes following the functional analysis lecture course. Core results covered:

Hahn–Banach: every bounded functional on a subspace extends to the whole space without increasing its norm.
Open mapping: a surjective bounded operator between Banach spaces is open.
Uniform boundedness: pointwise-bounded families of operators are norm-bounded.

A first taste

A normed space $X$ is Banach if it is complete. The dual $X^* = B(X, \mathbb{R})$ is always Banach, even when $X$ is not, since

\|f\| = \sup_{\|x\| \le 1} |f(x)|

inherits completeness from $\mathbb{R}$ .

(Lecture-by-lecture notes to be added as separate pages in this section.)

Logic

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Propositional logic

Also called zeroth-order logic. To keep the metalanguage of the definition as plain as possible we define propositional logic as:

when we have:

Something to Write

We define symbols.

an alphabet $\Sigma=\mathcal{V}\cup\{\lnot,\land, (, )\}$ where $\mathcal{V}= \{p_1, p_2, \dots\}$ is the set of symbols called propositional variables

Rules How to Write

We define syntax.

a language $\mathcal{L}\subseteq \Sigma^*$ on $\Sigma$ generated by a formal grammar (in BNF)
$\Phi ::= p_1, p_2, \dots \vert\ ( \lnot \Phi ) \ \vert \ ( \Phi \land \Psi )$
we may construct all others $(\lor, \implies, \iff)$ from these.
or equivalently a language $\mathcal{L}\subseteq \Sigma^*$ satisfying:

\frac{}{p \in \mathcal{L}} \ (p \in \mathcal{V}) \quad \frac{\varphi \in \mathcal{L}}{(\lnot \varphi) \in \mathcal{L}} \quad \frac{\varphi \in \mathcal{L}\ \psi \in \mathcal{L}}{(\varphi \land \psi) \in \mathcal{L}}

The language $\mathcal{L}$ is called the language of all well-formed formulas.

Meaning of What We Wrote

We define semantics.

we have two-element set containing True and False
a valuation or truth assignment is any function:
$v:\mathcal{V} \to \{0,1\}$
extending truth function to well-formed formulas:
$l: \mathcal{L} \to \{0,1\}$
by recursion:
$l(\varphi) = \begin{cases} v(\varphi) & \text{if } \varphi \text{ is } p\in\mathcal{V} \\ 1-l(\psi) & \text{if } \varphi \text{ is } (\lnot \psi) \\ \min(l(\psi),\ l(\chi)) & \text{if } \varphi \text{ is } ( \psi\land\chi) \\ \max(l(\psi),\ l(\chi)) & \text{if } \varphi \text{ is } ( \psi\lor\chi) \\ \max(1-l(\psi),\ l(\chi) & \text{if } \varphi \text{ is } (\psi\implies\chi) \\ 1\text{ iff } l(\psi)=l(\varphi)\text{ else }0& \text{if } \varphi \text{ is } (\psi\iff\chi) \end{cases}$

4. Semantic Notions

Let $\Gamma \subseteq \mathcal{L}$ , $\varphi \in \mathcal{L}$ .

Notion	Definition
$v$ satisfies $\varphi$	$\bar{v}(\varphi) = 1$
$v$ satisfies $\Gamma$	$\forall\,\gamma \in \Gamma,\; \bar{v}(\gamma) = 1$
Tautology $\models \varphi$	$\forall\, v : \mathcal{V} \to \{0,1\},\; \bar{v}(\varphi) = 1$
Contradiction	$\forall\, v,\; \bar{v}(\varphi) = 0$
Satisfiable	$\exists\, v,\; \bar{v}(\varphi) = 1$
Semantic consequence $\Gamma \models \varphi$	Every $v$ satisfying $\Gamma$ also satisfies $\varphi$
Logical equivalence $\varphi \equiv \psi$	$\Gamma = \emptyset,\; \{\varphi\} \models \psi$ and $\{\psi\} \models \varphi$

5. Proof System — Hilbert-Style Calculus

5.1 Axiom Schemas (over $\{\neg, \to\}$ )

\begin{align} \mathbf{A1}&: (\varphi \to (\psi \to \varphi)) \\ \mathbf{A2}&: ((\varphi \to (\psi \to \chi)) \to ((\varphi \to \psi) \to (\varphi \to \chi))) \\ \mathbf{A3}&: ((\neg\,\psi \to \neg\,\varphi) \to (\varphi \to \psi)) \end{align}

Any substitution instance of A1–A3 is an axiom.

5.2 Rule of Inference

Modus Ponens (MP):

\frac{\varphi \qquad (\varphi \to \psi)}{\psi}

5.3 Proof and Derivability

A proof of $\varphi$ from $\Gamma$ is a finite sequence $\varphi_1, \varphi_2, \ldots, \varphi_n$ where $\varphi_n = \varphi$ and each $\varphi_i$ is either:

An axiom, or
A member of $\Gamma$ , or
Obtained from earlier $\varphi_j, \varphi_k$ $(j,k < i)$ by MP.

Write $\Gamma \vdash \varphi$ (“ $\varphi$ is syntactically derivable from $\Gamma$ ”).

6. Fundamental Metatheorems

Theorem	Statement
Soundness	$\Gamma \vdash \varphi \;\Rightarrow\; \Gamma \models \varphi$
Completeness (Gödel/Post)	$\Gamma \models \varphi \;\Rightarrow\; \Gamma \vdash \varphi$
Compactness	$\Gamma \models \varphi \;\Rightarrow\; \exists\,\Gamma_0 \subseteq_{\text{fin}} \Gamma,\; \Gamma_0 \models \varphi$
Deduction Theorem	$\Gamma \cup \{\varphi\} \vdash \psi \;\Leftrightarrow\; \Gamma \vdash (\varphi \to \psi)$
Decidability	There exists an algorithm deciding $\models \varphi$ for any $\varphi$ (truth tables; PSPACE-complete in general for $\models$ ; co-NP-complete for $\not\models$ )

Together, Soundness + Completeness give:

\boxed{\Gamma \vdash \varphi \;\iff\; \Gamma \models \varphi}

meaning syntactic provability and semantic truth-in-all-models coincide exactly.

Summary Structure

\text{Propositional Logic} = \underbrace{(\Sigma,\, \mathcal{L})}_{\text{syntax}} + \underbrace{(\{0,1\},\, \bar{v})}_{\text{semantics}} + \underbrace{(\text{Axioms},\, \text{MP})}_{\text{proof system}}

with the guarantee that these three components are perfectly aligned via the Soundness and Completeness theorems.

Vector Fields

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Definition

A vector field on a smooth manifold $M$ is a smooth section $X : M \to TM$ of the tangent bundle: an assignment of a tangent vector $X_p \in T_pM$ to each point, varying smoothly. Equivalently, $X$ is a derivation on $C^\infty(M)$ :

X(fg) = f\,X(g) + g\,X(f).

Flows

A vector field generates a flow: the unique family of diffeomorphisms $\varphi_t$ with

\frac{d}{dt}\Big|_{t=0} \varphi_t(p) = X_p.

Existence and uniqueness of integral curves is the Picard–Lindelöf theorem in coordinates.

The Lie bracket

Two vector fields combine into a third via

[X, Y] = XY - YX,

which measures the failure of their flows to commute. The bracket makes the vector fields on $M$ into an infinite-dimensional Lie algebra, and $[X,Y] = 0$ exactly when the flows commute.

Turing Machines

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The model

A Turing machine is a tuple $(Q, \Gamma, \delta, q_0, F)$ : a finite set of states, a tape alphabet, a transition function

\delta : Q \times \Gamma \to Q \times \Gamma \times \{L, R\},

an initial state, and accepting states. Despite its austerity, the model captures — by the Church–Turing thesis — everything we mean by effective computation.

Undecidability

The halting problem asks: given a machine $M$ and input $w$ , does $M$ halt on $w$ ? Turing’s diagonal argument shows no machine decides this. Suppose $H$ decided halting; build $D$ that, on input $\langle M \rangle$ , runs $H(M, \langle M \rangle)$ and does the opposite. Then

D(\langle D \rangle) \text{ halts} \iff D(\langle D \rangle) \text{ does not halt},

a contradiction. Rice’s theorem generalises: every non-trivial semantic property of programs is undecidable.

An interactive machine

The machine below computes binary increment: state scan walks right to the end of the input, then carry flips trailing 1s to 0s until it can write the carried 1. Try 111 to watch a full carry chain.

State	Read	Write	Move	Next
scan	0 or 1	same	R	scan
scan	␣	␣	L	carry
carry	1	0	L	carry
carry	0 or ␣	1	—	halt

The cost model here (one cell, one step) is what makes statements like those in Asymptotic Notation meaningful.

Vector Spaces

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Axioms

A vector space over a field $k$ is an abelian group $(V, +)$ with a scalar multiplication $k \times V \to V$ satisfying, for all $a, b \in k$ and $u, v \in V$ :

a(u + v) = au + av, \quad (a+b)v = av + bv, \quad (ab)v = a(bv), \quad 1v = v.

Bases and dimension

Every vector space has a basis (a maximal linearly independent set), and any two bases have the same cardinality — the dimension of $V$ . The proof in the infinite-dimensional case requires Zorn’s lemma.

A linear map $T : V \to W$ between finite-dimensional spaces satisfies the rank–nullity theorem:

\dim V = \operatorname{rank} T + \dim \ker T.

Thesis Overview

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Working title

(To be filled in.)

Structure

Introduction and background
Main construction
Results
Outlook

Use this section for chapter drafts, references, and supervisor meeting notes — each as its own markdown file.

Breakdown of the Paper

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Introduction

TODO asddasasd

Introduction

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Introduction

TODO

Volume Reconstruction

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Medical Scans

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(30.10 - )

Medical Scans

This chapter serves as a introduction to the concepts of medical scans. The bread and butter of our thesis as the source of input data.

CT Scans

CT scan : computed tomography scan

In this section we will look at the fundamentals of ct scans. As our primary source of data it is important to get a sense of the historical motivation behind the technology and the science that drove it.

We will briefly cover its beginnings, advances and the current state-of-art solutions that maximize the quality of the resulting scan but at the same time minimize the exposure to radiation. Then, briefly, we will examine the physics and mathematics behind the process of acquiring and processing ct scans so that a full and informative picture can be made as to where our sources come from.

Thoughts

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Thoughts

(30.10.2025)

will contain a dynamically growing collection of thoughts with dates to track mental process and progress throughout the writing

(1.11.2025)

Roadmap

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Roadmap

(PLACEHOLDER MD TEST)

Castle Building Roadmap

Overview

A high-level, phase-driven roadmap for planning and executing a castle construction project. Adapt the timelines, resources and deliverables to your scope (historical reconstruction, fantasy build, real-world fortified manor, etc.).

Vision & Objectives

Primary vision: Build a durable, functional castle that meets safety, aesthetic and program requirements.
Objectives:
- Structural integrity and longevity
- Defensible layout (if applicable)
- Period-appropriate aesthetics / chosen style
- Visitor access and safety
- Budget and schedule adherence

Success Criteria

Structural inspection passed
Key spaces (keep, great hall, gatehouse, towers) functional and approved
Systems (water, drainage, basic utilities) operational
Final budget within agreed variance
Handover documentation and maintenance plan complete

Phases & High-Level Timeline

Concept & Feasibility — 2–6 weeks
Site Survey & Permits — 4–12 weeks (parallel tasks possible)
Schematic & Detailed Design — 8–16 weeks
Site Preparation & Foundations — 4–12 weeks
Superstructure (walls, towers) — 12–36 weeks
Roof, Floors & Openings — 8–20 weeks
Interiors & Systems — 8–20 weeks
Defenses, Landscaping & External Works — 6–16 weeks
Commissioning & Inspections — 2–6 weeks
Handover & Maintenance Setup — 2–4 weeks

(Adjust durations according to scale and resources.)

Example Milestones

M1: Feasibility report approved
M2: Planning permission granted
M3: Foundation completed and inspected
M4: External walls closed / weather-tight
M5: Roof complete
M6: Systems commissioned
M7: Final inspection passed
M8: Handover complete

Each milestone should include acceptance criteria, owner, target date, and required documents.

Roles & Responsibilities

Project Sponsor — funding and strategic decisions
Project Manager — schedule, budget, coordination
Lead Architect / Historian — design and authenticity
Structural Engineer — foundations and load-bearing elements
Site Manager / Contractor — daily site operations
MEP Engineer — utilities and systems
Safety Officer — site safety and compliance
QA / Inspector — quality checks and sign-offs

Resource Plan

Materials: stone/brick, lime mortar, timber, metalwork, roofing materials
Equipment: excavation, scaffolding, lifting equipment
Labor: masons, carpenters, laborers, specialists
Consultants: archaeologist (if needed), conservation specialist, planners
Budget categories: design, permits, materials, labor, contingencies, professional fees

Risk Register (sample entries)

Ground conditions worse than expected — mitigant: pre-construction geotechnical survey, contingency budget
Permit delays — mitigant: early engagement with authorities, buffer in schedule
Material shortages / price increases — mitigant: early procurement, alternate suppliers
Weather impacts — mitigant: phased scheduling, temporary weather protection

Deliverables per Phase

Concept: vision statement, feasibility study, preliminary cost estimate
Design: schematic plans, detailed drawings, structural calculations, specification
Construction: daily logs, inspection reports, material certificates
Commissioning: system test reports, safety certificates
Handover: as-built drawings, O&M manuals, maintenance schedule

Communication Plan

Weekly site report (to stakeholders)
Bi-weekly progress meeting (project team)
Monthly executive summary (sponsor)
Change control process: log changes, impact assessment, formal approval

Acceptance Checklist (before Handover)

Structural certificate obtained
Fire & life-safety checks passed
Utilities functional (water, power where applicable)
Drainage and erosion control verified
Accessibility / paths inspected
As-built drawings and documentation delivered

Maintenance & Lifecycle

Routine inspections (annual)
Masonry repointing schedule (as required)
Roof and drainage maintenance (bi-annual)
Record-keeping for repairs and conservation

Templates & Notes

Milestone template: Title, Owner, Target date, Acceptance criteria, Status
Task template: ID, Description, Assignee, Start/End, Dependencies, Estimate, Status
Change request template: ID, Description, Impact (schedule/budget/scope), Decision

Appendix

Glossary of terms
Reference standards and local regulations
Contact list (authorities, suppliers, consultants)

Customize this roadmap to your project’s scale, style and legal context. Use the milestones and templates to convert the plan into a detailed project schedule and task list.

Motivation

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Motivation

Taking a step back, modern Western medicine can be thought of as the center of mass for all interactions and exchanges of data between researchers and clinicians. As progress is made, this center shifts and so does the paradigm. Clinicians gather data from patients with diseases, subjecting them to treatments synthesized by researchers. Researchers, motivated by the chorus of clinicians signaling the need for cures, or by the insufficient effectiveness of existing treatments, shift their focus and resources toward creating new, improving, or maintaining these protocols.

Looking through a magnifying mirror into this web of data collection and analysis and fixing our focus to a single instance of doctor-patient interaction we can see a certain trend. It seems that the contact between the provider of treatment and the recipient has become a metaphor for measurement. We extract blood or tissue samples; we measure temperature, and analyze the chemical composition of various bodily fluids. As each new iteration of treatment aspires to take action as early as possible, the ability to recognize signals that predict the imminent coming of a disease has become of greatest importance to modeling and executing treatment effectively.

However, humans have long hit their limits in their ability to probe large sets of data while simultaneously performing calculations on them. We have turned to machines that collect, analyze, and output data only some of which we may fully comprehend. It does not appear that this disconnection between the actual corpus of the patient and the hands and eyes of the doctor will slow down. On the contrary, even for medical fields like orthopedics, the physical touch of the doctor and the subsequent communication about the patient’s pains have been, to a large degree, shifted to non-invasive scans and visualizations. One effectively no longer needs to move a knee joint to determine which knee hurts. The need to see a tumor to diagnose a cancer slowly fades in the face of blood analysis.

This thesis builds upon a crucial subtlety within this trend. The acceleration of data processing does not translate to a obsolescence of human perception. While the tactile link may be fading, the visual link is staying and through technology is becoming stronger. Specific domains, namely surgery, such as craniofacial surgery, orthopedics, and a behemoth that is medical education, refuse to distance themselves from the act of looking.

In these fields, “looking with eyes” is not a metaphor, it is a mechanical necessity. A surgeon cannot cut based on a spreadsheet of probabilities. They must see and internalize the spatial relationship of the animal anatomy. An orthopedic specialist must see the fracture within the context of the muscle and tissue. A medical student cannot learn the complexities of the human body through raw data points alone. They require a visual model to bridge the gap between theory and reality.

And as such, as we distance ourselves from the physical patient, the medical scan becomes the proxy for the patient’s body. The burden of reality presentation shifts from the flesh to the pixel. If the doctor is to rely on the screen rather than the touch, the image on that screen must not just be a data visualization. It should aim to be a faithful optical reconstruction of reality.

Literature

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Literature

Primary Literature

Optical properties of biological tissues: a review : https://doi.org/10.1088/0031-9155/58/11/r37

@article{Jacques_2013,
doi = {10.1088/0031-9155/58/11/R37},
url = {https://doi.org/10.1088/0031-9155/58/11/R37},
year = {2013},
month = {may},
publisher = {IOP Publishing},
volume = {58},
number = {11},
pages = {R37},
author = {Jacques, Steven L},
title = {Optical properties of biological tissues: a review},
journal = {Physics in Medicine & Biology},
}

3D Gaussian Splatting for Real-Time Radiance Field Rendering : https://doi.org/10.48550/arXiv.2308.04079

@misc{kerbl20233dgaussiansplattingrealtime,
      title={3D Gaussian Splatting for Real-Time Radiance Field Rendering}, 
      author={Bernhard Kerbl and Georgios Kopanas and Thomas Leimkühler and George Drettakis},
      year={2023},
      eprint={2308.04079},
      archivePrefix={arXiv},
      primaryClass={cs.GR},
      url={https://arxiv.org/abs/2308.04079}, 
}

Real-Time Monte-Carlo Path Tracing of Medical Volume Data : Local Copy

@inproceedings{Engel2016,
  author    = {Engel, Klaus},
  title     = {Real-Time Monte-Carlo Path Tracing of Medical Volume Data},
  booktitle = {Proceedings of the GPU Technology Conference (GTC)},
  year      = {2016},
  address   = {San Jose Convention Center, San Jose, CA, USA},
  month     = {April 4--7},
  note      = {GPU Technology Conference},
}

Physically Based Rendering: From Theory to Implementation : https://www.pbrt.org/

@book{Pharr2023,
  author    = {Pharr, Matt and Jakob, Wenzel and Humphreys, Greg},
  title     = {Physically Based Rendering: From Theory to Implementation},
  edition   = {4th},
  year      = {2023},
  publisher = {MIT Press},
  address   = {Cambridge, MA, USA},
  isbn      = {9780262048026},
  url       = {https://pbrt.org/}
}

Other Literature

(Subject to change)

Direct Volume Rendering

Articles

https://cgl.ethz.ch/teaching/former/scivis_07/Notes/stuff/StuttgartCourse/VIS-Modules-06-Direct_Volume_Rendering.pdf

General Rendering

Books

Physically Based Rendering (PBR),
Real-Time Rendering, CRC

Monte Carlo Path Tracing

Papers

Mathematical Basics of Monte Carlo Rendering Algorithms

3D Gaussian Splatting

https://github.com/MrNeRF/awesome-3D-gaussian-splatting - Curated list of resources on 3DGS

Papers

https://arxiv.org/pdf/2308.04079 - Original 3DGS paper
https://arxiv.org/pdf/2311.16043 - Original Relightable 3D Gaussians
https://arxiv.org/pdf/2311.17977 - 3DGS with shading functions
https://arxiv.org/pdf/2311.16473 - Another Relightable 3DGS
https://arxiv.org/pdf/2507.20512 - Complex 3DGS shadowing
https://arxiv.org/pdf/2412.19282 - Reflective 3DGS

Introduction

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Introduction

Motivation and Overview

The visualization of animal, particularly human, anatomy through non-invasive means is one of the most significant achievements in modern medicine.

Note: Or rather we should say one of the most significant accidental gifts of physics to modern medicine.

Medical scans allow clinicians to understand pathology and enable them to gather knowledge about the internal state of a human body without making a single incision. However, as data acquisition hardware has improved, it has begun producing increasingly complex datasets with higher resolution and the challenge has shifted from the question of how we capture data to how do we display it effectively.

The oldest technique for imaging the internal human body is projectional radiography, commonly known as X-ray. To image a common pathology, such as a broken bone, a subject is placed between a source of electromagnetic radiation and a detector. As radiation passes through the body, it is attenuated by various tissues, losing intensity. This results in a non-uniform intensity distribution on the detecting plane, allowing internal structures and organs to be distinguished.

A different approach, sonography, utilizes sound waves. A transducer emits high-frequency sound waves through contact with the skin. These waves travel into the body, where they are either scattered or reflected. The reflected waves are detected as echoes and algorithmically interpreted. To reconstruct an image, the scanner determines how long did the echo take to return and how strong the echo was. These two variables determine the spatial location and brightness of the corresponding pixels in the resulting visual.

Not all medical scans are equivalent, and the choice which to use depends on the needs of the clinician and on the availability of the given technology. Different scanning techniques produce distinct types of output data.

This thesis focuses on scans that produce spatially complete volumetric data.

Note: Such that it covers the whole volume uniformly.

The availability of 3D spatial information provides the freedom to project the data at arbitrary orientations and scales onto a 2D monitor, or even view it through virtual reality systems. This unlocks the full potential of human visual perception and allows for real-time interaction, creating an ideal environment for both education and clinical analysis.

Specifically, we are interested in those methods, where the body is mapped onto an euclidean space of data points, each containing sufficient information to determine the optical properties of that physical location.

In Computed Tomography (CT), the output data is created by the measurement of X-ray attenuation from multiple angles. The mathematical reconstruction of this data relies on the Radon transform, an integral transform which reconstructs a function from its line integrals.

Note: An integral transform where is some function defined on a plane and is a function defined on a space of line in the plane whose values are defined as the line integral of that function over that line. The reconstruction of the original object is then achieved by taking the inverse of this transform.

This results in a scalar field of attenuation coefficients that must be interpreted by the rendering algorithm.

Magnetic Resonance Imaging (MRI) utilizes nuclear magnetic resonance principles. Here, the output data is derived from the relaxation times of hydrogen nuclei in a magnetic field. The reconstruction of these spatial frequencies into an image is performed via the Fourier transform.

Both techniques result in volumetric grids of datapoints. The challenge of this thesis lies in translating these datasets into a physically based optical model that simulates the behavior of light and subsequently, after defining the lighting conditions, projecting the resulting light transport onto a viewing plane. This requires establishing a correlation between each data point and the tissue it represents. Modern approaches may utilize transformer-based neural networks for this task. Tissues can be classified using the global context of the data point’s position, geometric flow, or state-of-the-art pretrained segmentation models.

Once the biological tissue represented by a data point is identified, we can proceed to model its specific optical properties (Jacques, 2013).

Goals

The primary objective of this thesis is to develop a rendering framework that transforms the raw volumetric data into a visual representation both intuitive for non-professionals and functionally valuable for professionals.

Note: Or at least a prototype of rendering framework. Features such as full user interface are outside of the core research scope of this work. Note: An experienced radiologist might see a fracture where unskilled eye sees only a gray overlapping shapes.

By moving beyond traditional transfer functions toward photorealistic path tracing and splatting, we aim to provide clinicians and educators with a tool that improves spatial comprehension and diagnostic quality. The goal here is to demonstrate that anatomical realism does not have to be sacrificed for performance and specifically, we aim to enable a degree of interactivity previously reserved for lower-fidelity models. This includes dynamic lightning where users can manipulate light sources in real-time allowing for better understanding of depth and surface texture, linear and non-linear transformations, the ability to move, rotate and scale specific anatomical structures and temporal interactivity, where scans containing temporal dimension could be used to animate organs such as heart.

For surgeons and medical students, the benefit of photorealistic visualization lies in the depth of perception and material differentiation. Traditional rendering often results in ‘flat’ or ‘plastic’ appearances that lack both the obvious and subtle anatomical details. By simulating the complex behavior of light, such as subsurface scattering and realistic shadowing, our goal is to produce visualizations that mimic the look and feel of a physical human body dissection. This level of visual fidelity allows for better understanding of the relative distance and orientation of overlapping structures. Together with an UI that could envelop the internal rendering systems, it could provides students with a ‘virtual cadaver’ with realistic lighting, aiding in the transition from reading textbook theory to seeing the operating theater as close to in person as technology allows.

A core goal of this framework is to alow the user to manipulate the digital volume. Rather than viewing a static image, users will be able to interact with the anatomy through:

Segmentation - using neural network-assisted classification to isolate specific organs or tissue layers.
Volumetric editing - enabling the ‘peeling back’ of tissue layers, creating slices through the volume or scaling specific structures to reveal underlying pathology.
Variable transparency - adjusting the opacity of structures (like the skin or musculature) to make them semi-transparent, allowing for a ‘see-through’ effect that visualizes the internal relations within the body.

The nuances of this goal, however, lie in the tradeoff between what is achievable if high-end computing clusters are available and what can be done with a consumer-grade hardware found in clinical institutions and teaching faculties.

Note: This metric will be determined by an indirect survey.

Achieving path-traced realism in real-time requires clever data structures and pruning algorithms to minimize redundant calculations and the waste of available compute power by rendering elements outisde the line of sight. Optimizing how the splatting engine traverses the volumetric grid to ensure that only visible and relevant data points contribute to the final image. Implementing efficient ways to handle high-resolution medical datasets without exceeding the VRAM limits of standard GPUs. Balancing the stochastic character of path tracing with the rasterization speed of Gaussian splatting to maintain high-frame-rate experience without inducing sickness in the user.

Note: A common problem in VR systems with low framerates.

The motivating drive behind the proposal of this thesis is supported by the rapid improvement and increasing accessibility of computing hardware capable of such feats. Over the last several years, the sector has moved beyond large, costly solutions requiring flagship hardware toward self-contained, ergonomic systems. Virtual Reality (VR) still dominates the immersive experience of medical training. It provides a fully controlled environment for risk-free procedural rehearsals with a active proprietary software provider sector. There has also been a significant surge caused by a push for in the development of augmented reality (AR) and mixed reality (MR) devices that allow user to mix the light coming from the real environment with rendered objects. [TODO]

Modern, eyeglass-like wearables, such as smart glasses, are increasingly viewed as the primary global development target for augmented reality. These systems enable surgeons to overlay volumetric data directly onto the surgical field or allowing students to collaborate around a shared holographic specimen in a physical classroom, or potentially, from home. As the industry moves toward miniaturized, lightweight form factors with higher pixel densities and wider fields of view, the demand for rendering frameworks that can deliver photorealistic, low-latency results increases. This thesis addresses this need by optimizing the path-tracing and splatting pipeline specifically for the performance profiles of these emerging miniaturized wearable platforms.

Note: We will research the capabilities of such systems already in market and in development.

Finally, the development of this framework will not occur in a communication vacuum. A significant goal of this research is to ensure the output meets the standards of the medical community. We will be gathering qualitative critiques from clinical professionals to assess the diagnostic utility and accuracy of the rendered images and educational faculties to evaluate the efficacy of the tool as a pedagogical resource for medical training.

This iterative process of consultation and improvement aims to ensure that our rendering optimizations translate into tangible benefits for the clinical field.

As the field of medical visualization has moved from simple cross-sectional slices to complex, interactive 3D environments, achieving the current state of the art requires a clever combination of classical volume rendering, high-end cinematic techniques, and the results of recent explosion of interest in differentiable rendering.

Note: Such that those used in movie making and computer game development.

Historically, the visualization of volumetric grids has relied on two primary methods: isosurface reconstruction and direct volume rendering (DVR).

DVR uses ray marching method to simulate a ray attutenation from arbitrary viewing angle. The rays are extended from the viewing plane according to the desired type of projection (orthographic, perspective) and for each datapoint hit, the resulting dimming of the ray is calculated. Using transfer function to map scalar values to color and opacity, DVR allows for a semi-transparent view of internal structures. This method can be combined with precalculated gradient to simulate basic shading on the normals. While versatile, standard DVR often lacks realistic lighting, resulting in images that struggle to convey depth and material properties properly, acting more like a x-ray with arbitrary user-specifiec rotation and scale.

Isosurface reconstruction, most notably the marching cubes algorithm implementation, extracts a polygonal mesh from a volume at a specific intensity threshold. While efficient for hardware-accelerated rendering, it discards the internal ‘fuzzinnes’ of the data and tends to create choppy, smoothed out plastic-like surfaces.

The current industry gold standard for photorealistic medical visualization is Cinematic Rendering first introduced to mainstream by Siemens Healthineers (niedermayr-2024). In contrast to simplistic lighting models of DVR this model employs monte carlo path tracing to simulate the complex physics of light transport, including global illumination, soft shadows, and ambient occlusion. While Cinematic Rendering results in state-of-the-art visual quality, it is computationally expensive. It continues to struggle to maintain high framerate interactivity on consumer-grade hardware, often requiring specialized workstations or offline rendering.

In the pursuit of real-time performance, various novel view synthesis approaches building on the advances in differentiable rendering have been explored.

Note: Essentially outsourcing the problem of calculating high volumes of linear algebra appearing in ray optics mechanics to linear algebra of machine learning. Note: A rendering pipeline that is differentiable with respect to the parameters of the rendered scene. This allows for a gradient flow from a loss calculated using the ground truth and some scene with flexible enough components back to the scene, adjusting the components to represent the ground truth faithfuly.

Neural radiance fields (Mildenhall, 2020) (NeRFs) introduced a way to represent scenes within neural networks. However, NeRFs often suffer from long training times and slow inference speeds, making them not suitable for real-life performance and the interactive needs of a clinical environment.

The introduction of 3D Gaussian Splatting (3DGS) (Kerbl, 2023) has marked a significant shift in the field, offering the high fidelity of neural methods with the speed of point-based rasterization. Current research is rapidly expanding the capabilities of this approach. Recent advances have introduced relightable Gaussians (Gao, 2023; Jiang, 2023), allowing for dynamic lighting changes by incorporating BRDF-like properties into each splat.

Note: Bidirectional reflectance distribution function describes the way light behaves ones it hits the surface of given texture. It is a core part of physically based rendering mathematics.

Advances in shading functions (Liang, 2023) and reflective properties (Yao, 2024) are moving splatting toward true physically based rendering (PBR). New methods are addressing the challenge of shadows and complex occlusions within splatted scenes (Bai, 2025), which is vital for understanding the spatial depth of anatomical structures. Leveraging these advancements in Gaussian Splatting and combining them with path-tracing principles, this thesis aims to deliver a framework that matches the visual quality of Siemens’ Cinematic Rendering while maintaining the real-time performance necessary for modern medical practice.

Virtual Reality

While 2D monitors are sufficient for standard diagnostics, VR provides a 1:1 spatial mapping that aligns with how humans naturally perceive the physical world. This is particularly important in two key areas.

Traditional medical training often relies on textbooks, 2D atlases, and limited access to cadaver specimens. VR overcomes these problems by providing a ‘digital cadaver’ that is infinitely reusable and free from biological decay. By allowing students to ‘walk through’ a heart or navigate the vascular system at an exaggerated scale, VR enables for a deeper understanding of complex spatial relationships that are often lost in cross-sectional imaging. Students can perform repeated manipulations such as isolating a specific nerve or simulating a surgical pathway, allowing them to learn from mistakes without patient risk.

For surgeons, VR acts as a bridge between the digital scan and the operating theater. Surgeons can use the proposed rendering framework to interact with patient-specific anatomy, testing different surgical approaches by manipulating and ‘hiding’ tissue layers to find the most efficient route. Our framework could offer a multi-user environment where a team (e.g., a radiologist and a surgeon) can simultaneously view the same model in a shared virtual space, regardless of their physical location.

Progress so Far

The chosen topic covers both medical and computer science. The initial phase involved a study of the medical literature to understand the specific requirements. A review of current standards in medical imaging (e.g., DICOM, CT/MRI segmentation) was conducted where the limitations of traditional 2D slice viewing were discovered. Parallel to the medical study, a review of state-of-the-art computer science literature was conducted. Investigating the mathematical framework of 3D Gaussians scene reconstruction and studying how gradient-based optimization can be used to refine medical models.

Bibliography

Jacques, Steven L (2013). Optical properties of biological tissues: a review. Physics in Medicine & Biology, 58(11), R37.
Engel, Klaus (2016). Real-Time Monte-Carlo Path Tracing of Medical Volume Data. Proceedings of the GPU Technology Conference (GTC).
Pharr, Matt; Jakob, Wenzel; Humphreys, Greg (2023). Physically Based Rendering: From Theory to Implementation (4th ed.). MIT Press.
Kerbl, Bernhard et al. (2023). 3D Gaussian Splatting for Real-Time Radiance Field Rendering. arXiv:2308.04079.
Zhu, Zuoliang et al. (2025). GS-ROR2: bidirectional-guided 3DGS and SDF for reflective object relighting and reconstruction. ACM Transactions on Graphics, 45(1).
Gao, Jian et al. (2023). Relightable 3D Gaussians: Realistic Point Cloud Relighting with BRDF Decomposition and Ray Tracing. arXiv:2311.16043.
Jiang, Yingwenqi et al. (2023). GaussianShader: 3D Gaussian Splatting with Shading Functions for Reflective Surfaces. arXiv:2311.17977.
Liang, Zhihao et al. (2023). GS-IR: 3D Gaussian splatting for inverse rendering. arXiv:2311.16473.
Bai, Haiyang et al. (2025). GaRe: Relightable 3D Gaussian Splatting for Outdoor Scenes from Unconstrained Photo Collections. arXiv:2507.20512.
Yao, Yuxuan et al. (2024). Reflective gaussian splatting. arXiv:2412.19282.
Niedermayr, Simon et al. (2024). Application of 3D Gaussian splatting for cinematic anatomy on consumer class devices. arXiv:2404.11285.
Mildenhall, Ben et al. (2020). NERF: Representing scenes as neural radiance fields for view synthesis. arXiv:2003.08934.

Details

2026-06-27T22:02:55Z

Details

Master’s Thesis

Department - Faculty - Institute : Faculty of Mathematics, Physics and Informatics - Department of Computer Science - Comenius University Bratislava

Supervisor : doc. RNDr. Pavel Chalmovianský, PhD.

Title : Modern Techniques for Medical Data Imaging on Consumer-Grade Hardware

Abstract : Volumetric data from medical scans (e.g. CT, MRI) is essential for clinical analysis, medical education, and patient communication. However, achieving photorealistic rendering of this data using traditional methods often either compromises real-time rendering times or requires expensive hardware, limiting its accessibility.

: This thesis proposes a novel rendering framework to address this limitation. We will investigate the use of path tracing combined with splatting methods to render medical volumetric data. Together with targeted performance optimizations, we aim to achieve photorealistic and real-time results on consumer-grade GPUs.

: The efficacy of the proposed method will be evaluated through its performance and visual fidelity will be benchmarked against traditional volume rendering techniques. The diagnostic quality of the resulting visualizations will be qualitatively assessed in consultation with medical professionals.

Main

The Angel in the Withywindle

Constancy of the Speed of Light

The postulate

Where it comes from

What it costs

The light cone

Conditional Probability

Functional Analysis

Scope

A first taste

Algebra 1

Scope

A first taste

Logic

Propositional logic

Something to Write

Rules How to Write

Meaning of What We Wrote

4. Semantic Notions

5. Proof System — Hilbert-Style Calculus

5.1 Axiom Schemas (over {¬,→}\{\neg, \to\}{¬,→})

5.2 Rule of Inference

5.3 Proof and Derivability

6. Fundamental Metatheorems

Summary Structure

Vector Fields

Definition

Flows

The Lie bracket

Turing Machines

The model

Undecidability

An interactive machine

Vector Spaces

Axioms

Bases and dimension

Thesis Overview

Working title

Structure

Breakdown of the Paper

Introduction

Introduction

Introduction

Volume Reconstruction

Medical Scans

Medical Scans

CT Scans

Thoughts

Thoughts

Roadmap

Roadmap

Castle Building Roadmap

Overview

Vision & Objectives

Success Criteria

Phases & High-Level Timeline

Example Milestones

Roles & Responsibilities

Resource Plan

Risk Register (sample entries)

Deliverables per Phase

Communication Plan

Acceptance Checklist (before Handover)

Maintenance & Lifecycle

Templates & Notes

Appendix

Motivation

Motivation

Literature

Literature

Primary Literature

Other Literature

Direct Volume Rendering

Articles

General Rendering

Books

Monte Carlo Path Tracing

Papers

3D Gaussian Splatting

5.1 Axiom Schemas (over $\{\neg, \to\}$ )