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Not great. This one is quite different. We can immediately see that it has above-unit gain for a significant fraction of its spectrum, peaking around 0.

On the plus side, it has a much wider passband, and is in general much closer to an ideal interpolation filter than basic linear interpolation is.

This one very slightly pokes above unit gain around 0. Other than that, we have some passband ripple, but are really quite close to unit gain until at least half-Nyquist, after which our performance starts to deteriorate, as it does for all 6-tap filters.

Our final 6-tap filter in this round is a 6-tap Lagrange interpolator. Lagrange interpolators are a classic family of interpolating filters closely related to Lagrange polynomials and have extremely flat passbands:.

On the good side, yes, the passband really is flat. The trade-off is that the filter response visibly starts to dip around 0. On the other hand, this type of filter is not going to explode with repeated application.

Some ripple in the passband and an above-unit peak around 0. However it also manages to pass through frequencies all the way out to nearly 0. Passband straight as a ruler, but not much use past 0. First, adding more taps gives us frequency responses closer to our ideal, which is not exactly surprising. The trade-offs are that filters with more taps are more expensive to evaluate, and that codecs also care about things other than frequency response.

Among other things, we generally want reduced-precision fixed-point approximations to the filter coefficients both for efficiency particularly in hardware decoders and to ensure different codec implementations agree which, for various reasons, gets significantly nastier once floating-point is in the mix.

They are completely stable under repeated application but also tend to lose high frequencies fairly quickly. In the other direction, we have filters like the Lanczos-derived ones or those used in H. This last part is something I originally intended to do a more in-depth analysis of, which is the reason this post is so delayed.

No proper analysis here. The gist of it is this: if your samples are quantized to say 8-bit fixed point after every step, it seems plausible that you should be able to get away with slightly above unit gain at some frequencies.

Essentially, all the inputs being quantized are integers, which means they need to change by at least 0. I might do another post in the future if I do come up with a nice proof. Finally: how much does it matter?

The stability issue is similar: if we care about stability above all else, there is a known answer, which is to use Lagrange interpolators, which give us perfect stability but have subpar response at the high frequencies.

There is another thing to consider here: in a video codec, motion compensation is part of the coding loop. The encoder knows exactly what the decoder will do with a given frame and set of motion vectors, and will use that result to code further updates against.

In short, the interpolation filter is not left to its own devices in a feedback loop. The encoder is continuously monitoring what the current state of the frame in a compliant decoder will be. If there is a build-up of high-frequency energy over time, all that really happens is that at some point, the error will become high enough for the encoder to decide to do something about it, which means sending not just a pure motion vector, but instead a motion vector with a residual a correction applied to the image data after motion compensation.

Said residual is also coded in frequency space essentially , most commonly using a DCT variant. In short, the end result of using a filter that has resonances in the high frequencies is that a few times per second, the encoder will have to send residuals to cancel out these resonances. This costs extra bits, and correcting errors in the high frequencies tends to be more expensive than for the lower frequencies image and video codecs can generally code lower frequencies more cheaply than high frequencies.

In short, instead of steadily gaining energy in the high frequencies, we end up steadily losing energy in the high frequencies. And this too ends up with the encoder having to send high-frequency heavy residuals periodically, this time to add in the missing high frequencies instead of removing excess ones. Neither of these is obviously preferable to, or cheaper than, the other.

Sending high-frequency heavy residuals is relatively expensive, but we end up having to do it periodically regardless of which type we choose. For example, H. In that context, a somewhat sharpening motion interpolation filter can help mitigate some of the damage, whereas a Lagrange interpolator would make things even more one-sided.

The decision is made in context and depends on what the rest of the codec does. Casey Muratori posted on his blog about half-pixel interpolation filters this week, where he ends up focusing on a particular criterion: whether the filter in question is stable under repeated application or not. There are many things about filters that are more an art than a science, especially where perceptual factors are concerned, but this particular question is both free of tricky perceptual evaluations and firmly in the realm of things we have excellent theory for, albeit one that will require me to start with a linear algebra infodump.

Take any vector space V over some field and any linear map from that space to itself. An eigenvector v of T is a nonzero element of V such that for some — that is, the result of applying the map T to v is a scaled version of v itself. The best possible case is that we find lots of eigenvectors — enough to fully characterize the map purely by what it does on its eigenvectors.

That is, in the right basis made of eigenvectors , T is just a diagonal matrix, which is to say, a non-uniform scale. This makes analysis of repeated applications of T easy, since: and in general and : viewed in the basis made of eigenvectors, repeated application of T is just repeated scaling, and behaviour over lots of iterations ultimately just hinges on what the eigenvalues are.

Not every matrix can be written that way; ones that can are called diagonalizable. But there is a very important class of transforms and now we allow infinite-dimensional spaces again that is guaranteed to be diagonalizable: so called self-adjoint transforms. In the finite-dimensional real case, these correspond to symmetric matrices matrices A such that. Such transforms are guaranteed to be diagonalizable, and even better, their eigenvectors are guaranteed to be pairwise orthogonal to each other, meaning the transform Q is an orthogonal matrix a rotation or reflection , which among other things makes the whole process numerically quite well-behaved.

If a matrix is symmetric, it it diagonalizable, which allows us to build an iterative process to solve linear equations that we can analyze easily and know will converge if we do it right.

Which could be quite pathological. But what does any of this have to do with filtering? That means that if we fix either of the arguments, we get a linear map.

Suppose we have a FIR filter f given by its coefficients. Then we can define an associated linear map on a suitable space, say something like writing for the set of bounded sequences of complex numbers by setting. Luckily the notions of eigenvectors eigensequences in this case and eigenvalues generalize just fine.

Namely, define the family of sequences by:. Anyway, if we apply our FIR filter f to that signal, we get this is just expanding out the definition of discrete convolution for our filter and input signal, using the convention that unqualified summation is over all values of k where the sum is well-defined.

The first line is just expanding the definition; then in the second line we use the properties of the exponential function and the linearity of sums to pull out the constant factor of. The final line states exactly what we wanted, namely that the result of applying to is just a scaled copy of itself—we have an eigensequence with eigenvalue.

Yup, we started out with a digital FIR filter, asked what happens when we iterate it a bunch, did a brief detour into linear algebra, and ended up in Fourier theory. Long story short, if you want to know whether a linear digital filter is stable under repeated application, you want to look at its eigenvalues, which in turn are just given by its frequency response.

The proof for all three cases is simply observing that k-fold application of the filter f to the signal will result in the signal. Therefore, all you need to know about the stability of a given filter under repeated application is contained in its Fourier transform.

Hash tables are the most popular probabilistic data structure, by quite a margin. You compute some hash code then compute an array index from that hash code. In return, the worst-case lookup cost becomes bounded.

In conventional hash tables, we have different probing schemes, or separate chaining with different data structures. For a cache table, we have our strict bound P on the number of entries we allow in the same bucket, and practical choices of P are relatively small, in the single or low double digits. Having storage for all entries in the same row right next to each other leads to favorable memory access patterns, better than most probe sequences used in open addressing or the pointer-heavy data structures typically used in separate chaining.

Then we check whether there is a matching entry in that row, by looping over all P columns. Inserting an entry means replacing the previous Partial Wrap Cabinet Hinges 02 entry at that location. One of the benefits of keeping P entries per row is that it allows us to have two different keys with identical hash values i. So what is this type of data structure useful for? This is a rehash of something I wrote in a forum post something like 10 years ago.

It turns out that forum prohibited archiving in its robots. The original operator of said forum hosted a copy with broken formatting of the original posts for a while, but at a different URL, breaking all links. Quaternions are a common rotation representation in several fields including computer graphics and numerical rigid-body dynamics for reasons beyond the scope of this post.

To apply a rotation to a vector, one computes the quaternion product , where is implicitly identified with the quaternion with real scalar part 0 and as its imaginary part, and denotes the conjugate of. Anyway, the result of the above product is another pure imaginary quaternion, corresponding to the rotated vector.

Now generally, you often want to apply the same transformation to many vectors. You can look up the formula elsewhere or work it out from the expression above with some algebra.

The conjugate of is given by. The product of two quaternions and is given by. Anyway, armed with this formula, we can compute our final product without too much trouble. Not so bad. This proves that is indeed a pure imaginary quaternion again, just like the we started out with. This is what you get if you just plug everything into the formulas and apply basic algebraic simplifications until you run out of obvious things to try which is exactly what we did.

In terms of scalar operation count, this boils down to two cross products at 6 multiplies and 3 additions well, subtractions each; one 3D dot product at 3 multiplies and 2 additions; 3 vector-by-scalar multiplications at 3 multiplies each; two scalar multiplies to form and , although the latter can also be computed via addition if you prefer ; and finally 3 vector additions at 3 adds each.

The total operation count is thus 26 scalar multiplies and 17 additions, unless I miscounted. We can do better than that. We can thus plug in for , which yields:.

The two remaining terms involving both multiply by two, so we use a slightly different shared temporary for the final version of our formula:. Final scalar operation count: without multiply-adds, the two cross products sum to 12 multiplies and 6 adds, scaling t by two takes 3 multiplies or adds, your choice , and the final vector-by-scalar multiply and summation take 3 multiplies and 6 adds, respectively.

With multiply-adds, I count a total of 3 multiplies and 9 multiply-adds for the cross products the first cross product has nothing to sum to, but the second does ; either 3 multiplies or 3 adds your choice again for the doubling; and another 3 multiply-adds for the portion. Furthermore some GPUs can fold a doubling straight into the operand without counting as another operation at all; in that case we get 3 multiplies and 12 multiply-adds.

If so, great! My goal here is just to explain the basic idea, show how it plays out in practice, and maybe get you thinking about other applications. The mission statement for lossless data compression is pretty clear.

Lossy compression is another matter. The way we do this is by coming up with some error metric that measures the distance between the original image and the result the decoder will actually produce given a candidate bitstream. Now our bitstream has two associated values: its length in bits, the bit rate , usually called R in formulas, and a measure of how much error was introduced as a result of the lossy coding process, the distortion , or D in formulas.

R is almost always measured in bits or bytes; D can be in one of many units, depending on what type of error metric is used. Given a pair and another pair , how do we tell which one is better?

Suppose what we have 8 possible candidate solutions we are considering, each with their own rate and distortion scores. If we prepare a scatter plot of rate on the x-axis versus distortion on the y-axis, we might get something like this:. That is, some other candidate point has the same or lower rate, and also the same or lower distortion as the grayed-out points.

In the scatterplot, this means that there is at least one other point that is both to the left lower rate and below lower distortion. For any of the points in the diagram, imagine a horizontal and a vertical line going through it, partitioning the plane into four quadrants.

Any point that ends up in the lower-left quadrant lower rate and lower distortion is clearly superior. Likewise, any point in the upper-right quadrant higher rate and higher distortion is clearly inferior. Which brings us to the other four points: the three fat black points, and the red point. These are all points that have no other points to the left and below them, meaning they are pareto efficient.

The set of points that are pareto efficient constitutes the pareto frontier. These points are not all the same, though. The three fat black points are not just pareto efficient, but are also on the convex hull of the point set the lower left contour of the convex hull here is drawn using blue lines , whereas the red point is not.

The points that are both pareto and on the convex hull of the pareto frontier are particularly important, and can be characterized in a different way. It will look something like this:. The shading here is meant to suggest the motion of the green line; we keep sliding it up until it eventually catches on the middle of our three fat black points. If we change the angle of our line to something else, we can manage to hit the other two black points, but not the red one.

This has nothing to do with this particular problem and is a general property of convex sets: any vertices of the set must be extremal in some direction. Sliding our straightedge until we hit a point corresponds to the minimization problem.

But looking at the characterization above does at least give us some idea on what looking at these linear combinations ends up doing, and exactly what we end up giving up namely, the pareto points not on the convex hull. If we take our linear combination above and divide through by w 1 or Partial Wrap Cabinet Hinges Generator w 2 assuming they are non-zero; dividing our objective by a scalar constant does not change the results of the optimization problem , respectively, we get:.

Generalizing one step further, allowing not just equality but also inequality constraints i. Since the whole process makes sense even without explicitly mentioning Lagrange multipliers at all, that seemed like the most natural way to handle it. Neither our direct derivation nor the Lagrange multiplier version tells us what to set our mystery weight parameters to, however.

No matter which one we pick, we are going to end up finding an optimal point for some trade-off, namely one that is pareto; it just might not be the exact trade-off we wanted. We can then look at the R and D values of the solution we got back. With this kind of process, even knowing Partial Wrap Cabinet Hinges Github nothing else about the problem, you can systematically explore the options along the pareto frontier and try to find a better fit.

Yes, pretty much. In general, the more frequently something is done or the bigger the search space is , the more willing codecs are to make compromises with their distortion measure to enable more efficient searches.

In general, good results require both decent metrics and doing a good job exploring the search space, and accepting some defects in the metrics in exchange for a significant increase in search space covered in the same amount of time is often a good deal. This gives you points on the lower left convex hull of the pareto frontier, which is what you want. This applies in other settings as well.

For example, the various lossless compressors in Oodle are, well, lossless, but they still have a bunch of decisions to make, some of which take more time in the decoder than others. Either way, the details of this quickly get complicated, but the basic idea is really quite simple. I hope this post de-mystifies it a bit. Continued from part 2. Last time, we saw a whole bunch of different bit reader implementations. Note that if count is a compile-time constant, the computation for rBitMask can be entirely constant-folded.

The single cycle depicted in the graph is OK for very simple cases; more complicated cases will often need multiple cycles here, for example because they perform a table lookup to figure out the count. The variant 4 refill does take 3 extra instructions, but we can immediately see that we get more latent instruction-level parallelism ILP in return:. On the other hand, the key decode steps are still very linear.

Especially so if most of the latency in question comes from integer instructions that already have a single cycle of latency. Over the past 30 years, the number of executions units and instructions per cycle in mainstream CPU parts have steadily, if slowly, increased.

But if we want to see any benefit from this, we need to write code that has a use for these extra execution resources. As is often the case, the best solution to this problem is the straightforward one: if decoding from a single bitstream is too serial, then why not decode from multiple bitstreams at once? Even with a very serial decoder like variant 3 above, you can come a lot closer to filling up a wide out-of-order machine as long as you use enough streams.

To a first-order approximation, using N streams will also give you N times the latent ILP—and given how serial a lot of the direct decoders are, this will translate into a substantial usually not quite N-times, but still very noticeable speed-up in the decoder on wide-enough processors. There are several:. So how many streams should you use?

It depends. Even if your decoder has a lot of other stuff going on computations with the decoded values etc. Having two streams adds a relatively small amount of overhead to the bitstream format to signal the start of the data for stream 2 in every coding unit, or something equivalent , needs a modest amount of extra state for the second bit decoder, and tends to result in sizeable wins on pretty much any current CPU. Using more than 2 streams can be a significant win in tight loops that do nothing but bitstream decoding, but is overkill in most other cases.

Before you commit to a specific high number, you ideally want to try implementations on at least a few different target devices; a good number on one device may be past a big performance cliff on another, and having that kind of thing enshrined in a protocol or file format is unfortunate.

Yes, you can, but the trade-offs get a bit icky here. Vectorizing the simple decoders outlined above directly is, generally speaking, not great. To hide this overhead, and the associated latencies, you generally still need to be running multiple instances of your SIMD decoder in parallel, so your total number of streams ends up being the number of SIMD lanes times two or more, if you need more instances.

The same thing goes for GPUs, but even more so. At that scale, the extra overhead for signaling individual stream boundaries is definitely not negligible anymore, and the magic numbers are different between different GPU vendors; striking a useful compromise between the needs of different GPUs while also retaining the ability to decode on a CPU if no suitable GPU is available starts to get quite tricky.

This is definitely a challenge going forward. The beginning of stream 0 is implicit—right after the end of the header—and the end of the final stream is often supplied by an outer framing layer, but the initial positions of bitptr1 and bitptr2 need to be signaled in the bytestream somehow, usually by encoding the length of streams 0 and 1 in the header.

Compressed data is normally untrusted since all the channels you might get that data from tend to be prone to either accidental error during storage or in transit or intentional malicious attacker trying to craft harmful data corruption, so careful input validation is not optional.

The overhead of this can be reduced in various ways. For example, one popular method is to unroll loops a few times and check at the top that there are enough bytes left for worst-case number of bytes consumed in the number of unrolled iterations, then only dropping to a careful loop that checks every single byte access at the very end of the stream.

But why am I mentioning this here? Because it turns out that with multiple streams laid out sequentially, the overhead of bounds checking can be reduced. A direct range check for 3 streams that checks whether there are at least K bytes left would look like this:. But when the three streams are sequential, we can use a simpler expression.

And second, we can relax the check in the inner loop to use a much weaker test:. The idea is that bitptr1 starts out pointing at bitend0 , and only keeps increasing from there. Another interesting option is to reverse the order of some of the streams which flips endianness as a side effect , and then glue pairs of forward and backward streams together, like shown here for streams 1 and I admit this sounds odd, but this has a few interesting properties.

One of them is that it shrinks the amount of header space somewhat: in the image, the initial stream pointer for stream 2 is the same as the end of the buffer, and if there were 4 streams, the initial read pointers for stream 2 and 3 would start out in the same location but going opposite directions.

In general, we only need to denote the boundaries between stream pairs instead of individual streams. Once encoding is done, the two parts can be compacted in-place by moving the second half downwards. Between the various tricks outlined so far, the size overhead and the extra CPU cost for wrangling multiple streams can be squeezed down quite far. But we still have to deal with the increased number of live variables that multiple streams imply.

So how does that work? That allows us to Partial Wrap Cabinet Hinges Method get rid of the bit count and the instructions that manipulate it. Other than the strangeness going on in the refill, where we first have to figure out the number of bits consumed from the old marker bit, then insert a new marker bit and do a final shift to consume the partial bits from the first byte, this is like a hybrid between variants 3 and 4 from last time.

The latter means that we equalize with the regular bit extract form in terms of instruction count when we perform at least 3 decodes per refill, and start to pull ahead if we manage more than 3. Several of the fastest and hottest decoder loops in Oodle use it. Which brings me to my final topic for this post.

Both are broadly equivalent and have efficient algorithms. But having now belabored that point sufficiently, if we can make both of them work, which one should we choose? There are definitely differences that push you into one direction or another, depending on your intended use case.

Here are some you might want to consider, in no particular order:. Instead, my plan is to write about some more down-to-earth case studies soon, so I can be less hand-wavy, and maybe even do some actual assembly-level analysis for an actual real-world CPU instead of an abstract idealized machine.

This is that post. A good first step is to profile it, which will give us some hints which parts are slow, but not necessarily why. You should be able to make informed decisions up front from an algorithm sketch without having to actually write a fleshed-out implementation.

This goes especially for microbenchmarks. Real-world systems are generally too complex and intertwined to understand from surface measurements alone. Microbenchmarks are a useful tool to confirm that an existing model is a good approximation to reality, but not very helpful in building these models to begin with. Code will be written in a pseudo-C form, at most one instruction per line. This model requires us to make the distinction between values kept in registers and memory accesses explicit, and flattens down control flow to basic blocks connected by branches.

This model can be understood as approximating either a dataflow architecture , an out-of-order machine with a very large issue window and infinitely fast front-end , or a statically scheduled in-order machine running code compiled with a Sufficiently Smart Scheduler. The kind that actually exists; e. When these guesses are correct, the branches are effectively free other than still taking an instruction slot, during which time the machine checks whether its prediction was correct.

When the guess was incorrect, the machine reverts all computations that were down the incorrectly guessed path, and takes some number of clock cycles to recover. Dataflow, on the other hand—the dependencies between instructions, and how long it takes for these dependencies to be satisfied—is front and center. Why this emphasis? Dataflow and keeping track of the shape of data dependencies is an organizing principle of both the machines themselves and the compilers that target them.

And these dependencies are naturally expressed in graph form, with individual operations being the nodes and data dependencies denoted by directed edges. Note I group nodes into ranks by which cycle they can execute in, at the earliest, assuming we can issue as many instructions in parallel as we want, purely constrained by the data dependencies.

As we can see, during the first four cycles, all we do is keep issuing more loads and advancing the pointer. It takes until cycle 4 for the results of the first load to become available, so we can actually do some summing. After that, one more load completes every cycle, allowing us to add one more integer to the running sum in turn. This is a crucial point: the longest-latency instruction in this loop is definitely the load, at 4 cycles.

The actual problem here is with the pointer advance; every single instruction that comes after it in program order depends on it either directly or indirectly, and therefore, its 1 cycle of latency determines when the next loop iteration can start.

While many machines can issue a lot more instructions than that in short bursts, a steady state of 4 instructions per cycle is definitely good. We store first a bit integer and then a bit pointer to the next element, with the end of the list denoted by a special value stored in rEndPtr as before. Go to Home Page. Mobile Navigation. Same Day Shipping Find a Store. Search Go. American Pro Decor. Hickory Hardware.

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