Started working on reconstruction

content/SMHD.typ

@@ -172,3 +172,38 @@ Each measurement will be quantized with out quantizer $cal(E)$, returning a tupl
 $ K_i = cal(E)(s, m tilde(x_i)) = (k, h)_i $ <eq:smhd_quant>
 
 Performing the operation of @eq:smhd_quant for our whole set of measurements will yield a vector of tuples $bold(K)$.
+#pagebreak()
+=== Reconstruction
+
+We already demonstrated the basic principle of the reconstruction phase in section @sect:tmhd, more specifically with @fig:tmhd_example_enroll and @fig:tmhd_example_reconstruct, which show the advantage of using more than one quantizer during reconstruction. 
+
+We will call our repeated measurement of $tilde(x)$ that is subject to a certain error $tilde(x^*)$.
+To perform reconstruction with $tilde(x^*)$, we will first need to find all $s$ quantizers for which we generated the helper data in the previous step. 
+
+We have to distinguish two different cases for the value of $s$: 
+- $s$ is odd 
+- $s$ is even
+
+==== Even number of metrics 
+
+If $s$ is even, we need to move our quantizer $s/2$ times some distance to the right and $s/2$ times some distance to the left.
+We can define the ideal position for the quantizer bounds based on its corresponding metric as centered around the center of the related metric.
+
+We can find these new bounds graphically as depicted in @fig:smhd_find_bound_graph. We first determine the x-values of the centers of a metric (here M1, as shown with the arrows). We can then place the quantizer steps with step size $Delta$ (@eq:delta) evenly spaced around these points.
+
+
+
+#grid(
+  columns: (1fr, 0.1fr, 1fr),
+  [#scale(x: 70%, y: 70%)[
+  #figure(
+    include("../graphics/quantizers/s-metric/2_2_find_quantizer.typ"),
+    caption: [Ideal centers and bounds for the M1 quantizer]
+  )<fig:smhd_find_bound_graph>]],
+  [#align(center)[#align(horizon)[#text(25pt)[$arrow.r.double$]]]],
+  [#scale(x: 70%, y: 70%)[
+  #figure(
+    include("../graphics/quantizers/s-metric/2_2_found_quantizer1.typ"),
+    caption: [Quantizer for the first metric]
+  )]]
+)

content/introduction.typ

@@ -37,6 +37,6 @@ This transformation can be performed using the function $xi = tilde(x)$. The key
 Considering a normal distribution, the CDF is defined as 
 $ xi(frac(x - mu, sigma)) = frac(1, 2)[1 + \e\rf(frac(x - mu, sigma sqrt(2)))] $
 
-=== ECDF
+==== ECDF
 
 The eCDF is constructed through sorting the empirical measurements of a distribution @dekking2005modern. Although less accurate, this method allows a more simple and less computationally complex way to transform real valued measurements into the Tilde-Domain. We will mainly use the eCDF in @chap:smhd because of the difficulty of finding an analytical description for the CDF of a Gaussian-Mixture.

graphics/quantizers/s-metric/2_2_find_quantizer.typ

@@ -0,0 +1,46 @@
+#import "@preview/cetz:0.2.2": canvas, plot, draw, decorations, vector
+
+#let line_style = (stroke: (paint: black, thickness: 2pt))
+#let dashed = (stroke: (dash: "dashed"))
+#let dashed2 = (stroke: (dash: "dotted"))
+#let markings = (stroke: (paint: red, thickness: 2pt), fill: red)
+#canvas({
+  plot.plot(size: (8,6), name: "plot",
+    x-tick-step: none,
+    x-ticks: ((3/16, [3/16]), (7/16, [7/16]), (11/16, [11/16]), (15/16, [15/16])),
+    y-label: $cal(E)(2, 2, tilde(x))$,
+    x-label: $tilde(x)$,
+    y-tick-step: none,
+    y-ticks: ((0.125, [M1]) ,(0.25, [M2]),(0.375, [M1]) ,(0.5, [M2]),(0.625, [M1]) ,(0.75, [M2]),(0.875, [M1]) ,(1, [M2])),
+    axis-style: "left",
+    x-min: 0,
+    x-max: 1,
+    y-min: 0,
+    y-max: 1,{
+    plot.add(((0,0), (0.125, 0.125), (0.25,0.25), (0.375, 0.375),(0.5,0.5), (0.625, 0.625),(0.75,0.75), (0.875, 0.875),(1, 1)), line: "vh", style: line_style)
+    plot.add-hline(0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1, style: dashed)
+    plot.add-vline(3/16, 7/16, 11/16, 15/16, style: dashed2)
+    plot.add-anchor("q00", (1/16, 1/8))
+    plot.add-anchor("h00", (1/16, 0))
+    plot.add-anchor("q01", (5/16, 3/8))
+    plot.add-anchor("h01", (5/16, 0))
+    plot.add-anchor("q10", (9/16, 5/8))
+    plot.add-anchor("h10", (9/16, 0))
+    plot.add-anchor("q11", (13/16, 7/8))
+    plot.add-anchor("h11", (13/16, 0))
+  })
+  decorations.brace((-0.9,0.63), (-0.9,1.63), name: "00")
+  decorations.brace((-0.9,2.13), (-0.9,3.13), name: "01")
+  decorations.brace((-0.9,3.63), (-0.9,4.63), name: "10")
+  decorations.brace((-0.9,5.13), (-0.9,6.13), name: "11")
+
+  draw.content((v => vector.add(v, (-0.3, 0)), "00.west"), [00])
+  draw.content((v => vector.add(v, (-0.3, 0)), "01.west"), [01])
+  draw.content((v => vector.add(v, (-0.3, 0)), "10.west"), [10])
+  draw.content((v => vector.add(v, (-0.3, 0)), "11.west"), [11])
+
+  draw.line("plot.q00", ((), "|-", "plot.h00"), mark: (end: ">"))
+  draw.line("plot.q01", ((), "|-", "plot.h01"), mark: (end: ">"))
+  draw.line("plot.q10", ((), "|-", "plot.h10"), mark: (end: ">"))
+  draw.line("plot.q11", ((), "|-", "plot.h11"), mark: (end: ">"))
+})

graphics/quantizers/s-metric/2_2_found_quantizer1.typ

@@ -0,0 +1,22 @@
+#import "@preview/cetz:0.2.2": canvas, plot
+
+#let line_style = (stroke: (paint: black, thickness: 2pt))
+#let dashed = (stroke: (dash: "dashed"))
+#canvas({
+  plot.plot(size: (8,6),
+    x-tick-step: none,
+    x-ticks: ((3/16, [3/16]), (7/16, [7/16]), (11/16, [11/16]), (15/16, [15/16])),
+    y-label: $cal(Q)(2, 1, tilde(x))$,
+    x-label: $tilde(x)$,
+    y-tick-step: none,
+    y-ticks: ((3/16, [00]), (7/16, [01]), (11/16, [10]), (15/16, [11])),
+    axis-style: "left",
+    x-min: 0,
+    x-max: 1,
+    y-min: 0,
+    y-max: 1,{
+    plot.add(((0,3/16), (3/16,3/16), (7/16,7/16), (11/16,11/16), (15/16, 15/16), (15/16, 3/16), (1, 3/16)), line: "vh", style: line_style)
+    plot.add-hline(3/16, 7/16, 11/16, 15/16, style: dashed)
+    plot.add-vline(3/16, 7/16, 11/16, 15/16, style: dashed)
+  })
+})

template/conf.typ

@@ -46,7 +46,8 @@
       x: 2cm,
     ),
     header: [],
-    footer: []
+    footer: [],
+    //numbering: "1"
   )
 
   set par(justify: true)
@@ -89,10 +90,35 @@
     ]
   })
 
+  
+  
+  set page(numbering: none)
+
   contents_page()
   
+  set page(numbering: none)
+
   pagebreak()
 
+  set page(
+    paper: "a4",
+    margin: (
+      top: 3cm,
+      bottom: 3cm,
+      x: 2cm,
+    ),
+    header: [],
+    footer: none,
+  )
+
+  set page(footer: locate(
+  loc => if calc.even(loc.page()) {
+    align(right, counter(page).display("1"));
+  } else {
+    align(left, counter(page).display("1"));
+  }
+  ))
+
   doc
 }