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Aside from just being complicated, each possible combination would require its own set of prediction matrices, some quite large. The combinatorics alone render this scheme impractical. It would avoid the whole problem!

As it turns out, they TF allows us to split apart or merge together blocks while staying in the frequency domain.

By altering the blocksizes, we can eliminate the nonhomogeneity and reduce both the size and number of required prediction matrices.

For example, we could TF surrounding blocks to match the size of our input block, or always run prediction at a 4x4 blocksize.

Constraints Wishlist

Obviously, we want TF to be as cheap to compute as possible. Figure 1: A simplified illustration of the butterflies in an eight-point FFT leaving out the weighting factors. This underlying structure is the same whether the FFT is decimation-in-frequency, normally drawn with the inputs in ascending order, or decimation-in-time, normally drawn with the inputs bitreversed and the frequency coefficients in ascending order.

Similarly, merging two blocks requires adding a new stage to the beginning of the computation, not end. As such, merging two spatial blocks in the frequency domain also requires first inverting the original FFT.

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  • To find this approximation, we ask ourselves "What are we really trying to do? These blocks may be frequency coefficients, but the relationship between coefficients in neighboring blocks is spatial. We want to transform that spatial relationship into the frequency domain. Splitting is the converse, converting a frequency relationship into a spatial relationship.


    Ironically enough, we do this with a DCT! Taking four colocated frequency coefficients from neighboring blocks, a 2x2 forward DCT converts their spatial relationship into frequency. Similarly, taking a 2x2 block of frequency coefficients from a single block and using an inverse DCT, we convert their frequency relationship into a spatial relationship. Figure 2: Frequency-domain blocks can be split or merged by iterative application of the 2x2 TF transform to groups of four frequency coefficients at a time.

    The diagram above shows the location of input and output coefficients in the source and destination blocks. Note that coefficient ordering in the large block alternates in a folded pattern because the DCT is a symmetric rather than circular transform.

    An implementation can achieve the same effect by adjusting the signs of the coefficients of the small blocks rather than reordering the coefficients of the large block. We also need a four-point transform to transform 4x4 blocks to 16x16 blocks and vice versa. Unfortunately, a four-point DCT requires multiplies. Of course, we could use a two-point DCT to transform from 4x4 to 8x8 and then use a two-point DCT again from 8x8 to 16x As transform size increases, the WHT remains cheap to compute no multiplies.

    In Daala, of course, we need only the 2x2 and 4x4 WHT. Our 2x2 transform is pretty simple. The dimensions are separable, so we can run a 2 point WHT on the rows, and then run it again on the columns.

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    The basis functions for the two point WHT are [1,1] and [1,-1], scaled as we see appropriate. This is the choice we appear to be stuck with: the outputs being scaled up, or output scale asymmetry. We can also push the asymmetry to the other side and design lifting stages that take asymmetrically scaled inputs and produce symmetrically scaled outputs:.

    At this point, we have a little collection of four asymmetric lifting stages. Then we realize these asymmetries are all complementary! Since the starting inputs are implicitly symmetrically scaled, we start the 2x2 transform using the lifting stages that expect symmetric inputs. The first transform applies to the even column, and the second to the odd column.

    This produces asymmetrically scaled intermediate values:.

    Some easy transformations get the operation count down to seven adds and one shift per four pixels if we use an extra register. Starting by reordering the operations:.

    And now we see that we can arithmetically collapse several operations if we use an extra register, e :. And with that, we have a fast, reversible 2x2 WHT with unit scaling, the minimum possible range expansion doubled , and no multiplies. And, of course, we can use this exact same filter implementation to split blocks as well as merge them. Note that this is not the only possible implementation.

    We can shuffle the asymmetries and sequence of operations to implement versions with slightly different integer rounding behavior. The implementation shown here matches the Daala source as of August 2, How good is this approximation?

    We start with a grid of 4x4 blocks, with alternating blocks containing one of the 4x4 DCT basis functions.

    We transform each block into the frequency domain with a 4x4 DCT. The output blocks contain a single non-zero frequency coefficient each, confirmation that our inputs are basis functions. Now we start with the same input values, but grouped together into larger 8x8 blocks. We transform them to the frequency domain using 8x8 DCTs; note that the frequency domain looks quite different:. Finally we split each 8x8 frequency domain block into four 4x4 blocks using TF.

    We begin this time with the 8x8 DCT basis functions, so that our ideal 8x8 frequency-domain blocks contain one nonzero frequency coefficient each. Skipping straight to the results:.

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    Neither the split nor the merge are that bad of an approximation for such a cheap transform Completely recomputing the DCT is cheaper. However, the error pattern in the TF-merged output suggests we can improve our cheap approximation, and in fact we can. A simple secondary lifting filter applied to the rows and then columns of the initial TF output block improves the mean squared error by more than an order of magnitude:.


    We build a two-stage TF split transform as expected; use the inverse of the second stage filter as a prefilter to the one-stage splitting transform. The additional filter stage improves splitting as much as it improves merging. We can continue improving the filter to come arbitrarily close to the exact DCT output if we want. However, the cost of larger filters yields steeply diminishing returns. This is only a single image example, but the results are representative of general behavior.

    Single-stage TF using a WHT is a rough enough approximation that lapped or not makes relatively little difference. Our new-and-improved TF, or a variant of it, may show sufficient coding gain that we could potentially use it to replace larger DCT blocksizes entirely. That would increase speed and solve all our various heterogeneous blocksize problems-- if it actually works. Update: Introducing Daala part 4 now posted!

    Figure 3: The basis functions of the 4x4 Discrete Cosine Transform, the 4x4 Walsh-Hadamard Transform, and the 4x4 Haar Wavelet Transform, illustrating the similarities between the three transforms.

    How do you keep your coffee cup on the table? Figure 4: Illustration of 4x4 blocks of pixels transformed into 4x4 blocks of frequency coefficients using the DCT. Alternating blocks of pixels hold the 4x4 DCT basis functions. When transformed, the frequency-domain output of these basis blocks has a single non-zero frequency coefficient, making comparison to the TF output in Figure 6 easier.

    Figure 5: Beginning with the same input pixels as above, we instead group them into 8x8 blocks and transform them into the frequency domain using an 8x8 DCT. We will then split each 8x8 block of frequency coefficients above right into four 4x4 blocks of frequency coefficients using TF below. Figure 6: Illustration of TF splitting 8x8 frequency-domain blocks into 4x4 frequency domain blocks.

    However, our TF implementation is approximate; we see that our 4x4 frequency-domain blocks are similar but not identical to the output produced directly from a direct 4x4 DCT Figure 4, right. Figure 7: Comparison of frequency domain blocks produced by an 8x8 DCT above center , and manufactured from 4x4 blocks using TF merging above right.

    The spatial domain pixels that were used for the transform input appear above left.

    how it works

    Figure 8: Schematic representation of a second-stage for the TF transform. The second stage lifting filter applies in-place to the rows and then the columns of the first stage output. When lifting the rows, the filter processes only odd columns. When lifting columns, the filter processes only odd rows.

    The same filter is used for rows and columns. The additional TF stage also improves splitting, as expected. Figure The left and center images above illustrate how well a single-stage and two-stage TF merge respectively approximate a DCT of a larger block size.

    The image was blocked and converted to the frequency domain using unlapped 4x4 DCTs, the blocks merged into 8x8 frequency-domain blocks by TF, and then converted back into pixels using larger 8x8 DCTs.

    The original image appears on the right. All three images appear here magnified by 2x. Figure The left and center images above illustrate how well a single-stage and two-stage TF split respectively approximate a DCT of a smaller block size. The image was blocked and converted to the frequency domain using unlapped 8x8 DCTs, the blocks split into 4x4 frequency-domain blocks by TF, and then converted back into pixels using smaller 4x4 DCTs.

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