@ -54,6 +54,6 @@ Aliasing is also a common problem in [computer graphics](computer_graphics.md).
The same thing may happen in [ray tracing](ray_tracing.md) if we shoot a single sampling ray for each screen pixel. Note that [interpolation/filtering](interpolation.md) of textures won't fix texture aliasing. What can be used to reduce texture aliasing are e.g. by [mipmaps](mip.md) which store the texture along with its lower resolution versions -- during rendering a lower resolution of the texture is chosen if the texture is rendered as a smaller size, so that the sampling theorem is satisfied. However this is still not a silver bullet because the texture may e.g. be shrink in one direction but enlarged in other dimension (this is addressed by [anisotropic filtering](anisotropic_filtering.md)). However even if we sufficiently suppress aliasing in textures, aliasing can still appear in geometry. This can be reduced by [multisampling](multisampling.md), e.g. sending multiple rays for each pixel and then averaging their results -- by this we **increase our sampling frequency** and lower the probability of aliasing.
**Why doesn't aliasing happen in our eyes and ears?** Because our senses don't sample the world discretely, i.e. in single points -- our senses [integrate](integration.md). E.g. a rod or a cone in our eyes doesn't just see exactly one point in the world, it sees an averaged light over a small area, and it also doesn't sample the world at specific moments like cameras do, its excitation by light falls off gradually which averages the light over time, preventing temporal aliasing.
**Why doesn't aliasing happen in our eyes and ears?** Because our senses don't sample the world discretely, i.e. in single points -- our senses [integrate](integration.md). E.g. a rod or a cone in our eyes doesn't just see exactly one point in the world but rather an averaged light over a small area (which is ideally right next to another small area seen by another cell, so there is no information to "hide" in between them), and it also doesn't sample the world at specific moments like cameras do, its excitation by light falls off gradually which averages the light over time, preventing temporal aliasing.
So all in all, **how to prevent aliasing?** As said above, we always try to satisfy the sampling theorem, i.e. make our sampling frequency at least twice as high as the highest frequency in the signal we're sampling, or at least get close to this situation and lower the probability of aliasing. This can be done by either increasing sampling frequency (which can be done smart, some methods try to detect where sampling should be denser), or by preprocessing the input signal with a low pass filter or otherwise ensure there won't be too high frequencies.
@ -96,12 +96,6 @@ It also has the nice side effect of making this less likely to be used by corpor
The culture of being offended is [bullshit](bullshit.md), it is a [pseudoleftist](pseudoleft.md) (fascist) invention that serves as a weapon to justify censorship, canceling and bullying of people. Since I love all people, I don't support any weapons against anyone (not even against people I dislike or disagree with). People are offended by language because they're taught to be offended by it by the propaganda, I am helping them unlearn it.
### WTF your 404 page called me a retard :( ;_;
Sorry, please don't take it personally and allow a bit of politically incorrect humor on this site.
Keep in mind that in reality I love you with all my heart :) <3
### But how can you so pretentiously preach "absolute love" and then say you hate capitalists, fascists, bloat etc.?
OK, firstly we do NOT love *everything*, we do NOT advocate against hate itself, only against hate of living beings (note we say we love *everyone*, not *everything*). Hating other things than living beings, such as some bad ideas or malicious objects, is totally acceptable, there's no problem with it. We in fact think hate of some concepts is necessary for finding better ways.
The field of formal languages tries to [mathematically](math.md) and rigorously examine and describe anything that can be viewed as a language, which probably includes most structures we can think of, from human languages and computer languages to visual patterns and other highly abstract structures. Formal languages are at the root of theoretical [computer science](compsci.md) and are important e.g. for [computability](computability.md)/decidability, computational complexity, [security](security.md) and [compilers](compiler.md), but they also find use in linguistics and other fields of [science](science.md).
The field of formal languages tries to [mathematically](math.md) and rigorously view problems as languages; this includes probably most structures we can think of, from human languages and computer languages to visual patterns and other highly abstract structures. Formal languages are at the root of theoretical [computer science](compsci.md) and are important e.g. for the theory of [computability](computability.md)/decidability, computational complexity, [security](security.md) and [compilers](compiler.md), but they also find use in linguistics and other fields of [science](science.md).
A **formal language** is defined as a (potentially infinite) set of strings (which are finite but unlimited in length) over some alphabet (which is finite). I.e. a language is a subset of E* where E is a finite alphabet (a set of *letters*). (* is a *Kleene Star* and signifies a set of all possible strings over E). The string belonging to a language may be referred to as a *word* or perhaps even *sentence*, but this word/sentence is actually a whole kind of *text* written in the language, if we think of it in terms of our natural languages.
A **formal language** is defined as a (potentially infinite) set of strings (which are finite but unlimited in length) over some alphabet (which is finite). I.e. a language is a subset of E* where E is a finite alphabet (a set of *letters*). (* is a *Kleene Star* and signifies a set of all possible strings over E). The string belonging to a language may be referred to as a *word* or perhaps even *sentence*, but this word/sentence is actually a whole kind of *text* written in the language, if we think of it in terms of our natural languages. The [C](c.md) programming language can be seen as a formal language which is a set of all strings that are a valid C program that compiles without errors etc.
**For example**, given an alphabet [a,b,c], a possible formal language over it is [a,ab,bc,c]. Another, different possible language over this alphabet is an infinite language [b,ab,aab,aaab,aaaab,...] which we can also write with a [regular expression](regex.md) as a*b. We can also see e.g. English as being a formal language equivalent to a set of all texts over the English alphabet (along with symbols like space, dot, comma etc.) that we would consider to be in English as we speak it.
@ -13,7 +13,7 @@ A **formal language** is defined as a (potentially infinite) set of strings (whi
We usually classify formal languages according to the **[Chomsky](chomsky.md) hierarchy**, by their computational "difficulty". Each level of the hierarchy has associated models of computation ([grammars](grammar.md), [automatons](automaton.md), ...) that are able to compute **all** languages of that level (remember that a level of the hierarchy is a superset of the levels below it and so also includes all the "simpler" languages). The hierarchy is more or less as follows:
- **all languages**: This includes all possible languages, even those that computers cannot analyze (e.g. the language representing the [halting problem](halting_problem.md)). These languages can only be computed by theoretical computers that cannot physically exist in our universe.
- **type 0**, **recursively enumerable languages**: Most "difficult"/general languages that computers in our universe can analyze. These languages can be computed e.g. by a **[Turing machine](turing_machine.md)**, [lambda calculus](lambda_calculus.md) or a general unrestricted [grammar](grammar.md). Example language: all strings encoding a [Game of Life](game_of_life.md) run which ends in finite time. { At least I think :) ~drummyfish }
- **type 0**, **recursively enumerable languages**: Most "difficult"/general languages that computers in our universe can analyze. These languages can be computed e.g. by a **[Turing machine](turing_machine.md)**, [lambda calculus](lambda_calculus.md) or a general unrestricted [grammar](grammar.md). Example language: a^n where *n* is not a [prime](prime.md).
- **type 1**, **context sensitive languages**: Computed e.g. by a linearly bounded non-deterministic Turing machine or a context sensitive grammars. Example language: a^(n)b^(n)c^(n), n >= 0 (strings of *n* *a*s, followed by *n* *b*s, followed by *n* *c*s).
- **type 2**, **context free languages**: Computed by e.g. non-deterministic pushdown automata or context free grammars. (Deterministic pushdown automata compute a class of languages that is between type 2 and type 3).
- **type 3**, **regular languages**: The *easiest*, *weakest* kind of languages, computed e.g. by [finite state automata](finite_state_automaton.md)s or [regular expressions](regexp.md). This class includes also all finite languages.
