Talk:Discrete cosine transform

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partially normalized, scaling its result by 1/sqrt(2*n). This happens in line 44 of dct.m (Matlab R2008b), and corresponds to what the article terms as the unitary modification. In contrast, the fft function is not implemented in this manner, and instead matches the unnormalized naive DFT. In any case, rather than try to pick apart Matlab's code, it would be better to look at an alternate implementation. I have recreated the example both with FFTW and with a direct implementation of the naive form (I can provide the code on request). Both are equivalent to each other, and appear to match the DFT example shown, but the DCT example shown in the figure is very different, and looks significantly closer to the DFT result. That is to say, the clear advantage of DCT as shown in the example - which was, in fact, what generated me to try to recreate it in the first place - is not at all evident.

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outside the sample window (periodic vs. even extension). While this is heuristically useful in 8x8 image blocks, in which case the true extension is likely to resemble an even-mirrored function rather than a periodic one, in this whole-image example a periodic extension would appear to work just as well (all the borders have similar frequencies and colors). Now, I'd like to stress that I don't want to appear as denegrating Mr. Damato's hard work and good intentions. But unless I am wrong (and I am open to that possibility, though after having considered and investigated this for a while I believe it to be an unlikely one) it will be necessary to promptly remove the example. Best,

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difference; part of the problem with this example may be that it has so many sharp edges from the blades of grass. (For the same reason, JPEG compression is well known to have the hardest time with sharp lines and edges.) To see a really dramatic difference, it would be good to use a fairly smooth (soft-focus?) image, with very different brightness at the boundaries (e.g. sky above and ground below) as you point out. (The nice thing about 8×8 blocks is that, even if the overall image has many sharp edges, normally there will be many smooth sub-blocks within the image.)

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sequence is zero", which is what can be see in the diagram of DCT-III's input (I am assuming that this was your interpretation). My best guess is that the zeros are added in order to allow the mid-point to fall at an integer index. This is required because the boundaries are fractional (n=-1/2 and n=N-1/2). Note that the diagram for DCT-III clearly shows that the boundaries are not fractional - they are the first value of the sequence (n=0), and the value past the end of the sequence (n=N) which is set to zero. --

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prevents the cos^2() in the 2D DCT which is computationally more efficient and the math equation appearance is simpler which allows a better fundamental understanding. I assume the FFT applied to the DFT comes out the same as using FFT applied to 2D DCT. The DFT allows a direct view of the effect of the FT phasors as in the youtube video of a 2D homer simpson line drawing reproduced from the phasors of the DFT.

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symmetries (you can put a sine term into the first sum to get a DFT, because the odd sine terms will sum to zero for even data). ...and because the DFT is unitary (with appropriate normalization), all of the DCT types are orthogonal (with appropriate scaling). Again, however, it seems like you need more extensive tutoring than you can reasonably expect on a Talk page (or from an encyclopedia article).

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I should get the exact waveform back (within rounding errors of the computer), but I don't instead I get a new waveform that is so big it goes out of the range of my waveform display (easily in 1000s or even 1000000s range)! And I made sure my implementation of DCT-II and DCT-III used the EXACT algorithms as shown on the DCT Knowledge article here. I think something is wrong with your formulas here.

3342:] is a brief summary of relatively obscure article that says "While data compression has made significant contribution to the digital revolution in computer science and telecommunications..." It is simply not reliable source for such grandeur claim. And it does not mention DCT directly. Finally third source says nothing about Digital revolution.So again I am going to delete this claim. 22: 1570:

The factor of 2 in the cosine argument is due to the fact that you can cut the integration interval in half by the even symmetry.) There are many fine textbooks on discrete cosine transforms (e.g. the Rao and Yip book) that you can read to help clear up any misunderstandings—Knowledge talk pages are not someplace where you can expect to get extensive tutoring.

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and partial differential equations and Chebyshev approximation and numerous other applications (its energy compaction property, its diagonalization of convolutions with certain boundary conditions, its properties under differentiation, its relationship to a Fourier cosine series) all stem directly from its relationship to a DFT of even symmetry.

1613:, with a much better structure, and much less assumption about readers knowledge ! It is unnecessarily complicated, despite a DCT can be very simple to understand. Thanks, when I learned it, there was still no Internet, no Knowledge. But academi resources were much better written and much more accessible than this very poor article. 3763:

definition and also the fact that there are 8 slightly different variants. I have read in many places that DCT performs better in data compression, but I have never seen any quantification of this performance. Are there are some rigorous results showing that DCT performs better or is this always subject to our perception?

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When I perform a DCT-II (aka "the DCT") and then perform its supposed inverse DCT-III (aka "the IDCT") I do NOT get back my original input. It's a HUGE output. If my input waveform (that is in the range of +/- 100) I then transform with a DCT-II, and then immediately transform it back with a DCT-III,

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just to be sure I made a test: I have seen the DCT and FFT of a lossless image and of the same image heavily JPEG compressed (10% quality). I have noticed that the compressed picture had a wider spectrum than the original (I think because of the blocking effect). You can try to do it as well with any

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2) if cosine function starting at zero, ending at T - cosine transform get us only one element, rank two times highest than Fourier. Moreover, if we have only cosines different ranks starting at 0, ending at T (no phases), we obtain in cosine transform this maplitudes in even spectrum elements, zero

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it is not exactly correct: JPEG compression removes higher frequencies in each 8x8 block, while the picture shows the transform of the whole image; then JPEG causes the blocking artifact that introduces sharp changes in intensity on the edge of blocks, thus introducing other higher frequency. In any

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You make a very good point about the issues that this image's properties would have on the DCT. But I want to stress that the problem I am mentioning here is not a subjective issue with the presentation of the figure, but specifically that the two subfigures are not showing the same thing, because

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You're incorrect to state that they do not have an orthogonal basis. All of the DCTs I-VIII, appropriately normalized, are orthogonal matrices (i.e. form an orthogonal basis). (The relationship between DCT and DFT is analogous to the relationship between the cosine series and the Fourier series.

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or in P. Duhamel, “Implementation of ‘split-radix’ FFT algorithms for complex, real, and real-symmetric data,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-34, no. 2, pp. 285–295, 1986. The equivalence for the DCT II was commented on in e.g. M. Vetterli and H. J. Nussbaumer, “Simple FFT

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Well it's my understanding that the image is trying to illustrate that because the DCT is good at compressing the image energy into one small corner of the transform-space, it is useful for image compression. But, in order to show this, you need to demonstrate that DCT does this on an uncompressed

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A square matrix can be invertible without being orthogonal; the inverse may just be more complicated to compute. And there are infinitely many invertible real matrices, so the fact that it is real is hardly the defining property either. The things that make the DCT valuable to signal processing

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I believe there are several erroneous and exaggerated claims in the article. The claim that "Wavelet coding, the use of wavelet transforms in image compression, began after the development of DCT coding. The introduction of the DCT led to the development of wavelet coding, a variant of DCT coding

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All the formulas confusingly use N to refer to both the length of the input and output vectors, but these are typically different. And so what, for example, does it mean to have " PI / N " in the formula??? Which N??? This is VERY confusing -- why not use n/N and k/K which is a much more standard

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The example of the DFT and DCT of the image of the dandelion clock is highly misleading, because the original image was a JPEG-compressed image which will have thrown away information from the high-frequency DCT coefficients - so of course the DCT will have very little energy there! I think a new

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This article fails in all areas, is extremely badly structured. It should first present the simple DCT, present how it is computed, what it gives by an example, and then present how it can be inverted. Then the properties should be studied, and later the link to the more general theory of Discret

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After reading this article from a beginners point of view, and despite I have used and implemented various DCT variants, that were used in proiduction priduct form image and audio processing (since 1984!), I must admit that this article is a real nightmare, that just mixes opinions about what are

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unless I'm missing something, the figure that lists DCT I, II, III, and IV doesn't match with the text. The text describes the DCT-II as "...where the even-indexed elements are zero. That is, it is half of the DFT of the 4N inputs y_n, where y_{2n}=0 ... ". This matches DCT-III in the figure.

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well, in general I agree that real, unprocessed data have the highest quality, but I'm not sure we really need it for our purpose. I mean: a test image has to have some features (sharp edges, flat surfaces, details), like Liechtenstein, moreover it should have power in any frequency, and I think

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I should also point out that I believe the division by max(C) in the Matlab code may further exacerbate the problem, as well as the fact that the image is cropped to the lower-left corner, which obscures the fact that the DFT has negative frequencies. But these are minor compared to the larger

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In retrospect, there is no reason to assume that the DCT should show the such an improvement: as the FFTW website points out repeatedly, the DCT is equal to a real-even DFT of twice the length. As this article points out, the difference is specifically in the assumptions placed on the behavior

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The part that says "y_{2n}=0" refers to the input that would have to be provided to do an equivalent DFT, whereas the diagram shows the way that the DCT's input is effectively interpreted. Also, note that y_{2n}=0 means "every even element is zero", rather than "the element past the end of the

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You have the incorrect limits on the second sum (which should go from 0 to N, for an even extension), and there is no factor of 2 on the n=0 and n=N cases. However, the general idea is in the right direction: all 8 types of DCT can be converted to larger DFTs of real-even data with appropriate

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The DCT is more useful than the DFT only for non-complex. This is mentioned in the 2nd paragraph. The DFT is efficient than the 2D DCT if the 2nd D is assigned to the complex line; i.e. let the 2D x,y data be x+i*y in the DFT and use eq 3 on the DFT page. The inclusion of the sin() in the DFT

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Hey, I came here because I'm confused about the relation between DCT and DFT. It seems that they are closely related, but they are not quite the same (although for real signals they seem to be really closely related). DFT is conceptually simple. DCT is more complex, it suffices to look at the

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While the example makes a powerful case for the DCT over the DFT, after looking into it in detail I'm afraid that it is irreparably incorrect due to an odd ambiguity in the difference between Matlab's dct and dft functions. Specifically, the dct function is implemented in such a way that is

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that uses wavelets instead of DCT's block-based algorithm." is referenced by Hoffman. The problem is Hoffman does not say anything like this, that is that DCT led to wavelet. Moreover I suspect that is a typo or poorly worded sentence. On next page Hoffman says that "Fractal coding provides

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And finally the claim that "The wide adoption of DCT compression standards led to the emergence and proliferation of digital media technologies, such as digital images, digital photos, digital video, streaming media, digital television, streaming television, video-on-demand (VOD)," is also

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the image, then a real-even extension will also be continuous, whereas a periodic extension will not in general be continuous at the boundaries, leading in general to better "energy" compaction for the DCT. If the image itself is discontinuous, however, one doesn't expect to see much of a

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the DCT image is scaled down by a significant amount. This is therefore not an "apples-to-apples" comparison. Since it is objectively incorrect, it needs to be removed. I would replace it with an image with equal scale factor, but the DFT actually appears to be better than the DCT.

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Which mathematician (or group of mathematicians) coined this term? What year was it? Was it used before the year 1900? The main page of this article could be improved by giving a brief history behind the term, and what alternative terms were used before this term stuck.

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Actually, this seems like a confusion over the disambig pages, of which there are two. OP is looking at the page for "Dct", which has only this article and the DCT File Format. Responder is referring to the page for "DCT", which does in fact have quite a few more entries.

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problem. Can someone please confirm or refute my statement of the problem? (again, I would be happy to be shown wrong, if that is truly the case) If there is no objection, I should erase it. I'd especially like to hear from the original author, btw, if he has the time.

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the place for tutorial. Sending people to other resources first is really not the role of Knowledge, which is to introduce any subject with the simplest approach, and then add some development in other more detailed articles, and then send them to other external advanced

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I agree that a clearer example would be better, in any case—actually, it might be good to have an example where the DCT is better along with an example where it is about the same (e.g. an image with sharp edges), just to dispel any notion that the DCT is magically

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I want to locate precise boundary: what transform can be called DCT and what transform can`t. What I am seeking for is a precise definition of DCT. When I have such a definition, I can (or can`t) see all the mentioned above properties - either obvious or complex.

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in odd. If no phases, cosine transform not litters on other frequencies. If we have discrete transform, fo example 8 ssmples : x0..x7, x8 must be equal x0, and x9..x15 equal x7..x1. If I compute with plain Fourier this double samples I get 1) and 2)

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I just tried typing in the formula for the DCT 2 and 3 into my compiler to see what would happen. It wasn't inverting properly. It looks like it should be $ -\frac{1}{2} x_0$ instead of positive. Adding in the negative sign made it work for me.

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Cosine transform means : if we have signal on [0..T) and we add in interval [T..2T) mirror of this signal (I test, it is the same results if we mirror at zero, interval[-T..T) for -T the same as X0), next we apply _Fourier_transform_ we obtain:

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I'm not sure if this is the correct place to give feedback, but I'd just like to second this sentiment. The "Informal Overview" section of this article is excellent. It made me wish more Knowledge articles on mathematical topics had a similar

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This is not Mathpedia, it is an encyclopedia which is by definition supposed to contain content for the lay person, not serve as an academic textbook. I liken this to a dictionary that skips the definition to go right to usage and philology.

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I have never heard the terminology of "even" and "odd" ends used everywhere here without any definition ; I never needed that terminologogy to study DCT algorithms since 20 years, and I bet this comes from an abestract taken from an advanced

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one of the infinitely many other possible invertible real matrices, nor are they any one of the infinitely many possible real orthogonal matrices. I don't understand how you could have read what I wrote above to conclude that 2I is a DCT.

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Actually I do not entirely agree. I am not a physicist yet I understand what a nuclear weapon is (without understanding its mechanics) and Dr. Hawking has managed to simplify General Relatively for the masses without resorting to advanced

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I also note that the Liechtenstein test image is also JPEG-based. I accept that it isn't a terrible test image, and you have taken steps to mitigate JPEG artifacts, but surely the best test image deals with real, unprocessed data?

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Also, the part in the informal overview that talks about reducing discontinuities I find a bit misleading. We are dealing with a digital (discretised) signal here, right? So talking about discontinuities seems a bit out of place.

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I could go out for a walk and take a picture with my camera, storing it with TIFF to avoid any artifact, but I don't think it would be a featured picture... do you think we should try to take a picture by ourself to be used as

2901:"since the corresponding DFT is of length 2(N−1) (for DCT-I) or 4N (for DCT-II/III) or 8N (for DCT-VIII)." I think it should refer to DCT-IV instead of DCT-VIII, from the context. Anyway, I could not verify these lengths. -- 2579:

Just wanted to echo the sentiment above - a lot of authors/editors of the Knowledge maths and sciences articles could do everyone a service by taking a close look at that section as a model for how it should be done.

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This page seems to mention changing scale factors to make the transform orthogonal. I am fairly sure the transform is orthogonal without the scaling factors, and that the scaling is done to make the transform

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to DCT -based transform coding". If you read the whole section in this book, it is pretty obviously that wavelets are alternative to DCT and not based on them. Similarly all sources that I've read like this

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Shifting by 1/N does not result in a DCT according to any of the standard definitions, and is not sensible because only shifts by integers or half-integers preserve the sampling points under mirror flips.

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basis is that it corresponds to a DFT + even symmetry. This even symmetry is broken by the "grid" of sampling points if you don't place your origin at either a sample or halfway between two samples.

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Ah yes, of course you're right that JPEG only locally removes higher frequencies. However I still think that the example does not show what it claims to show, and I'd definitely like to see another.

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It would be nice if some qualified person would provide a better lay summary of DCT to explain what it does without needing to understand necessarily how it specifically does it. For example:

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Even symetry is one property of DCT. But if some other transform have or have not even symetry what is the influence on DCT discussion? Notice that I am refering to your own definition:

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exaggerated. Most of these sources simply say that those technology use dct, but not that dct "led" to their emeregence. For instance this article, referenced for digital photos

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Whoever originally wrote and all those who helped improve the informal overview section, you know your stuff and you have a knack for explaining it well! Thank you very much.

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transform? There was groundbreaking research in the 1990s that said "yes" (for the case of the non-equispaced Fourier Transform), and hence probably "yes" for the cosine case.

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Nope, you must have mistyped them, or have a bug in your program. (To see that your proposed minus sign isn't correct, consider the simple case of a transform of all 1's.)

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Having an orthogonal basis is not really the critical issue; there are lots of non-orthogonal bases out there that are very important. The reason that a DCT is a

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However the section on the DCT-III shows no such DFT equivalent. Can someone here please fill in the information regarding calculating the DCT-III using a DFT?

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By the way, this is my implementation in VB6, which I believe is an exact implementation of the DCT-II and DCT-III as shown in this Knowledge article:

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Ok I just figured out my problem. I had needed to multiply the output of the DCT-III by 2/DCTsize in order to make it the true inverse of the DCT-II.

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Liechtenstein has. If you hadn't known the picture was generated from a JPEG, would you have been able to realize it was originally stored as a JPEG?

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I think I have recognized the source of my personal ignorance in all this topic. I see that none of DCT I-VIII do not have orthogonal basis function:

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This even symmetry is broken by the "grid" of sampling points if you don't place your origin at either a sample or halfway between two samples.

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This article is pretty poor IMHO. How about explaining what it is without resorting to algebra, and also explaining its applications... --

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Er. Do you realize that this article now has a ~1mbyte animated gif inline? Thats seriously not-nice to people with limited bandwidth. --

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and DCT algorithms with reduced number of operations,” Signal Processing, vol. 6, no. 4, pp. 267–278, 1984; see also refs 1 and 6-10 of

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image. Otherwise all you demonstrate is that DCT is effective at compressing the energy in images which have already been JPEG-encoded.

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For DCTs, the input and output vectors are always the same length; otherwise the transform would not be invertible. There is only one

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Would be nice if someone added an example of how the DCT saves memory, by showing an 8x8 color block and going thru the math. Thanks,

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Excellent question. First, the answer is yes, you can perform a DCT on data that is non-uniformly distributed. Basically, replace

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Sorry, if you don't even know algebra you have no hope of understanding a DCT. Don't blame the article for your own limitations.

3165:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section. 1013: 946: 879: 313: 631: 529: 2019: 1606:

This articles fails everywhere, it is not made for Wikipedians, it's just a badly written abstract of some external paper!

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Relation to DFT gives us to calculate some properties easily. But once it is calculated, it have own uses. For example,

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If I had some better English skills (this is not my native language, although I can understand it without difficulty),

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Great page, really helped me out. The summation given in the formula for DCT-III starts at n=1, should this be n=0?

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Is it possible to compute a DCT if the data points are non-uniformly distributed? i.e. if I only know the function

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thanks for your review here. We had some POV pushing on this topic a while ago and not all of it was backed out. ~

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https://web.archive.org/web/20150711105353/http://wisnet.seecs.nust.edu.pk/publications/tech_reports/DCT_TR802.pdf

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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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A square matrix can be invertible without being orthogonal; the inverse may just be more complicated to compute.

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The article already has a long discussion of how the 8 types of DCT arise from a DFT of real-symmetric data.

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DCT should go to Discrete Cosine Transform and then mention up top DCT can also refer to DCT File Format.

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Next the claim that "DCT data compression has been fundamental to the Digital Revolution. ". First source

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disambiguation page. There's no reason to suggest that this article is the most prominent or relevant.

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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SA-DCT: Shape adaptive DCT. A new section inside DCT or a new article about SA-DCT should be created.

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The equivalencies can be found in many sources. The equivalence for the DCT-I is commented on e.g.

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I have started converting the notation in this article to match the changes that have been made to

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real inputs of even symmetry where the even-indexed elements are zero. That is, it is half of the

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For information on the non-equispaced FFT, google "NFFT". There are now several fast algorithms.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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before doing mass systematic removals. This message is updated dynamically through the template

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referenced for digital video again simply mention usage of DCT. The source for digital Cinema

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calculated by DCT I (boundary effect is neglectable here) gives cleaner picture than the same

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invertible linear transformations corresponding to DFTs of real-even data. They are not just

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Please help fix the broken anchors. You can remove this template after fixing the problems. |

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considered the current strategies, and what should be first a simple but solid definition.

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does not even mention DCT at all. JPEG article on wikipedia also need to be towned down.

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mentioned Digital revolution at the beginning but does not connect it to DCT. The second

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Please, help me to understand calculation of Inverse DCT given non-orthogonal basis like

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DCT is used in lossy image and audio compression by converting bitmaps into frequency....

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is about JPEG and only mention briefly DCT among other algorithms, while second source

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If you found an error with any archives or the URLs themselves, you can fix them with

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Ok. I fixed that code now. By the way, your suggested fix was wrong. The proper fix is

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In the DCT-II section it says that it can be calculated in terms of a DFT like this:

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No, the n=0 term is handled by a separate term since its normalization is different.

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Is it possible to emphasize importance of DCT I-VIII variants over other shifted DCT?

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with the actual grid point. But the more important question is, can we do this in a

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Array DCT seems to be not defined inside the function, perhaps the line should be

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1) spectrum with zero phases (zero imaginary part on FFT), only amplitude elements.

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I suppose, similar conversions are possible for the rest of boundary assumptions.

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Last edited at 16:08, 10 June 2007 (UTC). Substituted at 02:00, 5 May 2016 (UTC)

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to represent the length of the sequences in both the time and frequency domains.

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This transform is exactly equivalent (up to an overall scale factor of 2) to a

3258:. No special action is required regarding these talk page notices, other than 2817:: Knowledge is not the right forum to promote changes in the DCT definitions. 2512: 2088:

picture: you can find the Matlab code in the description page of the picture.

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to represent the transformation of the data sequence in the frequency domain.

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example should be created based on a genuine, non-lossy-compressed original.

3386: 3363:], in fact mentions MDCT and not DCT. I am going to tone down those claims. 2939: 2931: 3220:

http://wisnet.seecs.nust.edu.pk/publications/tech_reports/DCT_TR802.pdf

1122:. If you know something not based on orthogonality, please point me on. 182:

This article links to one or more target anchors that no longer exist.

2770:

is equal 1 for even k, 0 for odd k To normalize with input amplitudes

1588:

Fourier transforms may be added, as well as variants of the algorithm.

1153: 1681:

However, inverse transform can be easily computed given this trick:

290:

to represent the index variable for the frequency domain sequence.

1118:

As far as I know orthogonality is the sole property that gives us

2991:

DCT(k) = DCT(k) + WaveForm(n) * Cos(Pi * k / DCTsize * (n + 0.5))

2265:

Which is correct? BTW, I like the figure a lot, very intuitive.

2001: 1929:{\displaystyle \sum _{n=0}^{2N-1}\left=2\sum _{n=0}^{N-1}\left} 3046:

IDCT(k) = IDCT(k) + DCT(n) * Cos(Pi * n / DCTsize * (k + 0.5))

3011:

IDCT(k) = IDCT(k) + DCT(n) * Cos(Pi * n / DCTsize * (k + 0.5))

2344: 1625:

Perhaps, I can not agree about orthogonality of base function

271:

to represent the index variable for the time domain sequence.

156: 15: 3225:

When you have finished reviewing my changes, please set the

2191:, since it's a topic about a more general subject than DCT) 1598:

And I am completely opposed to your opinion here: Knowledge

3210:

for additional information. I made the following changes:

1070:{\displaystyle e^{j2\pi {\frac {(m+{\frac {1}{4}})n}{N}}}} 1003:{\displaystyle e^{j2\pi {\frac {(m+{\frac {1}{N}})n}{N}}}} 936:{\displaystyle e^{j2\pi {\frac {(m+{\frac {1}{2}})n}{N}}}} 523:

What is impact on resulting DCT if instead of calculating

1444:

Despite this, I do see some superior properties of basis

1679:

We can check this for case N = 2 (or N = 4 if you want).

1611:

I would delete it and restart it completely from scratch

3203: 3157: 2067:

case, I was thinking about re-making the picture using

739:{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cos \left} 618:{\displaystyle X_{k}=\sum _{n=0}^{N-1}x_{n}\cos \left} 3707: 3658: 3619: 3585: 3536: 3498: 3468: 3445: 3418: 2841: 2776: 2747: 2618: 1740: 1689: 1631: 1513: 1450: 1396: 1337: 1245: 1016: 949: 882: 830: 756: 634: 532: 472: 424: 2797:

should be divided by N for k=0 and by N/2 for k: -->

1381:{\displaystyle cos\left({\frac {2\pi }{N}}nk\right)} 101:, a collaborative effort to improve the coverage of 3262:using the archive tool instructions below. Editors 2607:But formulas DCR I..DCT IV satisfy only 1), not 2) 2114:any suggestion?! what's wrong with that, then? :-) 1961:Under the formal definition, it is not stated what 1670:{\displaystyle cos\left({\frac {\pi }{N}}nk\right)} 1552:{\displaystyle cos\left({\frac {\pi }{N}}nk\right)} 1489:{\displaystyle cos\left({\frac {\pi }{N}}nk\right)} 1435:{\displaystyle cos\left({\frac {\pi }{N}}nk\right)} 3731: 3693: 3644: 3606: 3571: 3523: 3484: 3454: 3427: 3036:Below is my new code for implementing a true IDCT: 2867: 2789: 2762: 2729: 1928: 1724: 1669: 1551: 1488: 1434: 1380: 1258: 1069: 1002: 935: 864: 769: 738: 617: 486: 439: 2513:http://en.wikipedia.org/Dual-clutch_transmission 2249:) if each of its columns is of unit magnitude. 3161:, and are posted here for posterity. Following 2245:is only orthogonal (by the definition given at 3401:How is DCT-III calculated in terms of the DFT? 3248:This message was posted before February 2018. 3155:The comment(s) below were originally left at 3131:Fails to explain when DFT is better than DCT? 1152:Formally, the discrete cosine transform is a 822:I had to do some homework to get your point:) 8: 1309:I will not try to prove that 2I is a DCT :) 1235:In case you have something simple like 2 * 1223:If so, may be I don`t know what is exactly 414:DCT for non-uniform distributed data points 3198:I have just modified one external link on 2343:There are approximately 15 entries on the 2069:Image:Lichtenstein img processing test.png 1965:represents. Or have I missed something? -- 47: 3758:Explanation of how DCT is better than DFT 3706: 3685: 3663: 3657: 3624: 3618: 3584: 3563: 3541: 3535: 3503: 3497: 3473: 3467: 3444: 3417: 2916:You're right, it should have been DCT-IV. 2859: 2846: 2840: 2781: 2775: 2746: 2706: 2688: 2672: 2661: 2648: 2623: 2617: 2515:because that uses the abbreviation DCT. 2241:is orthogonal with any scale factor, the 1986:Would be nice if someone added an example 1898: 1878: 1857: 1846: 1800: 1780: 1756: 1745: 1739: 1707: 1694: 1688: 1646: 1630: 1528: 1512: 1465: 1449: 1411: 1395: 1352: 1336: 1246: 1244: 1043: 1031: 1021: 1015: 976: 964: 954: 948: 909: 897: 887: 881: 865:{\displaystyle e^{j2\pi {\frac {mn}{N}}}} 845: 835: 829: 757: 755: 715: 696: 679: 663: 652: 639: 633: 594: 577: 561: 550: 537: 531: 476: 471: 423: 398:Tolstoy143 - "Quos vult perdere dementat" 345:less algebra, and more explanation please 2879:a discrete cosine transform definition. 2187:(I think this discussion should move to 1595:Most people won't need that terminology. 3158:Talk:Discrete cosine transform/Comments 2469:kind of notation and is not confusing. 1504:, calculated by "orthogonal basis" DCT. 1239:you can know its inverse easily - it`s 186:] The anchor (#AppleInsider) has been 49: 19: 1288:No. "Discrete cosine transforms" are 301:: Change all instances of the symbol 282:: Change all instances of the symbol 244:: Change all instances of the symbol 3237:to let others know (documentation at 2549:2602:304:47EB:F9F9:21B:77FF:FEAD:46DE 2511:I would suggest adding a redirect to 263:: Change all instances of the symbol 7: 95:This article is within the scope of 2958:DCT-III is NOT an inverse of DCT-II 38:It is of interest to the following 14: 3796:Mid-priority mathematics articles 3202:. Please take a moment to review 3163:several discussions in past years 2189:the wikiproject about electronics 115:Knowledge:WikiProject Mathematics 3112:IDCT(k) = FreqDomainData(0) / 2 160: 118:Template:WikiProject Mathematics 82: 72: 51: 20: 3050:IDCT(k) = IDCT(k) * 2 / DCTsize 314:Talk:Discrete Fourier transform 135:This article has been rated as 3694:{\displaystyle y_{4N-n}=y_{n}} 3572:{\displaystyle y_{2n+1}=x_{n}} 2757: 2751: 2724: 2703: 2641: 2635: 2590:18:18, 27 September 2019 (UTC) 1725:{\displaystyle x_{n}=x_{2N-n}} 1259:{\displaystyle {\frac {1}{2}}} 1053: 1034: 986: 967: 919: 900: 770:{\displaystyle {\frac {1}{N}}} 725: 706: 434: 428: 220:06:50, 19 September 2010 (UTC) 1: 3732:{\displaystyle 0<n<2N.} 3607:{\displaystyle 0\leq n<N,} 3321:Exaggerations in the article. 3316:21:42, 13 December 2016 (UTC) 3083:IDCT(k) = FreqDomainData(0) 2889:20:45, 16 November 2013 (UTC) 2827:21:23, 15 November 2013 (UTC) 2808:20:56, 15 November 2013 (UTC) 2275:03:14, 18 November 2007 (UTC) 2254:19:17, 7 September 2007 (UTC) 2231:18:48, 7 September 2007 (UTC) 1951:17:22, 11 February 2007 (UTC) 1618:21:57, 16 February 2007 (UTC) 456:20:09, 26 November 2006 (UTC) 364:20:18, 19 November 2006 (UTC) 354:10:21, 19 November 2006 (UTC) 339:22:15, 1 September 2006 (UTC) 109:and see a list of open tasks. 3791:C-Class mathematics articles 3395:00:19, 18 October 2021 (UTC) 3373:00:45, 15 October 2021 (UTC) 2952:16:06, 30 January 2014 (UTC) 2911:10:08, 29 January 2014 (UTC) 2459:21:14, 31 January 2009 (UTC) 2441:21:14, 31 January 2009 (UTC) 2422:17:46, 29 January 2009 (UTC) 2392:02:54, 29 January 2009 (UTC) 2368:05:27, 27 January 2009 (UTC) 1575:19:09, 6 February 2007 (UTC) 1325:17:27, 4 February 2007 (UTC) 1302:20:57, 3 February 2007 (UTC) 1218:16:58, 1 February 2007 (UTC) 1103:00:46, 1 February 2007 (UTC) 876:set of functions as well as 814:14:19, 30 January 2007 (UTC) 801:13:17, 30 January 2007 (UTC) 513:00:10, 25 January 2008 (UTC) 3171:Possibly Mid-priority, cf. 2868:{\displaystyle x_{0}=x_{N}} 2574:16:28, 26 August 2016 (UTC) 2557:15:50, 31 August 2012 (UTC) 2397:If the image is continuous 1388:is indeed orthogonal basis. 324:16:09, 6 January 2006 (UTC) 3812: 3279:(last update: 5 June 2024) 3195:Hello fellow Wikipedians, 3173:Discrete Fourier transform 3146:12:37, 12 March 2016 (UTC) 3125:04:22, 21 April 2014 (UTC) 3099:07:48, 17 April 2014 (UTC) 3065:05:12, 17 March 2014 (UTC) 3028:04:55, 17 March 2014 (UTC) 2973:01:58, 17 March 2014 (UTC) 2763:{\displaystyle \delta (k)} 2373:Serious Problem in Example 2352:23:26, 14 April 2008 (UTC) 2259: 2045:17:44, 11 April 2007 (UTC) 312:For more information, see 231:Discrete Fourier transform 3753:22:11, 12 June 2022 (UTC) 3645:{\displaystyle y_{2N}=0,} 3524:{\displaystyle y_{2n}=0,} 3200:Discrete cosine transform 3180:16:08, 10 June 2007 (UTC) 3170: 2507:22:52, 3 April 2011 (UTC) 2487:04:43, 23 June 2009 (UTC) 2260:figure doesn't match text 2024:20:12, 5 April 2007 (UTC) 1995:18:38, 5 April 2007 (UTC) 1980:20:15, 5 April 2007 (UTC) 1970:23:41, 4 April 2007 (UTC) 1272:Discrete Cosine Transform 1225:Discrete Cosine Transform 406:21:38, 7 March 2008 (UTC) 233:. The four changes are: 134: 67: 46: 3777:06:12, 3 June 2024 (UTC) 3044:For n = 1 To DCTsize - 1 3040:For k = 0 To DCTsize - 1 3009:For n = 1 To DCTsize - 1 3005:For k = 0 To DCTsize - 1 2989:For n = 0 To DCTsize - 1 2985:For k = 0 To DCTsize - 1 2942:) and refs 1-2 therein. 2531:02:26, 17 May 2012 (UTC) 2291:12:50, 3 June 2008 (UTC) 2196:21:46, 29 May 2007 (UTC) 2147:14:58, 29 May 2007 (UTC) 2133:14:53, 29 May 2007 (UTC) 2119:20:55, 24 May 2007 (UTC) 2106:16:21, 24 May 2007 (UTC) 2093:14:33, 23 May 2007 (UTC) 2079:14:25, 23 May 2007 (UTC) 2061:11:35, 23 May 2007 (UTC) 1113:Sorry. I didn`t get you. 141:project's priority scale 3328:yet another alternative 3191:External links modified 2338:) 00:21, 15 April 2008 2073:Electronics wikiproject 519:Shifted variants of DCT 205:History behind the term 98:WikiProject Mathematics 3733: 3695: 3646: 3608: 3573: 3525: 3486: 3485:{\displaystyle y_{n},} 3456: 3429: 3070:Bug in the pseudocode? 2869: 2791: 2764: 2731: 2683: 2493:805 kbyte animated gif 1930: 1868: 1770: 1726: 1671: 1553: 1490: 1436: 1382: 1260: 1194: 1181:), or equivalently an 1140: 1071: 1004: 937: 866: 771: 740: 674: 619: 572: 488: 441: 188:deleted by other users 28:This article is rated 3734: 3696: 3647: 3609: 3574: 3526: 3487: 3457: 3430: 3077:IDCT(k) = DCT(0) / 2 2870: 2792: 2790:{\displaystyle X_{k}} 2765: 2732: 2657: 2010:comment was added by 1931: 1842: 1741: 1727: 1672: 1554: 1491: 1437: 1383: 1261: 1150: 1136: 1072: 1005: 938: 867: 787:comment was added by 772: 741: 648: 620: 546: 489: 442: 3705: 3656: 3617: 3583: 3534: 3496: 3466: 3443: 3416: 3260:regular verification 3042:IDCT(k) = DCT(0) / 2 3007:IDCT(k) = DCT(0) / 2 2839: 2774: 2745: 2616: 1738: 1687: 1629: 1511: 1448: 1394: 1335: 1243: 1014: 947: 880: 828: 754: 632: 530: 470: 460:de:Benutzer:RokerHRO 440:{\displaystyle x(t)} 422: 121:mathematics articles 3250:After February 2018 3229:parameter below to 2944:— Steven G. Johnson 2835:Really, definition 2819:— Steven G. Johnson 1177:denotes the set of 1129:usually works with 487:{\displaystyle n/N} 3729: 3691: 3642: 3604: 3569: 3521: 3482: 3455:{\displaystyle 4N} 3452: 3428:{\displaystyle 4N} 3425: 3304:InternetArchiveBot 3255:InternetArchiveBot 3151:Assessment comment 2865: 2787: 2760: 2727: 2609:I derive formula: 2479:—Steven G. Johnson 2414:—Steven G. Johnson 2051:Misleading example 2042:—Steven G. Johnson 1977:—Steven G. Johnson 1948:—Steven G. Johnson 1926: 1722: 1667: 1572:—Steven G. Johnson 1549: 1486: 1432: 1378: 1322:—Steven G. Johnson 1299:—Steven G. Johnson 1256: 1215:—Steven G. Johnson 1100:—Steven G. Johnson 1067: 1000: 933: 862: 811:—Steven G. Johnson 767: 736: 615: 484: 437: 361:—Steven G. Johnson 336:—Steven G. Johnson 305:to the new symobl 286:to the new symbol 267:to the new symbol 248:to the new symbol 225:Change in notation 90:Mathematics portal 34:content assessment 3280: 3185: 3184: 3089:comment added by 2923:Numerical Recipes 2722: 2547:comment added by 2521:comment added by 2339: 2326:comment added by 2247:Orthogonal matrix 2233: 2221:comment added by 2027: 1957:Formal definition 1911: 1818: 1654: 1536: 1473: 1419: 1365: 1254: 1127:signal processing 1063: 1051: 996: 984: 929: 917: 858: 804: 765: 748:Am I calculating 723: 704: 602: 202: 201: 177:in most browsers. 155: 154: 151: 150: 147: 146: 3803: 3738: 3736: 3735: 3730: 3700: 3698: 3697: 3692: 3690: 3689: 3677: 3676: 3651: 3649: 3648: 3643: 3632: 3631: 3613: 3611: 3610: 3605: 3578: 3576: 3575: 3570: 3568: 3567: 3555: 3554: 3530: 3528: 3527: 3522: 3511: 3510: 3491: 3489: 3488: 3483: 3478: 3477: 3461: 3459: 3458: 3453: 3434: 3432: 3431: 3426: 3384: 3314: 3305: 3278: 3277: 3256: 3244: 3168: 3167: 3160: 3101: 2874: 2872: 2871: 2866: 2864: 2863: 2851: 2850: 2796: 2794: 2793: 2788: 2786: 2785: 2769: 2767: 2766: 2761: 2736: 2734: 2733: 2728: 2723: 2718: 2707: 2693: 2692: 2682: 2671: 2653: 2652: 2628: 2627: 2566:Adamlegumetaylor 2559: 2533: 2321: 2316:Default DCT Page 2216: 2005: 1935: 1933: 1932: 1927: 1925: 1921: 1920: 1916: 1912: 1907: 1899: 1883: 1882: 1867: 1856: 1835: 1831: 1830: 1826: 1819: 1817: 1809: 1801: 1785: 1784: 1769: 1755: 1731: 1729: 1728: 1723: 1721: 1720: 1699: 1698: 1676: 1674: 1673: 1668: 1666: 1662: 1655: 1647: 1558: 1556: 1555: 1550: 1548: 1544: 1537: 1529: 1495: 1493: 1492: 1487: 1485: 1481: 1474: 1466: 1441: 1439: 1438: 1433: 1431: 1427: 1420: 1412: 1387: 1385: 1384: 1379: 1377: 1373: 1366: 1361: 1353: 1265: 1263: 1262: 1257: 1255: 1247: 1076: 1074: 1073: 1068: 1066: 1065: 1064: 1059: 1052: 1044: 1032: 1009: 1007: 1006: 1001: 999: 998: 997: 992: 985: 977: 965: 942: 940: 939: 934: 932: 931: 930: 925: 918: 910: 898: 871: 869: 868: 863: 861: 860: 859: 854: 846: 782: 776: 774: 773: 768: 766: 758: 745: 743: 742: 737: 735: 731: 724: 716: 705: 697: 684: 683: 673: 662: 644: 643: 624: 622: 621: 616: 614: 610: 603: 595: 582: 581: 571: 560: 542: 541: 493: 491: 490: 485: 480: 446: 444: 443: 438: 196:Reporting errors 164: 163: 157: 123: 122: 119: 116: 113: 92: 87: 86: 76: 69: 68: 63: 55: 48: 31: 25: 24: 16: 3811: 3810: 3806: 3805: 3804: 3802: 3801: 3800: 3781: 3780: 3760: 3703: 3702: 3681: 3659: 3654: 3653: 3620: 3615: 3614: 3581: 3580: 3559: 3537: 3532: 3531: 3499: 3494: 3493: 3469: 3464: 3463: 3441: 3440: 3414: 3413: 3403: 3378: 3323: 3308: 3303: 3271: 3264:have permission 3254: 3238: 3208:this simple FaQ 3193: 3156: 3153: 3133: 3113: 3102: 3084: 3078: 3072: 3054: 3017: 2997: 2960: 2899: 2855: 2842: 2837: 2836: 2777: 2772: 2771: 2743: 2742: 2708: 2684: 2644: 2619: 2614: 2613: 2597: 2542: 2539: 2516: 2495: 2466: 2375: 2318: 2299: 2262: 2223:208.215.227.190 2208: 2053: 2034: 2006:—The preceding 1992:Daniel.Cardenas 1988: 1967:Sam Pablo Kuper 1959: 1900: 1897: 1893: 1874: 1873: 1869: 1810: 1802: 1799: 1795: 1776: 1775: 1771: 1736: 1735: 1703: 1690: 1685: 1684: 1645: 1641: 1627: 1626: 1527: 1523: 1509: 1508: 1496:. 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77: 65: 64: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 3808: 3797: 3794: 3792: 3789: 3788: 3786: 3779: 3778: 3774: 3770: 3764: 3757: 3755: 3754: 3750: 3746: 3742: 3739: 3726: 3723: 3720: 3717: 3714: 3711: 3708: 3686: 3682: 3678: 3673: 3670: 3667: 3664: 3660: 3639: 3636: 3633: 3628: 3625: 3621: 3601: 3598: 3595: 3592: 3589: 3586: 3564: 3560: 3556: 3551: 3548: 3545: 3542: 3538: 3518: 3515: 3512: 3507: 3504: 3500: 3479: 3474: 3470: 3449: 3446: 3438: 3422: 3419: 3411: 3406: 3400: 3396: 3392: 3388: 3382: 3377: 3376: 3375: 3374: 3370: 3366: 3362: 3360: 3357: 3355:This sources 3353: 3351: 3348: 3343: 3341: 3339: 3335: 3333: 3329: 3320: 3318: 3317: 3312: 3307: 3306: 3295: 3291: 3288: 3284: 3283: 3282: 3275: 3269: 3265: 3261: 3257: 3251: 3246: 3242: 3236: 3232: 3228: 3221: 3217: 3213: 3212: 3211: 3209: 3205: 3201: 3196: 3190: 3188: 3181: 3178: 3174: 3169: 3166: 3164: 3159: 3150: 3148: 3147: 3143: 3139: 3130: 3126: 3122: 3118: 3115: 3114: 3107: 3106: 3105: 3100: 3096: 3092: 3088: 3081: 3075: 3069: 3067: 3066: 3062: 3058: 3038: 3034: 3030: 3029: 3025: 3021: 3003: 3001:DCT-III code: 2983: 2981:DCT-II code: 2979: 2975: 2974: 2970: 2966: 2957: 2953: 2949: 2945: 2941: 2937: 2933: 2929: 2924: 2920: 2919: 2915: 2914: 2913: 2912: 2908: 2904: 2896: 2890: 2886: 2882: 2878: 2860: 2856: 2852: 2847: 2843: 2834: 2833: 2832: 2831: 2828: 2824: 2820: 2816: 2812: 2811: 2810: 2809: 2805: 2801: 2782: 2778: 2754: 2748: 2719: 2715: 2712: 2709: 2700: 2697: 2694: 2689: 2685: 2679: 2676: 2673: 2668: 2665: 2662: 2658: 2654: 2649: 2645: 2638: 2632: 2629: 2624: 2620: 2612: 2611: 2610: 2594: 2592: 2591: 2587: 2583: 2575: 2571: 2567: 2562: 2561: 2560: 2558: 2554: 2550: 2546: 2536: 2534: 2532: 2528: 2524: 2520: 2514: 2509: 2508: 2504: 2500: 2492: 2488: 2484: 2480: 2476: 2472: 2471: 2470: 2463: 2461: 2460: 2456: 2452: 2442: 2438: 2434: 2429: 2428: 2427: 2426: 2423: 2419: 2415: 2411: 2406: 2405: 2400: 2396: 2395: 2394: 2393: 2389: 2385: 2379: 2372: 2370: 2369: 2365: 2361: 2353: 2350: 2346: 2342: 2341: 2340: 2337: 2333: 2329: 2328:203.129.33.32 2325: 2315: 2313: 2310: 2308: 2304: 2296: 2292: 2288: 2284: 2279: 2278: 2277: 2276: 2272: 2268: 2255: 2252: 2248: 2244: 2240: 2236: 2235: 2234: 2232: 2228: 2224: 2220: 2214: 2205: 2197: 2194: 2190: 2186: 2185: 2184: 2183: 2182: 2181: 2172: 2171: 2170: 2169: 2168: 2167: 2158: 2157: 2156: 2155: 2154: 2153: 2148: 2145: 2140: 2139: 2138: 2137: 2134: 2131: 2126: 2125: 2120: 2117: 2113: 2112: 2111: 2110: 2107: 2104: 2100: 2099: 2094: 2091: 2086: 2085: 2084: 2083: 2080: 2077: 2074: 2070: 2065: 2064: 2063: 2062: 2059: 2050: 2046: 2043: 2039: 2038: 2037: 2031: 2025: 2021: 2017: 2013: 2009: 2003: 1999: 1998: 1997: 1996: 1993: 1985: 1981: 1978: 1974: 1973: 1972: 1971: 1968: 1964: 1956: 1952: 1949: 1944: 1943: 1942: 1922: 1917: 1913: 1908: 1904: 1901: 1894: 1890: 1887: 1884: 1879: 1875: 1870: 1864: 1861: 1858: 1853: 1850: 1847: 1843: 1839: 1836: 1832: 1827: 1823: 1820: 1814: 1811: 1806: 1803: 1796: 1792: 1789: 1786: 1781: 1777: 1772: 1766: 1763: 1760: 1757: 1752: 1749: 1746: 1742: 1734: 1733: 1732: 1717: 1714: 1711: 1708: 1704: 1700: 1695: 1691: 1663: 1659: 1656: 1651: 1648: 1642: 1638: 1635: 1632: 1619: 1616: 1612: 1608: 1605: 1601: 1597: 1594: 1590: 1586: 1582: 1581: 1580: 1579: 1576: 1573: 1568: 1567: 1566: 1564: 1545: 1541: 1538: 1533: 1530: 1524: 1520: 1517: 1514: 1503: 1499: 1482: 1478: 1475: 1470: 1467: 1461: 1457: 1454: 1451: 1428: 1424: 1421: 1416: 1413: 1407: 1403: 1400: 1397: 1374: 1370: 1367: 1362: 1358: 1355: 1348: 1344: 1341: 1338: 1326: 1323: 1319: 1318: 1317: 1315: 1303: 1300: 1295: 1291: 1287: 1286: 1285: 1284: 1280: 1276: 1273: 1269: 1251: 1248: 1238: 1234: 1229: 1228: 1226: 1222: 1221: 1220: 1219: 1216: 1212: 1209: 1202: 1198: 1193: 1191: 1190:square matrix 1188: 1184: 1180: 1176: 1172: 1167: 1166: 1162: 1159: 1156:, invertible 1155: 1149: 1148: 1146: 1145: 1139: 1135: 1134: 1132: 1128: 1124: 1121: 1117: 1116: 1112: 1104: 1101: 1097: 1093: 1092: 1091: 1090: 1089: 1088: 1082: 1078: 1060: 1056: 1048: 1045: 1040: 1037: 1028: 1025: 1022: 1018: 993: 989: 981: 978: 973: 970: 961: 958: 955: 951: 926: 922: 914: 911: 906: 903: 894: 891: 888: 884: 875: 855: 851: 848: 842: 839: 836: 832: 824: 821: 820: 819: 818: 815: 812: 807: 806: 805: 802: 798: 794: 790: 786: 777: 762: 759: 732: 728: 720: 717: 712: 709: 701: 698: 692: 688: 685: 680: 676: 670: 667: 664: 659: 656: 653: 649: 645: 640: 636: 628:we calculate 611: 607: 604: 599: 596: 590: 586: 583: 578: 574: 568: 565: 562: 557: 554: 551: 547: 543: 538: 534: 526: 525: 524: 518: 514: 510: 506: 502: 501: 495: 481: 477: 473: 465: 464: 463: 461: 457: 454: 450: 431: 425: 413: 407: 403: 399: 394: 393: 392: 391: 385: 381: 380: 379: 378: 371: 370: 369: 368: 365: 362: 358: 357: 356: 355: 352: 344: 340: 337: 333: 332: 331: 326: 325: 322: 317: 315: 310: 308: 304: 300: 296: 291: 289: 285: 281: 277: 272: 270: 266: 262: 258: 253: 251: 247: 243: 239: 234: 232: 224: 222: 221: 217: 213: 204: 197: 189: 185: 184: 183: 176: 172: 166: 159: 158: 142: 138: 132: 129: 128: 125: 108: 104: 100: 99: 91: 85: 80: 78: 75: 71: 70: 66: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 3765: 3761: 3743: 3740: 3407: 3404: 3354: 3344: 3336: 3327: 3324: 3302: 3299: 3274:source check 3253: 3247: 3234: 3230: 3226: 3224: 3197: 3194: 3186: 3177:Geometry guy 3154: 3134: 3103: 3085:— Preceding 3079: 3074:In the line 3073: 3055: 3035: 3031: 3018: 2998: 2980: 2976: 2961: 2922: 2900: 2876: 2740: 2598: 2578: 2543:— Preceding 2540: 2523:108.4.30.149 2517:— Preceding 2510: 2496: 2474: 2467: 2447: 2409: 2398: 2380: 2376: 2356: 2319: 2311: 2300: 2283:82.18.14.143 2263: 2242: 2238: 2209: 2054: 2035: 1989: 1962: 1960: 1940: 1624: 1610: 1599: 1329: 1308: 1293: 1289: 1271: 1224: 1213: 1210: 1206: 1186: 1182: 1179:real numbers 1174: 1170: 1164: 1160: 1151: 1137: 1131:real numbers 1119: 1095: 789:Arkadi kagan 749: 627: 522: 453:89.49.215.47 448: 417: 383: 373:mathematics. 348: 327: 318: 311: 306: 302: 298: 294: 292: 287: 283: 279: 275: 273: 268: 264: 260: 256: 254: 249: 245: 241: 237: 235: 228: 208: 181: 173:Anchors are 170: 137:Mid-priority 136: 96: 62:Mid‑priority 40:WikiProjects 3241:Sourcecheck 2903:GuenterRote 2322:—Preceding 2303:Diving hawk 2237:Whilst the 2217:—Preceding 2213:Orthonormal 1120:Inverce DCT 112:Mathematics 103:mathematics 59:Mathematics 3785:Categories 3311:Report bug 2987:DCT(k) = 0 2936:this paper 2928:this paper 2897:DCT V-VIII 2595:My formula 2537:THANK YOU! 2464:N != N !!! 2206:Orthogonal 1603:resources. 943:, but not 874:orthogonal 447:values at 386:and so on. 3294:this tool 3287:this tool 2349:Oli Filth 2251:Oli Filth 2239:transform 1600:should be 1390:However, 321:Metacomet 3381:DMKR2005 3365:DMKR2005 3300:Cheers.— 3087:unsigned 2940:on arxiv 2932:on arxiv 2564:section. 2545:unsigned 2519:unsigned 2412:better. 2336:contribs 2324:unsigned 2219:unsigned 2020:contribs 2008:unsigned 1975:Fixed. 1683:In case 1561:Thanks. 1502:spectrum 1498:spectrum 1290:specific 1277:Thanks. 1270:. Is it 1199:Thanks. 1158:function 1079:Thanks. 1010:and not 797:contribs 785:unsigned 750:DCT I + 297:becomes 278:becomes 259:becomes 240:becomes 3769:Jedreky 3745:Benhut1 3462:inputs 3439:of the 3227:checked 3204:my edit 3117:Benhut1 3057:Benhut1 3020:Benhut1 2965:Benhut1 2582:Norlesh 2451:Cdecoro 2433:Cdecoro 2384:Cdecoro 2360:Cdecoro 2012:Stevenj 1442:is not. 1173:(where 351:Rebroad 190:before. 139:on the 30:C-class 3492:where 3235:failed 3052:Next k 3048:Next n 3015:Next k 3013:Next n 2995:Next k 2993:Next n 2938:(also 2930:(also 2881:Borneq 2800:Borneq 2741:where 2410:always 2399:within 2297:SA-DCT 2267:Lavaka 2243:matrix 2144:Rhebus 2130:Rhebus 2103:Rhebus 2058:Rhebus 1592:paper. 1563:Arkadi 1314:Arkadi 1279:Arkadi 1201:Arkadi 1168:-: --> 1154:linear 1096:useful 1081:Arkadi 505:Lavaka 36:scale. 2815:WP:OR 2174:test? 3773:talk 3749:talk 3718:< 3712:< 3701:for 3652:and 3596:< 3579:for 3391:talk 3387:Kvng 3369:talk 3231:true 3142:talk 3138:Ywaz 3121:talk 3095:talk 3061:talk 3024:talk 2969:talk 2948:talk 2907:talk 2885:talk 2823:talk 2813:See 2804:talk 2586:talk 2570:talk 2553:talk 2527:talk 2503:talk 2483:talk 2455:talk 2437:talk 2418:talk 2388:talk 2364:talk 2332:talk 2307:talk 2287:talk 2271:talk 2227:talk 2032:n=1? 2016:talk 2002:JPEG 2000:See 793:talk 509:talk 496:fast 402:talk 293:4. 274:3. 255:2. 236:1. 216:talk 171:Tip: 3437:DFT 3412:of 3410:DFT 3358:], 3268:RfC 3245:). 3233:or 3218:to 2877:not 2875:is 2798:0. 2477:. 2345:DCT 2215:. 1294:any 872:is 686:cos 584:cos 319:-- 131:Mid 3787:: 3775:) 3751:) 3671:− 3590:≤ 3393:) 3371:) 3281:. 3276:}} 3272:{{ 3243:}} 3239:{{ 3175:. 3144:) 3123:) 3097:) 3063:) 3026:) 2971:) 2950:) 2909:) 2887:) 2825:) 2806:) 2749:δ 2710:π 2677:− 2659:∑ 2633:δ 2588:) 2572:) 2555:) 2529:) 2505:) 2485:) 2457:) 2439:) 2420:) 2390:) 2366:) 2334:• 2309:) 2289:) 2273:) 2229:) 2022:) 2018:• 2004:. 1914:π 1862:− 1844:∑ 1824:π 1764:− 1743:∑ 1715:− 1649:π 1565:. 1531:π 1468:π 1414:π 1359:π 1316:. 1266:* 1227:. 1185:× 1163:: 1133:. 1029:π 962:π 895:π 843:π 799:) 795:• 699:π 689:⁡ 668:− 650:∑ 597:π 587:⁡ 566:− 548:∑ 511:) 462:) 404:) 316:. 218:) 3771:( 3747:( 3727:. 3724:N 3721:2 3715:n 3709:0 3687:n 3683:y 3679:= 3674:n 3668:N 3665:4 3661:y 3640:, 3637:0 3634:= 3629:N 3626:2 3622:y 3602:, 3599:N 3593:n 3587:0 3565:n 3561:x 3557:= 3552:1 3549:+ 3546:n 3543:2 3539:y 3519:, 3516:0 3513:= 3508:n 3505:2 3501:y 3480:, 3475:n 3471:y 3450:N 3447:4 3423:N 3420:4 3389:( 3383:: 3379:@ 3367:( 3313:) 3309:( 3296:. 3289:. 3140:( 3119:( 3093:( 3059:( 3022:( 2967:( 2946:( 2905:( 2883:( 2861:N 2857:x 2853:= 2848:0 2844:x 2821:( 2802:( 2783:k 2779:X 2758:) 2755:k 2752:( 2725:) 2720:N 2716:k 2713:n 2704:( 2701:s 2698:o 2695:c 2690:n 2686:x 2680:1 2674:N 2669:1 2666:= 2663:n 2655:+ 2650:0 2646:x 2642:) 2639:k 2636:( 2630:= 2625:k 2621:X 2584:( 2568:( 2551:( 2525:( 2501:( 2481:( 2475:N 2453:( 2435:( 2416:( 2386:( 2362:( 2330:( 2305:( 2285:( 2269:( 2225:( 2026:. 2014:( 1963:k 1936:. 1923:] 1918:) 1909:N 1905:k 1902:n 1895:( 1891:s 1888:o 1885:c 1880:n 1876:x 1871:[ 1865:1 1859:N 1854:0 1851:= 1848:n 1840:2 1837:= 1833:] 1828:) 1821:2 1815:N 1812:2 1807:k 1804:n 1797:( 1793:s 1790:o 1787:c 1782:n 1778:x 1773:[ 1767:1 1761:N 1758:2 1753:0 1750:= 1747:n 1718:n 1712:N 1709:2 1705:x 1701:= 1696:n 1692:x 1677:. 1664:) 1660:k 1657:n 1652:N 1643:( 1639:s 1636:o 1633:c 1559:. 1546:) 1542:k 1539:n 1534:N 1525:( 1521:s 1518:o 1515:c 1483:) 1479:k 1476:n 1471:N 1462:( 1458:s 1455:o 1452:c 1429:) 1425:k 1422:n 1417:N 1408:( 1404:s 1401:o 1398:c 1375:) 1371:k 1368:n 1363:N 1356:2 1349:( 1345:s 1342:o 1339:c 1281:. 1274:? 1268:I 1252:2 1249:1 1237:I 1203:. 1192:. 1187:N 1183:N 1175:R 1171:R 1165:R 1161:F 1083:. 1061:N 1057:n 1054:) 1049:4 1046:1 1041:+ 1038:m 1035:( 1026:2 1023:j 1019:e 994:N 990:n 987:) 982:N 979:1 974:+ 971:m 968:( 959:2 956:j 952:e 927:N 923:n 920:) 915:2 912:1 907:+ 904:m 901:( 892:2 889:j 885:e 856:N 852:n 849:m 840:2 837:j 833:e 803:. 791:( 778:? 763:N 760:1 746:? 733:] 729:k 726:) 721:N 718:1 713:+ 710:n 707:( 702:N 693:[ 681:n 677:x 671:1 665:N 660:0 657:= 654:n 646:= 641:k 637:X 612:] 608:k 605:n 600:N 591:[ 579:n 575:x 569:1 563:N 558:0 555:= 552:n 544:= 539:k 535:X 507:( 482:N 478:/ 474:n 458:( 449:t 435:) 432:t 429:( 426:x 400:( 307:X 303:f 299:X 295:f 288:k 284:j 280:k 276:j 269:n 265:k 261:n 257:k 250:N 246:n 242:N 238:n 214:( 143:. 42::

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