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Marius Drechsler 2024-07-21 17:19:09 +02:00
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@ -230,14 +230,31 @@ As for metric 2, we can apply the same strategy and find the points for the vert
Analytically, the offset we are applying to $cal(E)(2, 2, tilde(x))$ can be defined as
$ phi = lr(frac(1, 2^n dot s)mid(|))_(n=2, s=2) = 1 / 8 $<eq:offset>
$ phi = lr(frac(1, 2^n dot s dot 2)mid(|))_(n=2, s=2) = 1 / 16 $<eq:offset>
This is also shown in @fig:smhd_2_2_reconstruction, as our quantizer curve is moved $1/8$ to the left and the right.
This is also shown in @fig:smhd_2_2_reconstruction, as our quantizer curve is moved $1/16$ to the left and the right.
==== Odd number of metrics
If a odd number of metrics is given, the offset can still be calculated using @eq:offset. Additionally, we will keep the original quantizer used during enrollment.
If a odd number of metrics is given, the offset can still be calculated using @eq:offset. Additionally, we will keep the original quantizer used during enrollment (@fig:smhd_3_2_reconstruction).
Comparing @fig:smhd_2_2_reconstruction, @fig:smhd_3_2_reconstruction and their respective values of @eq:offset, we can observe, that the offset $phi$ gets smaller the more metrics we use.
]
#figure(
table(
columns: (11),
inset: 7pt,
align: center + horizon,
[$m$],
[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],
[$phi$],[$1/8$],table.cell(fill: gray)[$1/16$], [$1/24$], table.cell(fill:gray)[$1/32$], [$1/40$], table.cell(fill:gray)[$1/48$], [$1/56$], table.cell(fill:gray)[$1/64$], [$1/72$], table.cell(fill:gray)[$1/80$]
),
caption: [Offset values for 2-bit configurations]
)<tab:offsets>
Before we can go deeper into the properties of the offset value $phi$, we will introduce a way to programmatically find the offset values for all s quantizers.
#figure(
kind: "algorithm",
supplement: [Algorithm],