symmetry (int) represents which additional symmetries of the input pattern are digested. This is incompatible with waves representing probabilities. Made for PROCJAM 2017 | Thanks to our Kickstarter backers . UnityWFC requires that it's tiles have an orientation with vertical Y and horizontal X, so if you're importing 3D assets you may have to reconfigure the mesh export. In this setting, each color corresponds to a position in space. Wait a minute Momenta are defined as the Fourier transform of the wave functions But do they still have the meaning they had in classical mechanics? It's especially exciting for game designers, letting us draw our ideas instead of hand coding them. In our demo, this will result in a blank canvas, and the success var being false. Humm. This principle doesnt involve any probability. UnityWFC requires that it's tiles have an orientation with vertical Y and horizontal X, so if you're importing 3D assets you may have to reconfigure the mesh export. It's . more replies. But I believe that the word uncertainty is misleading. NOTE: Oskar Stlberg's interactive web demo is a great way to get a feel for how the algorithm works. Start a new unity project and either import the .unitypackage or clone the github repository into the Assets directory. Because the spacing is larger than our N setting, all the algorithm knows is that coins are surrounded by empty space, and our output has a random scattering of coins. This corresponds to the relation $p=hk/(2\pi)$, where $p$ is the momentum, $h$ is the Planck constant and $k$ is the wavenumber. Click here for instructions on how to enable JavaScript in your browser. It's Training component should hook up to the input you've created. The Markov chain Monte Carlo sampling strategy sets up an irreducible, aperiodic Markov chain for which the stationary distribution equals the posterior distribution of interest. This choice is then propagated throughout the grid, eliminating adjacent possibilities that don't match the input model. We're using a modified version of kchapelier's javascript port, which has been edited into a single file to make things a little quicker to set up. There's also an aspect of collaboration with the machine, as the resulting output can be surprising and sometimes lead you to change your design. Words in a sentence. So how does Heisenbergs spreading principle translate into mathematics? This is because people often focus on the measurement part of quantum mechanics. The superpositions and collapses of spins can thus be described by simple vectors and matrices. But, overall, I really dont understand the measurement problem. Wavefunction Collapse can even accomodate additional constraints, like "this tile must be sea", or "this pixel must be red", or "there can only be one monster in the output." The way the quantum state collapses depends on whats measured. Solid logging system capable of handling a variety of data collection needs, Added more automated control scripts that can execute the desired runs under desired conditions, Added a simple UI to allow for check-box options to toggle specific things on and off. The eigenvalues and eigenvectors of the obtained Hamiltonian, especially the lowest eigenvalues, now correspond to possible orbits of electrons around the nucleus. The automation of kernel neighborhood (constraint) determination makes MkWFC a more scalable model with varying sizes. By following a minimum entropy heuristic, the algorithm resolves constraints from a point on the output sample and propagates outwards resolving entropy to create a final pattern. In this setting, the wavenumber corresponds to the number of cycles of the trigonometric wave in 1 meter. So, can actual wave functions be trigonometric waves? Approach 2: Modification of probability and entropy, 3.3.1. A chain is a sequence of events. When running the trials highlighted above using MkWFC and comparing the performance against TWFC, there was some interesting phenomenon worth noting. First task is to find a port of WFC in your language of choice. Although difficult to solve, this equation can be considered as a classical one. The results of the algorithm are convincing as the constraint solving texture synthesis maintains the constraints in output. So whats the probabilistic distribution of a wave function? Heres the best explanation Ive come up with. Check out this awesome video: Yes! We need to use the Fourier analysis. This is great for quick inspection or texture generation, but for a game project we usually have a specific data format we like to work with to describe our worlds. Classically, the position and the momentum of a particle are two distinct concepts which can be defined independently from each other. The final feature is backtracking. Then, the amplitude of the wave function at a position in space is represented by its associated dot in the complex plane. The name Wave Function Collapse refers to the quantum mechanics process of changing the superposition of the wave function due to the presence of the observer. While you can tweak your input to increase the chances for full solutions, another strategy is to retry the algorithm until it can complete. This means that the wave function is very localized around this position, and nearly nil anywhere else. MkWFC performs as expected with given constraints. Execution time for OWFC is very high due to the iterative nature of the algorithm. We're using a modified version of kchapelier's javascript port, which has been edited into a single file to make things a little quicker to set up. Its a wave function which will appear to be non-moving. The TWFC model, on the other hand, does not identify the kernels by splitting the input image into chunks, instead, it receives these kernels and their constraints in the form of meta-data. Depending on your tile dimensions, you may have to change the Gridsize property on the Training component. First, add this function to get a pixel value from the ImageData: Next let's make a lookup table to translate from a color value to a single integer. It's especially exciting for game designers, letting us draw our ideas instead of hand coding them. It's usefull for creating things like tileable textures, but also has a surprising influence on the output. 2016 par L Nguyn Hoang, avec Wordpress. These variations can help flesh out the patterns in your input, but aren't necessary. Quite often though, this idea is rejected because a restricted concept of Markov chain is discussed. This would give additional control over the algorithm and allow to specify the spatial distribution of tiles across the solution space. The spin is whats missing to describe quantum states. Note that the greater the wavenumber, the more energy is associated to it. Heres a GIF showing the algorithms constraint solving nature to resolve an output procedurally generated image: There are 2 models incorporated to generate procedural content using WFC, they are the Overlapping WFC (OWFC) and Tiling WFC (TWFC) models. Same here. These objects are studied by. There are also several physical measures which are combinations of momenta and positions, like angular momenta. Implementation of OWFC and TWFC models was done early using solid references to ensure the comparison model was in place. Since TWFC uses tiles on a fixed tile grid as opposed to pixels in an image, the user has control over the size of the tiles used as well as the complexity of the output generated as tiles can be simplified or made more detailed based on the users need. For example, if we represent this sentence as a chain: have an idea have ikea! Because the spacing is larger than our N setting, all the algorithm knows is that coins are surrounded by empty space, and our output has a random scattering of coins. While I encourage you to explore the relation between input and output, there are a few tricks discovered by the community that can give you more control on the results. seed (int) all internal random values are derived from this seed, providing 0 results in a random number. An assortment of functions and a class to efficiently and easily evaluate and analyze Markov Chains. It may happen that during propagation all the coefficients for a certain pixel become zero. I dont see what this has to do with momenta. This method, called the Metropolis algorithm, is applicable to a wide range of Bayesian inference problems. We'll start with a function that loads an image element, draws it to a temporary canvas, and extracts the ImageData. You can get around this problem by either uploading your project somewhere, or running a local server on your computer. WFC's special approach to constraint solving is a process of elimination. One of the interesting phenomena was that due to the increase in coverage with the MkWFC model, the number of constraints read would be almost double that of TWFC which will result in the increase of coverage of constraints in the output. These constraints determine the occurrence of specific information in the input data which can be visualized as neighboring pixel information in the case of 2D images. similar to OWFC, TWFC still requires the need to resolve the entropies of each tile in the output image as the WFC wave considers all tiles to have the highest possible entropies (i.e, the total number of tiles used by the algorithm) when generating the output. Then we'll walk through setting up WFC in javascript and the Unity game engine. While I encourage you to explore the relation between input and output, there are a few tricks discovered by the community that can give you more control on the results. Any stationary wave function is then a superposition of the wave functions which correspond to these values. The Most Beautiful Equation of Math: Euler's Identity The Most Beautiful Equation of Math: Euler's IdentityBy L Nguyn Hoang | Updated:2016-01 | Views: 90679In 1988, Euler's identity was elected most beautiful theorem of mathematics. In mathematical terms, this corresponds to saying that the collapsing of the wave function corresponds to a Markov chain. Below is an example of the meta-data fed as input to MkWFC: By simplifying the meta-data being used, we are able to eliminate the non-trivial meta-data data generation process of having to determine the constraints manually. This equation is the Schrdinger equation. Save my name, email, and website in this browser for the next time I comment. It will experience decoherence. From creating radio to hearing sounds, this concept is a translation between two mathematical worlds: Signals and Frequencies. Now let's feed the ImageData into an OverlappingModel. At least, that's how I like to think about this, since, apparently, at least on the web, this is controversial. The two commonly used WFC implementations are the Overlapping WFC (OWFC) and Tiling WFC (TWFC) implementations. But I dont see how they account for the measurement problem. If you find a way to combine my representations of wave functions with Schrdinger equation and the ideas of superposition, I would be very interested in hearing about it! But on the other hand, it can be probabilistic. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. The wave can then be seen as a colored path in the complex plane. And the decomposition is given by the Fourier transform. Once you're set up, you'll be able to create a variety of procedural pattern types in minutes. From creating radio to hearing sounds, this concept is a translation between two mathematical worlds: Signals and Frequencies. This mapping of wave functions with momentum space wave functions by the Fourier transform is displayed below. NOTE: Oskar Stlberg's interactive web demo is a great way to get a feel for how the algorithm works. This is where a quote Richard Feynman becomes reassuring: I think I can safely say that nobody understands quantum mechanics. Lets assume that the wave function has collapsed around the green position. Show that the Green's function is . Hey @selfsame! To simplify our understanding of wave functions, lets consider a 1-dimension space. This model solves some of the problems of OWFC, by changing some of the inputs fed to the algorithm while maintaining the process involved in resolving the constraints used. . For instance, the Hamiltonian is the operator associated to energy. The recursive nature of OWFC occurs when entropy resolution on a pixel requires propagation of this information to all neighboring pixels, who will then need to propagate their manipulation to their neighboring pixels and so on. Let's try this out for our demo, converting the ImageData into a simple array of arrays, which hold a single number representing their tile type. In this setting, whats surprising is that the measurement of the position of a wave localizes the wave around a position. Markov Chain Functions. To perform the entropy resolution, an arbitrary pixel is selected in the output and assigned one of its possible color values. I would need help to find out what I'm possibly missing or doing wrong. We'll take a look at the kinds of output WFC can produce and the meaning of the algorithm's parameters. Reply . This article does. Recall that the wave function is obtained by summing all momentum space wave functions of all momenta. If you're designing a dungeon, floor variants can be included to encourage the emergence of larger rooms. END. And the reason why such a reasoning holds is because of the most fundamental equation of quantum physics which describes the dynamics of wave functions. Each grid location holds an array of booleans for what tiles it can and cannot be. Besides the words, we take . Create an empty named input and add a Training component to it. OK But if physical wave functions arent trigonometric, how can we find their wavenumbers? By splitting an image into some uniform chunks called Kernels, the WFC algorithm can identify patterns that exist in the input sample and their rate of occurrence. The material mainly comes from books of Norris, Grimmett & Stirzaker, Ross, Aldous & Fill, and Grinstead & Snell. For example you could sprinkle some trees at random coordinates, draw roads with a brownian motion, and add rooms with a Binary Space Partition. Ex 4. Ex 5. The square of the norm of the value of the momentum space wave function is the probability density function of finding the momenta in case of observation. Approach 1: Generative Adversarial Networks, 3. Markov chains illustrate many of the important ideas of stochastic processes in an elementary setting. In this setting, waves are commonly represented as signals through space, as follows. Yes! Heres some of the code showing how TWFC was implemented using a templated class: When comparing the 2 models, OWFC model takes a sample image as input while the TWFC model takes a meta-data file as input which contains a set of tiles that can be used as subsets of the image or tiles that complete an image and some constraints that determine what tiles can be placed next to each other. Your email address will not be published. This classical subject is still very much alive, with important developments in both theory and applications coming at an accelerating pace in recent decades. Start a new unity project and either import the .unitypackage or clone the github repository into the Assets directory. Thus, let me introduce a new visualization of wave functions of my own creation. Below is the code I implemented to generate the WFC Propagator and to infer neighborhood constraints from a set of images having known the tiles being used and their fixed size beforehand. Click here for instructions on how to enable JavaScript in your browser. Child prefabs that are placed within the Training bounds will be read by the algorithm. The value corresponds to the overlap models internal pattern indexes, so some experimentation is needed to figure out a suitable value. . Fourier Analysis: Signals and Frequencies Fourier Analysis: Signals and FrequenciesBy L Nguyn Hoang | Updated:2016-01 | Views: 13077Fourier analysis is a fundamental theory in mathematics with an impressive field of applications. But its not the case in quantum mechanics! Here are some of the results we found when comparing TWFC with MkWFC with different input and output sizes: These results indicate that there was an average of 294% increase in the number of constraints identified by MkWFC as opposed to TWFC. In particular, the energy levels cannot contain too many electrons, which implies that there are sorts of blanks around nuclei that can each contain one electron only. Precisely! The js version uses ImageData for input and ouput. No indeed. Whats more the Euclidean norm of the coordinates is equal to 1. Maxim Gumin first implemented WFC using C# to incorporate a texture synthesis constraint solving PCG algorithm. Due to the nature of the constraints used by the algorithm, there are instances where entropy resolution cannot resolve the output pixel entropies. It let's you play the part of the observation phase and animates the propagation after you select a grid value. The type of this parameter is usually implementation specific. However, this also introduces a greater scope for conflicting constraints like in the case of generating circuits where sometimes the output wave would not be able to resolve entropies per pixel to a final value of 1 due to adjacency constraint conflict. The check involves overlapping the kernel with the neighbor pixel to resolve to determine if the entropies for the neighbor pixel are valid based on the occurrences of pixels in the input. Linear mapping based on the lowest and the highest weight, 3.3.2. A good starting image is something like: Load your index in a browser, then open the developer console and test the img_url_to_data function: If you'd opened the index.html file directly in your browser, then at this point you'll see an error message like: This is because it's a security violation for javascript to have access to resources from outside the same domain. Logging was done inefficiently which took up too much time during data analysis. This is the core of the measurement problem. This trick also works for the general structure of your patterns. Wave Function Collapse. Is the localized wave necessarily horizontal? If you can, please write an article to explain in more details what Im saying here succinctly. Approaches which seem more promising to me on a mathematical viewpoint involve quantum information theory or particles going back in time. On one hand, there is the deterministic revertible Schrdinger equation. Inefficient iterative check-in MkWFC model that does some repetition on the input sample during constraint identification. Wave Function Collapse(WFC) is a constraint solving algorithm named after the quantum physics wave superposition concept. It's also configured with components, so there's no code required for basic use. The traditional approach to this sort of output is to hand code algorithms that generate features, and combine them to alter your game map. Any idea how I can start? This corresponds to whats known as the time-independent Schrdinger equation: Exactly! Instead, the MkWFC model uses the kernel identification advantages of OWFC, while maintaining the control and performance advantages of TWFC. Since this point is one of the farthest for the origin, the square of the norm of the amplitude is relatively great. With an image editor, create a small input image, and save it as test.png in your project directory. I thought that quantum mechanics was a probabilistic description of the world. Yes! This phenomenon accounts for Niels Bohrs weird model of the hydrogen atom, as explained in this video from the Science Channel: Fortunately, no. In this article well see how a physics-based method can be used to secure online information. But anyways, the greatest applications of Schrdinger equation rather concern the calculations of stationary states. This is represented below: Conversely, were the wave function very localized, the momentum space wave function would be circling periodically, and would therefore be spread over all momentum space. my article on the essence of quantum mechanics, The Most Beautiful Equation of Math: Euler's Identity, Fourier Analysis: Signals and Frequencies. Yes! We can add a feature for this at the end of the start function like so: WFC usually creates some sort of image data type for it's output. Equivalently, we can focus on its momentum. I have a lot of trouble making sense out of it. Great question! N (int) represents the width & height of the patterns that the overlap model breaks the input into. The most likely collapse is the red one. These kernels are then overlapped to determine the possible occurrences of pixels in the output. It rather aims at providing you an image of what the intimate relationship between position space and momentum space wave functions. As you start using WFC, you'll start to get a feel for how it transforms input patterns into output. Beware that Im going to use the word spectrum to talk about the range in space described by the figure. More interestingly, in an atom, the positive charges of a nucleus affect the potential energies of wave functions of electrons, which in turns modify the Hamiltonian. Finally, call img_url_to_data with your image url and the start callback: If everything works, you should now see your output drawn to the canvas: NOTE: Depending on the input pattern, it's possible that WFC will not be able to finish solving an output. Greens Functions. Each of the neighboring pixels will check with observed kernels to determine what the possible colors could be for themselves given the previously resolved pixel with entropy 1. To conclude, let me recall the two major dynamics of wave functions. The non-trivial manual generation of the meta-data however, affects the scalability of TWFC with either addition or deletion of tiles. Quite often though, this idea is rejected because a restricted concept of Markov chain is discussed. For instance, the amplitude of the wave function drawn above at the green position in space is -1, that is, the point of distance 1 from the origin on its left. In this article well see how a physics-based method can be used to secure online information. By following a minimum entropy heuristic, the algorithm resolves constraints from a point on the output sample and propagates outwards resolving entropy to create a final pattern. Thus, any wave function can be uniquely written as a linear combination of these eigenvectors. So the energy of a wave is the speed at which it rotates? . This is powerful but time consuming, and your original vision can sometimes get lost during the process. For the input game object's Training component: For the output game object's Overlap WFC component: Periodic input enabled (was disabled by default), Periodic output enabled (was disabled by default), Incremental enabled (was disabled by default - not strictly necessary but convenient), Copied the pattern for input from the image showing both the input and output objects in the Scene view.