From 0cc1e27cb55db069b4bc7d9839ad9fae5df0efb8 Mon Sep 17 00:00:00 2001 From: Marius Drechsler Date: Mon, 22 Jul 2024 18:23:38 +0200 Subject: [PATCH] Big brain stuff --- content/SMHD.typ | 55 +++++++++++++++++++++++++++++++++++++++------- main.pdf | Bin 550256 -> 555146 bytes main.typ | 4 ++++ template/conf.typ | 2 +- 4 files changed, 52 insertions(+), 9 deletions(-) diff --git a/content/SMHD.typ b/content/SMHD.typ index d58fe2b..831b31b 100644 --- a/content/SMHD.typ +++ b/content/SMHD.typ @@ -239,7 +239,6 @@ This is also shown in @fig:smhd_2_2_reconstruction, as our quantizer curve is mo 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( @@ -254,12 +253,52 @@ Comparing @fig:smhd_2_2_reconstruction, @fig:smhd_3_2_reconstruction and their r caption: [Offset values for 2-bit configurations] ) -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], +Lets look deeper into the properties of the offset value $phi$. +As previously stated, we will need to move the enrollment quantizer $s/2$ times to the left and $s/2$ times to the right. +For example, setting parameter $s$ to $4$ means we will need to need to move the enrollment quantizer $lr(s/2 mid(|))_(s=4) = 2$ times to the left and right. +As we can see in @fig:4_2_offsets, $phi$ for the indices $i = plus.minus 2$ are identical to the offsets of a 2-bit 2-metric configuration. +In fact, this property carries on for higher even numbers of metrics. + +#grid( + columns: (1fr, 1fr), + [#figure( + table( + columns: (5), + inset: 7pt, + align: center + horizon, + [$bold(i)$], [$-2$], [$-1$], [$1$], [$2$], + [*Metric*], [M1], [M2], [M3], [M4], + [$bold(phi)$], [$-frac(1, 16)$], [$-frac(1, 32)$], [$frac(1, 32)$], [$frac(1, 16)$] + ), + caption: [2-bit 4-metric offsets] + ) +], + [#figure( + table( + columns: (7), + align: center + horizon, + inset: 7pt, + [$bold(i)$], [$-3$], [$-2$], [$-1$], [$1$], [$2$], [$3$], + [*Metric*], [M1], [M2], [M3], [M4], [M5], [M6], + [$bold(phi)$], [$-frac(1, 16)$], [$-frac(1, 24)$], [$-frac(1, 48)$], [$frac(1, 48)$], [$frac(1, 24)$], [$frac(1, 16)$] + ), + caption: [2-bit 6-metric offsets] + ) +] +) + +At $m=6$ metrics, the biggest offset we encounter is $phi = 1/16$ at $i = plus.minus 3$.\ +In conclusion, the maximum offset for a 2-bit configuration $phi$ is $1/16$ and we will introduce smaller offsets in between if we use a higher even number of metrics. More formally, we can define the maximum offset for an even number of metrics as follows: +$ phi_("max,even") = frac(frac(m,2), 2^n dot m dot 2) = frac(1, 2^n dot 4) $ + +Here, we multiply @eq:offset with the maximum offsetting index $i_"max" = s/2$. + +Now, if we want to find the maximum offset for a odd number of metrics, we need to modify @eq:max_offset_even, more specifically its numerator. +We know, that we need to keep the original quantizer for a odd number of metrics, besides that, the method stays the same. +For that reason, we will decrease the parameter $m$ by $1$: +$ +phi_"max_odd" &= frac(frac(m-1, 2), 2^n dot m dot 2)\ +&= lr(frac(m-1, 2^n dot m dot 4)mid(|))_(n=2, m=3) = 1/24 +$ - include("../pseudocode/find_quantizers.typ") -) -As shown in @alg:fancy diff --git a/main.pdf b/main.pdf index 656a2cd690c655004280821217569c84250d561f..7947345b00b59eeb717b850ab77b9e0129b5553b 100644 GIT binary patch delta 191744 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"@preview/quill:0.3.0" +#import "@preview/equate:0.2.0": equate + +#show: equate.with(breakable: true, sub-numbering: true) +#set math.equation(numbering: "(1.1)") #import "template/conf.typ": conf diff --git a/template/conf.typ b/template/conf.typ index 02c3c14..9b59b73 100644 --- a/template/conf.typ +++ b/template/conf.typ @@ -36,7 +36,7 @@ pagebreak() - set math.equation(numbering: "(1)") + //set math.equation(numbering: "(1)") set page( paper: "a4",