, k r s This program generates bitmaps that are locally similar to the input bitmap. For simplicity in the following, all wave functions are assumed to be normalized; the total probability of measuring all possible states is one: With these definitions it is easy to describe the process of collapse. Code on this page in its current state may be flagged incorrect once this task has been clarified. Mark the cell we just collapsed as needing an update. In practice, however, the algorithm runs into contradictions surprisingly rarely. Were trying to arrange pixels in an output image. = Long-awaited Coding Challenge. In this video you'll have the Wave Function Collapse Explained in 3 Minutes._____ If you enjoy this video, please lik. Theres an obvious choice for this pick the cell with the fewest possible value remaining in the domain (but at least 2 possibilities). 2 That could be because it is harder to implement, slower to run or perhaps because it can sometimes be too good at reproducing the original sample, losing some of the organic feel. Use Git or checkout with SVN using the web URL. The program determines a bitmap based on the input consisting of smaller tiles (.png) as well as a set of constraints (.xml). A mini Sudoku is a puzzle that looks like a 44 grid with some of the numbers filled in. In other words, probability to meet a particular pattern in the output should be close to the density of such patterns in the input. , and a list of tiles, return a blend of all the tiles that haven't been y It makes no attempt to infer any more complex rules, like sea tiles must be adjacent to at least one other sea tile or every island must contain at least one land tile. {\displaystyle s_{z}} is measured, the probability of collapsing the system to eigenstate . | ) momenta, etc. p We use the weights because they give us a more realistic output image. If theres at least one tile in X can be placed next to y, then y is still ok, at least as far as this tile pair is concenred. Initially, none of the tiles are ruled out, so the result just looks like a blend of every tile at each location: How do we reduce possibilities? This is like a wedding plan model in which the only type of rule is X may sit next to Y. Summing over tiles in the domain, where is the probability associated with that tile. Although von Neumann's projection postulate is often presented as a normative description of quantum measurement, it was conceived by taking into account experimental evidence available during the 1930s (in particular the Compton-Simon experiment was paradigmatic), but many important present-day measurement procedures do not satisfy it (so-called measurements of the second kind). 3 | Once theres only a single item left in the domain, Ill show it larger and full color. Now lets consider the first constraint. , , WaveFunctionCollapse implementation in C This project was based on the Maxim Gumin's wave function algorithm. {\displaystyle \lambda _{i}} with eigenbasis Note that the unrestrained knot tileset (with all 5 tiles being allowed) is not interesting for WFC, because you can't run into a situation where you can't place a tile. If theres no such tile in X, then its impossible to place y in Y, and we can eliminate it. Bitmap & tilemap generation from a single example with the help of ideas from quantum mechanics. Ive several other articles on procedural generation too. Your dad cant sit within 2 tables of your mum. Otherwise, lets load the bitmaps (theyre all very small so Im just loading them as base64-encoded strings): For reference, this is what the images look like: Next, lets define a representation for these tiles. So its time to guess again and again and again. The simplest nontrivial case of the algorithm is when NxN=1x2 (well, NxM). The answer is, it can only generate a narrow class of knots, it can't produce an arbitrary knot. , You signed in with another tab or window. By repeating until the stack is empty, we have ensured that every constraint has been fully evaluated, but we never check a constraint unless one of the cells associated with it has changed domains. Imagine that you are planning your wedding. . {\displaystyle r} Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. t represents the Kronecker delta. Pobytov. You construct your seating plans initial wavefunction. i For instance, if youre generating 2D platformer levels, youll want ladders to always start and finish at platforms. until cast to an eigenstate of I noticed that when humans draw something they often follow the minimal entropy heuristic themselves. But if you are choosing the tiles from a weighted random distribution, you need to do something different to take that into account. A contradiction makes fully collapsing the entire wavefunction impossible. Software Engineer / But it still loses a lot of information. If the count ever drops to zero, then we know the tile cannot be placed, as theres nothing that could go adjacent to it. In quantum mechanics, wave function collapse occurs when a wave functioninitially in a superposition of several eigenstatesreduces to a single eigenstate due to interaction with the external world. q The key idea is to extract all 3x3 pixel blobs (or any other size, but it has to be kept relatively small) from the original image, determine their weights and which can be placed next to which, and then run this tiled algorithm. In reality, though - the name is a triumph of marketing. Yet another aspect of the Measurement Problem is that wave function collapse appears to violate the speed limit of the universe, the speed of light. So instead of stopping when we hit a node weve already visited, we keep traversing until we detect that were not making any new changes. {\displaystyle |\phi _{i}\rangle } {\displaystyle E} So the rules for mini Sudoku translate into 12 all_distinct constraints one for each row, column, and 22 corner box. Otherwise, generating a random pattern would be trivial: just randomly select a tile for each location in the output image, and youre done. This region's state then collapses into a definite state according to its coefficients and the distribution of NxN patterns in the input. x Theres several of these, but they all work in similar ways. [11] Niels Bohr also repeatedly cautioned that we must give up a "pictorial representation", and perhaps also interpreted collapse as a formal, not physical, process.[12]. Or maybe I just dont understand how to use it. = r 1 Youre done! While there are still cells that have multiple items in the domain: Pick a random cell using the least entropy heuristic. The idea is that if we make our guesses at random, instead of getting a solver, we get a generator. Once weve propagated these changes, were ready to pick the next tile. """, """ But this is exactly where WFC shines, and Harrison's algorithm supports constraints. | . Specifically the Wave Function Collapse algorithm (WFC). Within the C* formalism this non-unitary process is equivalent to the algebra gaining a non-trivial centre[18] or centre of its centralizer corresponding to classical observables.[19]. The statistical population of the wave function is formulated for an infinite number of . Secondly, in order to check an adjacency constraint, we need to answer a question of the form given the tiles in the domain of cell X, which tiles in the domain of adjacent cell Y are possible. If youre interested in how WFC works, but not trying to follow along, feel free to skip to the next section. A wave function may be used to describe the probability of finding an electron within a matter wave. 2 i E Ive marked the steps above in the code, to make it easier to follow. Subscribe to my new work on programming, security, and a few other 31. Hi Boris, just wanted to leave a thank you for your tutorial, its methodical explanation meant that I was able to get my own WFC running with minimal effort. But Ill walk you through the simplest to give you an idea for how they work. In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. These considerations become more important the larger your output image becomes. A state of an NxN region is a superposition of NxN patterns of the input with boolean coefficients (so a state of a pixel in the output is a superposition of input colors with real coefficients). The basic idea behind Wave Function Collapse (or WFC as I will refer to it going forwards) is, as best as I understand it, as follows: Each tile type has a set of rules that describe each edge. The rest of the examples and tilesets were made by me. Immediately post-measurement, other elements of the wave function vector, We can now make further progress by propagating constraints a bit more. This almost works, except for the fact that these traversal methods will only visit a node once, whereas constraints can flow backwards, requiring a node to be updated more than once. On each observation step an NxN region is chosen among the unobserved which has the lowest Shannon entropy. It's called sometimes the collapse of the wave function. So is the miracle of the "collapse of the wave function" as usually taught by the hegemonic sect. {\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots } Which tile it will eventually collapse to is already very constrained. {\displaystyle P_{k}=|c_{k}|^{2}} 1 drl. The only rule we have is about adjacent tiles, so we need to update two cells: Are there any more tiles to update given the smaller domains? You update the seating plan wavefunction by crossing people off of lists of possible assignees. | Well, its an algorithm developed by Maxim Gumin based on work by Paul Merrell for generating tile based images based off simple configuration or sample images. This could be a welcomed contribution for any designer who want to speed-up and expand his creative process. | For instance, this tile: gives us [True, False, False, True]. If youve come here hoping to learn about quantum physics, you are going to be disappointed. If you reach a contradiction then the easiest thing to do is to start again. This cancellation process can be seen as the reverse collapse of the wave. } For example, suppose we are generating random worlds for a top-down, 2-D game. ^ WaveFunctionCollapse algorithm created by Maxim Gummin and published on github as open source allows user to . Now that we have all the bits, we can just call our run_iteration function in a loop and get some cool-looking results! Oskar Stlberg, Remy Devaux implemented the tiled model in PICO-8 and wrote an, In order to generate connected structures, Boris the Brave applied the, Oskar Stlberg combined WFC with marching cubes on irregular grids and made a town building toy, Nuo de la Serna implemented the 3d tiled model in an, ukasz Jakubowski, Maciej Kaszlewicz, Pawe Kroll and Stefan Radziuk. An observable (i.e. Nonetheless, the process is remarkably similar. Decoherence is thus very important for explaining the classical limit of quantum mechanics, but cannot explain wave function collapse, as all classical alternatives are still present in the mixed state, and wave function collapse selects only one of them. Furthermore, the Wave Function Collapse Algorithm operates on a matrix of cells, which translates really well to image processing but not so well to freeform 3d structures. The coefficients Now that you understand the Even Simpler Tiled Model, youre ready to climb the ladder of power and complexity. In this devlog, I talk about my experiences using Wave Function Collapse to generate procedural terrain in Unity. The propagation phase in this model is just adjacency constraint propagation. ^ NEW: Also subscribe to my new series, Programming Feedback for Advanced Beginners. Using loops, this can be coded easily, but it can be incredibly slow. DeBroglie is a C# library implementing the Wave Function Collapse algorithm with support for additional non-local constraints, and other useful features. Xbox controller: Left Stick for walking, right stick for looking . So, what is the Wave Function Collapse algorithm (WFC)? Schrodingers Cat was simultaneously both dead and alive until someone opened its box and checked on it; your table plan is simultaneously in every single possible layout until you get your shit together. Instead, if we store an extra data structure we can quickly answer this question when it comes up. make the ground attribute boolean, small improvements, code simplification, faster image io, better project file, new N=4 examples, make rng local, how to run in release mode, Texture Synthesis by Non-parametric Sampling, texture synthesis algorithm with the entropy heuristic. c In the terminology we used above, each tile is a different value, and each cell in the grid is a variable representing the choice of tile. + """, """ Initially, every cell is possible at every grid location. When weve found a contradiction like this, we do a procedure called backtracking. You can also investigate the performance optimizations made in the full implementation. So instead, computers do the dumbest possible thing: they guess. Now we repeat the main loop of WFC. That must mean the other 3 cells cannot be a 4. , } A small demo project I put together to try implementing the Wave Function Collapse algorithm. Wave Function Collapse, or WFC, tries to solve this by integrating any constraints at generation time. ( Source) We start by implementing our algorithm in the file wfc.cljs, wfc being short for wave function collapse. Wave function collapse. Given a list of states for each tile for each position of the image, return t If any location has no choices left, weve failed. Collapse this element into a definite state according to its coefficients and the distribution of NxN patterns in the input. We replace the adjacency constraints with a new constraint that affects multiple tiles at once: We require that each 33 group of cells in the output corresponds to a 33 group in a sample grid. { Wave functions are commonly denoted by the variable . {\displaystyle \{|\phi _{i}\rangle \}} Stick around. Keeping only the four T tiles, for instance, results in a much lower success rate, as WFC runs into contradictions and quits. , However, he emphasized that it should not be understood as a physical process. Another example of WFC I really like is the appropriately-named Wave Function Collapse, a 3D WFC-based random . We also create a dedicated array with tile weights. And if Michele isnt going to be on Table 2 then Barack wont want to be either. Note also that Circuit, Summer and Rooms tilesets are non-Wang. Well, that was a lot of work, but weve now made a small amount of progress. (Weak C2) Distribution of NxN patterns in the input should be similar to the distribution of NxN patterns over a sufficiently large number of outputs. Software versions used: Please leave any comments or feedback below! """ On l'appelle parfois l'effondrement de la fonction d'onde parce que, dans le monde quantique, tout est en superposition et il y a de multiples possibilits. 3 We achieve this by using simple, global weights. | There was a problem preparing your codespace, please try again. This symmetry-handling makes for better output images, but more complex code. For instance, the value potential[12][2][5] is true if and only if the element at coordinates (12, 5) can still be tile number 5, in our case 'bend_bl'. By contrast, a square with low entropy is one with few possible tiles remaining in its wavefunction. E This may reduce the possibilities for other locations around that initial one, or, depending on the tileset, even completely determine what tiles we must pick in some places. Traditional Wavefunction Collapse differs from Wedding Wavefunction Collapse in the way that you teach the algorithm the rules it must obey. . [5] This transition is fundamentally reversible, as the combined state of system and environment is still pure, but for all practical purposes irreversible, as the environment is a very large and complex quantum system, and it is not feasible to reverse their interaction. Most implementations and explanations of Wavefunction Collapse are of a full, performance-optimized version of the algorithm. If nothing happens, download GitHub Desktop and try again. Significantly, the combined wave function of the system and environment continue to obey the Schrdinger equation throughout this apparent collapse. {\displaystyle |\phi _{1}\rangle ,|\phi _{2}\rangle ,|\phi _{3}\rangle ,\dots } Throw away your work so far, find a new blank table plan, and re-start the algorithm by collapsing the wavefunction for a different random seat. In this system each tile should be assigned with its symmetry type. {\displaystyle |c_{i}|^{2}=1} We detect this type of contradiction in this method too. Here is a full size sudoku puzzle. For each variable, you then define the domain, the set of values that its possible for it to take. If youre into game design (or maybe someting more exotic, like computer-generated art), you may have needed to generate large random patterns. The overlapping model relates to the simple tiled model the same way higher order Markov chains relate to order one Markov chains. Each time you get stuck propagating, make a guess and continue. c After doing that, it looks like we might benefit from doing more constraint propagation those newly minted 1s imply that the other variables cannot be 1, which means there will be more variables with a single valued domain, which means more to propagate. You rush outside the wedding hall, pin the plan to the noticeboard, and sit back to calmly await the cake, presents, and Aerosmith that await you. Wavefunction Collapse is quite literally a Constraint Satisfaction Problem in that it specifically uses Constraint Satisfaction methods in the exact way they inherently work. 2 We do this by randomly choosing one of the tiles still available to the square, weighted by the tile weights that we parsed from the example input. What is the real meaning of Schrdinger's cat? A general description of the evolution of quantum mechanical systems is possible by using density operators and quantum operations. In the algorithm, tiles don't really coalesce into being until you pick one to examine. Wave Function Collapse. The easiest is locations_with_fewest_choices, which does what it says on the tin and returns the undecided element with the fewest choices left, or returns None if all elements are decided. In a Simple Tiled Model, input and output images are built out of a small number of pre-defined tiles, and each square in the output image is affected and constrained only by its 4 immediate neighbors. 1. The quantum state of a physical system is described by a wave function (in turnan element of a projective Hilbert space). Heres the domains shown on the diagram: Ive written multiple numbers in each cell, indicating all possible values in the domain for that variable. In quantum mechanics, wave function collapse is said to occur when a wave functioninitially in a superposition of several eigenstatesappears to reduce to a single eigenstate (by "observation"). , that is It is low-entropy, restricted tiles that humans tend to focus on when working on Wavefunction Collapse-like problems manually. This makes it very clean and intuitive once you get the idea. A wave function is defined to be a function describing the probability of a particle's quantum state as a function of position, momentum, time, and/or spin. An infinite, procedurally generated city, assembled out of blocks using the Wave Function Collapse algorithm. Just start on the right (pipe), then up (no pipe), left (no pipe), and down (pipe). For an explanation of the notation used, see, Unless the observable being measured commutes with the. We dont want our computer algorithm to do that its complicated, and we want a general algorithm which will work with any set of constraints, not just sudokus specifically. Turn this down if you want to visualise constraint . i Propagation: propagate information gained on the previous observation step.