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the larger matrix. This property is useful to reduce implementation costs as the
same multipliers can be reused for various transform sizes.
5. The DCT matrix can be specified by using a small number of unique elements.
By examining the elements
c
ij
of (
6.2
) it can be shown that the number of
unique elements in a DCT matrix of size 2
M
2
M
is equal to 2
M
1. As
further elaborated in Sect.
6.2.4
, this is particularly advantageous in hardware
implementations.
6. The even basis vectors of the DCT are symmetric, while the odd basis vectors
are anti-symmetric. This property is useful to reduce the number of arithmetic
operations.
7. The coefficients of a DCT matrix have certain trigonometric relationships that
allows for a reduction of the number of arithmetic operations beyond what is
possible by exploiting the (anti-)symmetry properties. These properties can be
utilized to implement fast algorithms such as the Chen's fast factorization [
7
].
6.2.2
Finite Precision DCT Approximations
The core transform matrices of HEVC are finite precision approximations of the
DCT matrix. The benefit of using finite precision in a video coding standard is that
the approximation to the real-valued DCT matrix is specified in the standard rather
than being implementation dependent. This avoids encoder-decoder mismatch and
drift caused by manufacturers implementing the IDCT with slightly different float-
ing point representations. On the other hand, a disadvantage of using approximate
matrix elements is that some of the properties of the DCT discussed in Sect.
6.2.1
may not be satisfied anymore. More specifically, there is a trade-off between the
computational cost associated with using high bit-depth for the matrix elements and
the degree to which some of the conditions of Sect.
6.2.1
are satisfied.
A straightforward way of determining integer approximations to the DCT matrix
elements is to scale each matrix element with some large number (typically between
2
5
and 2
16
) and then round to the closest integer. However, this approach does not
necessarily result in the best compression performance. As shown in Sect.
6.2.3
,for
a given bit-depth of the matrix elements, a different strategy for approximating the
DCT matrix elements results in a different trade-off between some of the properties
of Sect.
6.2.1
.
6.2.3
HEVC Core Transform Design Principles
The DCT approximations used for the core transforms of HEVC were chosen
according to the following principles. First, properties 4-6 of Sect.
6.2.1
were
satisfied without any compromise. This choice ensures that several implementation
friendly aspects of the DCT are preserved. Second, for properties 1-3 and 7 of
Sect.
6.2.1
, there were trade-offs between the number of bits used to represent each
matrix element and the degree by which each of the properties were satisfied.
