Since loading an image source is not instant, we'll use a callback function that is invoked when the data has been loaded. The set of all possible energies, which is called the spectrum of the Hamiltonian, is made of discrete values more often than not. 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. On the other hand, there is the probabilistic collapse. Rather, you should think of Heisenbergs principle as a spreading principle. This is the case, for instance, of the troubling double slit experiment and its variants you can read about in my article introducing quantum mechanics. Notify me of follow-up comments by email. What the difference between a 2D markov chain and Wave-Function-Collapse though? And this half accounts for plenty of counter-intuitive observations. If you can, please write an article to explain in more details what Im saying here succinctly. For instance, the Hamiltonian is the operator associated to energy. Instead of using image files for input and output, it works with arrangements of prefab GameObjects. There is very little control on the procedural-ness of the algorithm since the output is defined by the input sample, manipulating constraints can be very tricky. 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. At least, that's how I like to think about this, since, apparently, at least on the web, this is controversial. They emphasize that with increasing Markov Chain's . Markov Chains are a common approach for analyzing behaviors such as describing movements of packets in a net- work regarding the relationship between customers and the waiting time for receiving packets . Markov Chains - Simplified !! N (int) represents the width & height of the patterns that the overlap model breaks the input into. Greens Functions. Ex 6. Sorry for that. In this article well see how a physics-based method can be used to secure online information. The article de- scribes how Markov Chain models can be used to analyze priority data. Click here for instructions on how to enable JavaScript in your browser. Great question! Mathematically, this corresponds to saying that the Fourier transform is more or less a bijection from the Hilbert space of wave functions and the Hilbert space of momentum space wave functions. The following figure shows a wave function rotating. So the energy of a wave is the speed at which it rotates? That'd explain why when I couldn't find an explanation of what wave function collapse meant. OK But if physical wave functions arent trigonometric, how can we find their wavenumbers? A chain is a sequence of events. With python, this can be as simple as opening a terminal in the project directory and running python3 -m http.server 8999 or python2 -m SimpleHTTPServer 8999. Thus, stationary states involve only such rotations. In classical physics, wed only need to know about its velocity to know everything about it. Create an empty named input and add a Training component to it. artist Maxim Gumin proposed the wave function collapse (WFC)[23] algorithm in 2016, which is one of the texture synthesis techniques. To simplify our understanding of wave functions, lets consider a 1-dimension space. Basically, the Fourier transform provides a description of any wave, like the one on the left, by an infinite combination of trigonometric waves, like those on the right: In this figure, the greater the wavenumber, the more arrows I have put on the trigonometric functions. Execution time for OWFC is very high due to the iterative nature of the algorithm. Create an empty named output and add an Overlap WFC component to it. This is a direct consequence of their relationship through the Fourier transform. Below is a sample meta-data used as input to TWFC. Made for PROCJAM 2017 | Thanks to our Kickstarter backers . Wave Function Collapse (WFC) by @exutumno is a new algorithm that can generate procedural patterns from a sample image. 1.1 Specifying and simulating a Markov chain What is a Markov chain? In a nutshell: My implementation of the Wave Collapse Function algorithm in Python 2.7 is flawed but I'm unable to identify where the problem is located. @sbrl I'd say the difference is propagation and backtracking, so they'll both solve a tile the same way, but WFC will propagate that choice eliminating possibilities throughout the map that would be illegal, and if that propagation ends up with a tile where no more choices remain it will backtrack and try a new choice. This corresponds to a Markov dynamics, where the probabilities of transitions towards other quantum states are defined by the current state. In this example, we would like to have mario style coin features, a few tiles above the platforms. Due to the nature of the constraints used by the algorithm, there are instances where entropy resolution cannot resolve the output pixel entropies. Yes! Instead, to account for a rate of occurrence, TWFC requires this information to be some tile frequency or tile weight in the meta-data file. WFC's special approach to constraint solving is a process of elimination. Your can now load your index with the url http://localhost:8999, Now let's feed the ImageData into an OverlappingModel. Whats extremely weird is that this probabilistic distribution depends on whats being measured. 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. We'll start with a function that loads an image element, draws it to a temporary canvas, and extracts the ImageData. What about the Copenhagen interpretation? Your email address will not be published. Instantly share code, notes, and snippets. 2.1. It's mainly useful for "vertical" words, where you want a distinct ground and sky separation. Key focus: Markov chains are a probabilistic models that describe a sequence of observations whose occurrence are statistically dependent only on the previous ones. The following figure displays the possible collapses of a wave function, where bolder arrows correspond to more likely collapses. Does the quantum wave function represent reality? 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! Keywords: Wave Function Collapse, Procedural City Generation, Machine Learning, Generative Adversarial Network, 2.2 Artificial Intelligence application to Wave Function Collapse, 2. This trick also works for the general structure of your patterns. 1 is just the original input, 2-8 adds mirrored and rotated variations. In this setting, waves are commonly represented as signals through space, as follows. Enforcing the patterns building desirable content, Figure 2 On the left: Input image that indicates areas with specific desirable values, on the right: the solution that corresponds to the input image, Figure 3 Examples of the 2D patterns generated by Max Gumin with a use of WFC, Figure 4 Island generated with Wave Function Collapse in Bad North. In this setting, the wavenumber corresponds to the number of cycles of the trigonometric wave in 1 meter. Adding prefabs to it's palette property will display them below the image area, where they can be sampled by clicking them with the S key. The results of the algorithm are convincing as the constraint solving texture synthesis maintains the constraints in output. Here is an introduction to the theory. From creating radio to hearing sounds, this concept is a translation between two mathematical worlds: Signals and Frequencies. 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. Recent discoveries in the branch of physics known as quantum mechanics have powerful applications in the field of network security - they have the potential to break forms of internet security based on mathematics such as the RSA algorithm, and also present new ways to safely send information. In other words, Plancks constant is the energy of a wave which turns at the speed of 1 turn per second. If you're into this kind of thing ExUtumno notes that the Simple Tiled Model is analagous to an order-1 Markov chain, and more complex models are like those of higher order. Even worse, there is no mathematical basis to say what it means for something to be measured. As it turns out, in any defined system, not all energies can be reached by a stationary wave. Now, we have to be a little more careful before actually talking about frequencies. 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. Hummm Thats a tough one. In particular, there is no spreading relation between the position in the horizontal direction and the momentum in the vertical direction. According to this, any wave is uniquely decomposable into an infinite superposition of trigonometric waves of different frequencies. 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. By doing so, the tiling model can benefit from not having to use per-pixel entropies, instead, it can have a tile that can comprise some set of pixels that fit on a fixed tile grid. Yes! WFC consists of two parts, an input Model and a constraint Solver. This is important. Does this mean that a wave function is associated with several momenta? We'll take a look at the kinds of output WFC can produce and the meaning of the algorithm's parameters. In fact, by considering that the weight of trigonometric waves which dont appear on the right is 0, we can consider this output as a matching of every wavenumber with an amplitude. It is able to produce larger texture from a smaller input while The image below as an input provided to OWFC: If a kernel refects the occurrence of the entropies on the neighbor pixel, the entropy is maintained, otherwise, all the options that are unlikely to occur are eliminated from the neighbor pixel. This should sound surprising to you. Im not sure what the Fourier transform stands for in this picture? Implies the local constraints, a probabilistic approach and Maxim Gumin's passion for Markov's work. Does this mean that the representation as a signal done above is wrong? Yes! You calculate the evolution of the density matrix (instead of the wavefunction), and it's always a deterministic PDE, even during "measurement". During the observation phase, one tile is selected and given a single random solution from the remaining possibilities. In the figure above, the most likely collapses correspond to positions blue, green and red, because the distances from the origin to the value of the wave function at these points are greater. Show that the Green's function is . Then, the amplitude of the wave function at a position in space is represented by its associated dot in the complex plane. We'll start with a function that loads an image element, draws it to a temporary canvas, and extracts the ImageData. WFC is especially exciting for the game jam community, where time is limited. Depending on your tile dimensions, you may have to change the Gridsize property on the Training component. I invite everyone interested in some coding challenges to implement one . After you've included the WFC code in your project and made sure it compiles, you're ready to test some input. 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." Then we'll walk through setting up WFC in javascript and the Unity game engine. The most likely collapse is the red one. Wavefunction collapse is most accurately treated through decoherence theory (see quantum decoherence, einselection, etc.) It will experience decoherence. Its a wave function which will appear to be non-moving. Hey, nice tutorial! With python, this can be as simple as opening a terminal in the project directory and running python3 -m http.server 8999 or python2 -m SimpleHTTPServer 8999. Came here looking for a . Wave Function Collapse (WFC) is a constraint solving algorithm named after the quantum physics wave superposition concept. If it went clockwise, it would be negative. Wave Function Collapse (WFC) by @exutumno is a new algorithm that can generate procedural patterns from a sample image. Child prefabs that are placed within the Training bounds will be read by the algorithm. It's . 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. NOTE: For this tuturial, we'll be focusing on the Overlap Model, as it's easier to create input for. 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. But its not the case in quantum mechanics! Entropy resolution may not always take place in case there is too much detail in the image (i.e: Too many colors to resolve, too many intricate patterns to resolve for). symmetry (int) represents which additional symmetries of the input pattern are digested. Inefficient iterative check-in MkWFC model that does some repetition on the input sample during constraint identification. Hey @selfsame! Note that the trigonometric function in the figure turns anti-clockwise. This, however, comes at the cost of manually generating a meta-data file that feeds the relevant constraint information to TWFC which is non-trivial, this affects the scalability of TWFC as increasing the complexity of the input can result in the need for increased investment of resources in determining the constraints required to generate desired results. The wave can then be seen as a colored path in the complex plane. Here is an introduction to the theory. As quantum mechanics was approached with concepts of classical mechanics, this frustrating phenomenon was commented as the impossibility of knowing both position and momentum simultaneously. It's especially exciting for game designers, letting us draw our ideas instead of hand coding them. An assortment of functions and a class to efficiently and easily evaluate and analyze Markov Chains. The algorithm gained much popularity because of the variety of the outputs generated from only one input. A Markov chain is a stochastic process, but it differs from a general stochastic process in that a Markov chain must be "memory-less."That is, (the probability of) future actions are not dependent upon the steps that led up to the present state. This dynamics of wave functions is what will be discussed here. These objects are studied by. Depending on your tile dimensions, you may have to change the Gridsize property on the Training component. More precisely, wave functions are elements of the Hilbert space $L^2$. Create an empty named input and add a Training component to it. An operator is a Hermitian linear mapping of the Hilbert space into itself. You should see a round blue gizmo at the center of every tile within it's bounds. It implies that the wavenumber of the trigonometric wave is positive, and that the associated wave function goes from left to right. Now, I have been using the word probabilistic, but you dont need to mention probability to describe the spreading of the wave function. Child prefabs that are placed within the Training bounds will be read by the algorithm. This may at first be considered a bug with the algorithm but is actually a valid output due to constraint identification allowing the placement of castles next to the adjacency scenarios exhibited by the neighboring tiles on the map. This other half is what people often refer to as quantum weirdness. In the case of 2D, the OWFC model splits an input image into smaller NxN sized kernels. Classically, the position and the momentum of a particle are two distinct concepts which can be defined independently from each other. To perform the entropy resolution, an arbitrary pixel is selected in the output and assigned one of its possible color values. Because of Paulis exclusion principle, two electrons cannot be in the same quantum state. From the Fourier transform that provides a mapping of every wavenumber with an amplitude, we can easily deduce a mapping of every momentum with an amplitude. Recall that these waves are fields which map each point of space with a number. When running the trials highlighted above using MkWFC and comparing the performance against TWFC, there was some interesting phenomenon worth noting. But, overall, I really dont understand the measurement problem. This classical subject is still very much alive, with important developments in both theory and applications coming at an accelerating pace in recent decades. As pointed out by Derek, Heisenbergs principle only applies to spreadings along one dimension. As you start using WFC, you'll start to get a feel for how it transforms input patterns into output. It will remain in a superposition. Also, the farther to the origin a point is, the greater the speed must be. Wave Function Collapse (WFC) by @exutumno is a new algorithm that can generate procedural patterns from a sample image. Its a much more significative concept in physics because of the law of conservation of momentum, which states that the momentum of a closed system is always conserved. Click to share on Facebook (Opens in new window), Click to share on Twitter (Opens in new window), Click to share on LinkedIn (Opens in new window), Click to share on Pinterest (Opens in new window), Click to share on Reddit (Opens in new window), Click to share on Pocket (Opens in new window), Click to share on Skype (Opens in new window), Click to share on Tumblr (Opens in new window). my article on the essence of quantum mechanics, The Most Beautiful Equation of Math: Euler's Identity, Fourier Analysis: Signals and Frequencies. In this article well see how a physics-based method can be used to secure online information. Text Generation. It's especially exciting for game designers, letting us draw our ideas instead of hand coding them. 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. Markov Chains are a tool used to calculate probabilities of entering a state given their previous state. The final feature is backtracking. In fact, the square of the module of a coordinate is the probability that the wave function collapses to the eigenvector associated to the coordinate. This meta-data can be used by TWFC to propagate constraints and solve the entropies of kernels without having to determine them in a pre-compute step. This method, called the Metropolis algorithm, is applicable to a wide range of Bayesian inference problems. Recall that the wave function is obtained by summing all momentum space wave functions of all momenta. Instead of computing the kernels present in an input sample, the TWFC model relies on a meta-data file that can provide the constraints required to generate an output image. I'm not sure I quite understand that bit. It's especially exciting for game designers, letting us draw our ideas instead of hand coding them. But to understand this, Id need to introduce operators, which would take too much time. 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. This phenomenon accounts for Niels Bohrs weird model of the hydrogen atom, as explained in this video from the Science Channel: Fortunately, no. It starts with Kleineberg's core idea, and limits it to a one-dimensional grid a sentence, with grid locations for each word. Now, recall that the wave function is defined up to a rotation in the complex plane. If you're designing a dungeon, floor variants can be included to encourage the emergence of larger rooms. Without Paulis exclusion principle, electrons would not be able to combine atoms together, which means that there would be no chemistry. Sorry for the long detour. You can get around this problem by either uploading your project somewhere, or running a local server on your computer. Markov chains illustrate many of the important ideas of stochastic processes in an elementary setting. For instance, in the case of the figure above, where the momentum space wave function was localized around the pink-red momentum, while the wave function was spread all over the spectrum. It's usefull for creating things like tileable textures, but also has a surprising influence on the output. So how does Heisenbergs spreading principle translate into mathematics? The collapse of the wave function is a central problem which physicists continue to explore. Start a new unity project and either import the .unitypackage or clone the github repository into the Assets directory. This toolbox supplies functions for evaluating . It has been widely taught worldwide. Video of my thesis defense for my work on Markov Chain based Wave Function Collapse. These two conditions correspond to the Hamiltonian at position being proportional to the wave function at this position with a real-valued proportionality coefficient independent from this position. If an observation & propagation result in an unsolvable contradiction, they're reverted and a different observation is attempted. You can get around this problem by either uploading your project somewhere, or running a local server on your computer. Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UKEssays.com. 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. . NOTE: The Tile Painter component can speed up input creation, letting you click and draw tiles from a palette of prefabs. Its not wrong. When working with WFC, it's often a good idea to toggle Periodic Output on and off, checking if either setting influences the results in a favorable way. From creating radio to hearing sounds, this concept is a translation between two mathematical worlds: Signals and Frequencies. 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. more replies. So a vertical localized wave would be essentially identical to the one above! 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