<s>
Pyramid	B-Algorithm
vector	I-Algorithm
quantization	I-Algorithm
(	O
PVQ	O
)	O
is	O
a	O
method	O
used	O
in	O
audio	O
and	O
video	O
codecs	B-General_Concept
to	O
quantize	B-Algorithm
and	O
transmit	O
unit	O
vectors	O
,	O
i.e.	O
</s>
<s>
PVQ	O
may	O
also	O
be	O
used	O
as	O
part	O
of	O
a	O
gain/shape	O
quantization	B-Algorithm
scheme	O
,	O
whereby	O
the	O
magnitude	O
and	O
direction	O
of	O
a	O
vector	O
are	O
quantized	O
separately	O
from	O
each	O
other	O
.	O
</s>
<s>
PVQ	O
was	O
initially	O
described	O
in	O
1986	O
in	O
the	O
paper	O
"	O
A	O
Pyramid	O
Vector	O
Quantizer	B-Algorithm
"	O
by	O
Thomas	O
R	O
.	O
Fischer	O
.	O
</s>
<s>
Conversion	O
to/from	O
the	O
more	O
familiar	O
Euclidean	O
distance	O
(	O
L2-norm	O
)	O
is	O
possible	O
via	O
vector	B-Algorithm
projection	I-Algorithm
,	O
though	O
results	O
in	O
a	O
less	O
uniform	O
distribution	O
of	O
quantization	B-Algorithm
points	O
(	O
the	O
poles	O
of	O
the	O
Euclidean	O
n-sphere	O
become	O
denser	O
than	O
non-poles	O
)	O
.	O
</s>
<s>
No	O
efficient	O
algorithm	O
for	O
the	O
ideal	O
(	O
i.e.	O
,	O
uniform	O
)	O
vector	B-Algorithm
quantization	I-Algorithm
of	O
the	O
Euclidean	O
n-sphere	O
is	O
known	O
as	O
of	O
2010	O
.	O
</s>
<s>
This	O
non-uniformity	O
can	O
be	O
reduced	O
by	O
applying	O
deformation	O
like	O
coordinate-wise	O
power	O
before	O
projection	B-Algorithm
,	O
reducing	O
mean-squared	O
quantization	B-Algorithm
error	O
by	O
~	O
10%	O
.	O
</s>
<s>
PVQ	O
is	O
used	O
in	O
the	O
CELT	B-Algorithm
audio	O
codec	B-General_Concept
(	O
now	O
Opus	B-Algorithm
)	O
and	O
the	O
Daala	B-Application
video	I-Application
codec	I-Application
.	O
</s>
<s>
As	O
a	O
form	O
of	O
vector	B-Algorithm
quantization	I-Algorithm
,	O
PVQ	O
defines	O
a	O
codebook	O
of	O
quantization	B-Algorithm
points	O
,	O
each	O
of	O
which	O
is	O
assigned	O
an	O
integer	O
codeword	O
from	O
0	O
to	O
1	O
.	O
</s>
<s>
In	O
set-builder	B-Language
notation	I-Language
:	O
</s>
<s>
Increasing	O
the	O
parameter	O
results	O
in	O
more	O
quantization	B-Algorithm
points	O
,	O
and	O
hence	O
typically	O
yields	O
a	O
more	O
"	O
accurate	O
"	O
approximation	O
of	O
the	O
original	O
unit	O
vector	O
at	O
the	O
cost	O
of	O
larger	O
integer	O
codewords	O
that	O
require	O
more	O
bits	O
to	O
transmit	O
.	O
</s>
<s>
Suppose	O
we	O
wish	O
to	O
quantize	B-Algorithm
three-dimensional	O
unit	O
vectors	O
using	O
the	O
parameter	O
=	O
2	O
.	O
</s>
