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diff --git a/BachelorThesis/Software/PyPrecisionGUI/localization.py b/BachelorThesis/Software/PyPrecisionGUI/localization.py
index f81737c..c1223df 100644
--- a/BachelorThesis/Software/PyPrecisionGUI/localization.py
+++ b/BachelorThesis/Software/PyPrecisionGUI/localization.py
@@ -1,5 +1,6 @@
 import numpy as np
 import os
+from copy import deepcopy
 
 
 class Localization:
@@ -10,9 +11,10 @@ class Localization:
         #self.table = np.load('fingerprinting_tables/SmallManualFingerprint_Ferrite.npy') # manual fingerprinting table (Ferrite) [A bit off]
         #self.table = np.load('fingerprinting_tables/Julian_BThesis_table2_switchedAnt5&6 and 7&8.npy')    # Switched Ant5<->6 and 7<->8 in Excel (No Ferrite) [Does not work!]
         self.table = np.load('fingerprinting_tables/Julian_THIS_ONE_IS_IT.npy') # 2cm precision this definetly worked for (No ferrite)
+        self.normalized_table = deepcopy(self.table)
         self.data = np.load('recorded_data/current_recording.npy')
 
-    def localize(self, fv):
+    def localize(self, fv, exclude_frames=True):
         """ Input:  - fv [type = numpy array (mb list) [15, 1]]:
                         A feature vector contains the antenna voltages of one sample in following order:
                         frame1, frame2, ... frame 8, main1, main2, ... main8
@@ -29,22 +31,26 @@ class Localization:
 
         feature_vector = fv * self.scale_factor
 
-        # table_copy = self.table     # make a copy of the fingerprinting table
-        # table_copy[:, 17:25] = 0    # set all mains in the copy to 0
-        #
-        # max_idx = np.argmax(abs(fv[:8]))    # find highest Frame
-        #
-        # table_copy[:, max_idx+9] = 0        # set the row of highest Frame in fingerprinting to 0
-        #
-        # print("fv =", fv)
-        # print("max_idx=", max_idx)
-        # print("tablecopy= ", table_copy[0, :])
+        if exclude_frames:
+            table_copy = deepcopy(self.table)     # make a copy of the fingerprinting table
+            table_copy[:, 17:25] = 0    # set all mains in the copy to 0
 
+            max_idx = np.argmax(abs(feature_vector[:8]))    # find highest Frame
+            print("Feature_vector BEFORE taking highest frame out:\n", feature_vector)
+            print("Highest Frame:", max_idx+1)
+            feature_vector[max_idx] = 0
+            print("Feature_vector AFTER taking highest frame out:\n", feature_vector)
 
+            table_copy[:, max_idx+9] = 0        # set the row of highest Frame in fingerprinting to 0
 
+            print("First row of table", table_copy[0])
 
-        repeated_feature_vector = np.square(self.table[:, 9:] - feature_vector)   # Before changing anything on this function
-        #repeated_feature_vector = np.square(table_copy[:, 9:] - feature_vector)
+            # print("feature vector =", feature_vector)
+            # print("max_idx=", max_idx)
+            # print("tablecopy= ", table_copy[0, :])
+            repeated_feature_vector = np.square(table_copy[:, 9:] - feature_vector)
+        else:
+            repeated_feature_vector = np.square(self.table[:, 9:] - feature_vector)   # Before changing anything on this function
 
         euclidean_distances = np.sum(repeated_feature_vector, 1)
         order = np.argsort(euclidean_distances)
diff --git a/BachelorThesis/Software/PyPrecisionGUI/plot_precision_tests.py b/BachelorThesis/Software/PyPrecisionGUI/plot_precision_tests.py
index 0c5ed27..18eb654 100644
--- a/BachelorThesis/Software/PyPrecisionGUI/plot_precision_tests.py
+++ b/BachelorThesis/Software/PyPrecisionGUI/plot_precision_tests.py
@@ -428,88 +428,89 @@ def load_entire_ART(recording_names):
     #     z_list.append(pos[2])
 
 
+# Input here -----------------------------------------------------------------------------------------------------------
+recording_selection = ["horizontal", "diagonal"]
+sf_list = np.arange(0.0015, 0.0025, 0.0001)   # [0.002]
+k_list = [5]
+annotate_points = False
+plot_title = "Removing the highest frame receiving coil from localization entire area"
+label = "Highest frame removed"
+# End of input ---------------------------------------------------------------------------------------------------------
+
 # Make the plot look nice
 plot_surroundings()
 
-# Load ART positions
-ART_pos = load_entire_ART(["double_current_horizontal", "double_current_diagonal"])
+# Load data positions
+if len(recording_selection) == 1:   # if we only have to look at simply load ART and IndLoc
+    ART_pos = load_entire_ART([recording_selection[0]])     # Load ART positions
+    IndLoc_data = recording_dict_fs[recording_selection[0]]   # Load IndLoc data
+if len(recording_selection) == 2:   # if we want to look at multiple recordings its a little more complicated
+    ART_pos = load_entire_ART([recording_selection[0], recording_selection[1]])     # Load both ART positions
+
+    rec1 = recording_dict_fs[recording_selection[0]]    # Load IndLoc1
+    rec2 = recording_dict_fs[recording_selection[1]]    # Load IndLoc2
+
+    IndLoc_data = np.append(rec1, rec2, axis=0)     # Append both IndLoc arrays correctly
 
-# Load IndLoc data
-horizontal = recordings_dict["double_current_horizontal"]
-diagonal = recordings_dict["double_current_diagonal"]
-IndLoc_data = np.append(horizontal, diagonal, axis=0)   # shape (42, 2000, 19) = (P1-P42, 2k samples, fv)
 
 # Average IndLoc data
-IndLoc_data = np.average(IndLoc_data, axis=1)   # shape (42, 19) = (P1-P42, fv_avg)
+#IndLoc_data = np.average(IndLoc_data, axis=1)   # shape (42, 19) = (P1-P42, fv_avg)
 
 # Localize IndLoc data for each scaling factor in sf_list and each k in k_list
-sf_list = [0.001]   # [0.002]
-k_list = [3]
 IndLoc_pos = calc_multiple_positions(data=IndLoc_data,
-                                     output_filename="horizontal_diagonal",
+                                     output_filename=str(recording_selection),
                                      only_fs=True,
                                      sf_list=sf_list,
                                      k_list=k_list)
 
-# Find the best localization
+# Find the best localization parameters
 best_mean = 10000
 best_std_dev = 10000
 best_both = 10000
 for i in range((len(sf_list)*len(k_list))):
+    print(i)
     distances, curr_mean, curr_std_dev = eval_loc(ART_pos=ART_pos, IndLoc_pos=IndLoc_pos[i])
-
+    #print("- current k_nearest:", sf_list[i], " --> mean=", curr_mean, ", std_dev=",curr_std_dev)
     curr_both = curr_mean + curr_std_dev
 
-    if curr_mean <= best_mean:
+    if curr_both <= best_both:
         best_both = curr_both
         best_mean = curr_mean
         best_std_dev = curr_std_dev
         index = i
 
-best_sf = sf_list[index]
+if len(sf_list) != 1:   # it looks like we`re analyzing multiple sf`s
+    best_k = k_list[0]
+    best_sf = sf_list[index]
+if len(k_list) != 1:
+    best_sf = sf_list[0]
+    best_k = k_list[index]
+if (len(sf_list) == 1) and (len(k_list) == 1):
+    best_sf = sf_list[0]
+    best_k = k_list[0]
 
 print("\n----Best localization parameters were found!-----\n scaling_factor = ", best_sf, "\n mean_error = ", best_mean,
       "\n standard_deviation = ", best_std_dev)
 
 
 # Plot IndLoc pos
-plot_multiple_positions(positions_filename="horizontal_diagonal",
+plot_multiple_positions(positions_filename=str(recording_selection),
                         sf_list=[best_sf],
-                        k_list=k_list,
+                        k_list=[best_k],
                         only_label_once=True,
-                        xy_title="Statistical optimization of the scaling factor for the entire localization area",
-                        yz_title="Statistical optimization of the scaling factor for the entire localization area",
-                        annotate_points=False,
+                        xy_title=plot_title,
+                        yz_title=plot_title,
+                        annotate_points=annotate_points,
                         annotationsize=15,
-                        labels=iter(["scaling factor = " + str("{:.6f}".format(best_sf))]),
+                        labels=iter([label]),        # labels=iter(["k = " + str(best_k)]),
                         color_all_black=True,
                         markersize=100)
 
 # Plot ART positions
-plot_art_positions(ART_pos, annotate_flag=False, markersize=100, annotation_size=15)
-
-
-# print(IndLoc_pos)
-#recording_selection = "horizontal_and_diagonal"
-
-
-
-# scale_factor_list = [0.001755]
-#
-# calc_multiple_positions(recording_name=recording_selection,
-#                         only_fs=True,
-#                         sf_list=[0.001755],
-#                         k_list=[3])
-#
-# plot_multiple_positions(recording_name=recording_selection,
-#                         sf_list=[0.001755],
-#                         k_list=[3],
-#                         only_label_once=True,
-#                         xy_title="Statistical optimization of the scaling factor for an offcenter set of points",
-#                         yz_title="Statistical optimization of the scaling factor for an offcenter set of points",
-#                         annotate_points=False,
-#                         labels=iter(["scaling factor = " + str("{:.6f}".format(0.001755))]),
-#                         color_all_black=True)
+plot_art_positions(ART_pos,
+                   annotate_flag=annotate_points,
+                   markersize=100,
+                   annotation_size=15)
 
 xy_subplot.legend(bbox_to_anchor=(1.01, 1.00), loc=2, borderaxespad=0., fontsize=10)
 plt.show()
diff --git a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.aux b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.aux
index e3d578b..60fddd2 100644
--- a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.aux	
+++ b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.aux	
@@ -98,6 +98,7 @@
 \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The magnetic flux in the infinite wire loop 2($\Phi _{21}$) and it's dependency on the distances to wire loop 1.\relax }}{7}\protected@file@percent }
 \@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Magnetic flux in 2 due to the left wire of 1}}}{7}\protected@file@percent }
 \@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Magnetic flux in 2 due to the right wire of 1}}}{7}\protected@file@percent }
+\newlabel{eq:localization_function}{{6}{7}}
 \@writefile{toc}{\contentsline {section}{\numberline {3}The IndLoc system}{8}\protected@file@percent }
 \newlabel{sec:The IndLoc system}{{3}{8}}
 \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces An overview of the main components of the IndLoc system and how they are connected to each other.\relax }}{8}\protected@file@percent }
@@ -181,26 +182,49 @@
 \@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Heating up}}}{20}\protected@file@percent }
 \@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {setup is warm}}}{20}\protected@file@percent }
 \@writefile{tdo}{\contentsline {todo}{maybe cite the quickguide manual here?}{20}\protected@file@percent }
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-\pgfsyspdfmark {pgfid51}{39101366}{25502419}
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-\newlabel{fig:estimation_scaling_factor_1}{{14}{21}}
-\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces A zoomed in view of Figure \ref  {fig:estimation_scaling_factor_1} shows which scaling factors result in a localization closest to the correct positions.\relax }}{22}\protected@file@percent }
-\newlabel{fig:estimation_scaling_factor_2}{{15}{22}}
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-\newlabel{fig:estimation_scaling_factor_3}{{16}{23}}
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-\newlabel{fig:estimation_scaling_factor_4}{{17}{24}}
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-\newlabel{fig:estimation_scaling_factor_5}{{18}{24}}
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-\newlabel{fig:estimation_scaling_factor_6}{{19}{25}}
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-\@writefile{toc}{\contentsline {section}{\numberline {8}SUMMARY, OUTLOOK INTO THE FUTURE}{28}\protected@file@percent }
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+\newlabel{fig:estimation_scaling_factor_1}{{14}{22}}
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+\newlabel{sec:remove_highest_frame_receiving_coil_from_the_localization_test}{{5.4.3}{30}}
+\@writefile{lof}{\contentsline {figure}{\numberline {22}{\ignorespaces (sf= 0.001755 , k= 19) Placing the localization object in 25 positions near the centre of the localization area with removing the highest frame receiving coil, resulting in mean error $\mu = 5.56\textit  {m}m$ and standard deviation $\sigma = 3.2\textit  {m}m$.\relax }}{30}\protected@file@percent }
+\newlabel{fig:estimation_removing_highest_frame_1}{{22}{30}}
+\@writefile{lof}{\contentsline {figure}{\numberline {23}{\ignorespaces (sf= 0.0021 , k= 12) Placing the localization object in 25 positions near the corner of the localization area with a k-nearest value of 4, resulting in mean error $\mu = 13.32\textit  {m}m$ and standard deviation $\sigma = 8.24\textit  {m}m$.\relax }}{31}\protected@file@percent }
+\newlabel{fig:estimation_k_nearest_2}{{23}{31}}
+\@writefile{lof}{\contentsline {figure}{\numberline {24}{\ignorespaces (sf= 0.00204 , k= 4) Placing the object in 42 positions running from the centre to the edges of the localization area with a k-nearest value of 4, resulting in mean error $\mu = 15.21\textit  {m}m$ and standard deviation $\sigma = 14.33 \textit  {m}m$.\relax }}{31}\protected@file@percent }
+\newlabel{fig:estimation_k_nearest_3}{{24}{31}}
+\@writefile{toc}{\contentsline {subsubsection}{\numberline {5.4.4}Noise investigation}{32}\protected@file@percent }
+\newlabel{sec:Noise_investigation}{{5.4.4}{32}}
+\@writefile{toc}{\contentsline {section}{\numberline {6}RESULTS}{33}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\numberline {7}DISCUSSION}{34}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{\numberline {8}SUMMARY, OUTLOOK INTO THE FUTURE}{35}\protected@file@percent }
 \bibdata{Literatur_BA_2}
 \bibcite{HARRIEHAUSEN.2020b}{1}
 \bibcite{Albach.2014}{2}
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-cleartomark
-{restore}if
-%%EndFont 
-%%BeginFont: CMSY6
-%!PS-AdobeFont-1.0: CMSY6 003.002
-%%Title: CMSY6
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY6.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
-FontDirectory/CMSY6 known{/CMSY6 findfont dup/UniqueID known{dup
-/UniqueID get 5096647 eq exch/FontType get 1 eq and}{pop false}ifelse
-{save true}{false}ifelse}{false}ifelse
-11 dict begin
-/FontType 1 def
-/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMSY6 def
-/FontBBox {-4 -948 1329 786 }readonly def
-/PaintType 0 def
-/FontInfo 9 dict dup begin
-/version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY6.) readonly def
-/FullName (CMSY6) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY8.) readonly def
+/FullName (CMSY8) readonly def
 /FamilyName (Computer Modern) readonly def
 /Weight (Medium) readonly def
 /ItalicAngle -14.04 def
@@ -1084,7 +467,7 @@ FontDirectory/CMSY6 known{/CMSY6 findfont dup/UniqueID known{dup
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
-dup 6 /plusminus put
+dup 0 /minus put
 readonly def
 currentdict end
 currentfile eexec
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 {restore}if
 %%EndFont 
-%%BeginFont: CMBX6
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-%%Title: CMBX6
+%%BeginFont: CMTI10
+%!PS-AdobeFont-1.0: CMTI10 003.002
+%%Title: CMTI10
 %Version: 003.002
 %%CreationDate: Mon Jul 13 16:17:00 2009
 %%Creator: David M. Jones
 %Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMBX6.
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMTI10.
 % This Font Software is licensed under the SIL Open Font License, Version 1.1.
 % This license is in the accompanying file OFL.txt, and is also
 % available with a FAQ at: http://scripts.sil.org/OFL.
 %%EndComments
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-/FamilyName (Computer Modern) readonly def
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-{restore}if
-%%EndFont 
-%%BeginFont: CMR6
-%!PS-AdobeFont-1.0: CMR6 003.002
-%%Title: CMR6
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR6.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
-FontDirectory/CMR6 known{/CMR6 findfont dup/UniqueID known{dup
-/UniqueID get 5000789 eq exch/FontType get 1 eq and}{pop false}ifelse
-{save true}{false}ifelse}{false}ifelse
-11 dict begin
-/FontType 1 def
-/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
-/FontName /CMR6 def
-/FontBBox {-20 -250 1193 750 }readonly def
-/PaintType 0 def
-/FontInfo 9 dict dup begin
-/version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR6.) readonly def
-/FullName (CMR6) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMTI10.) readonly def
+/FullName (CMTI10) readonly def
 /FamilyName (Computer Modern) readonly def
 /Weight (Medium) readonly def
-/ItalicAngle 0 def
+/ItalicAngle -14.04 def
 /isFixedPitch false def
 /UnderlinePosition -100 def
 /UnderlineThickness 50 def
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
-dup 12 /fi put
+dup 14 /ffi put
+dup 38 /ampersand put
+dup 39 /quoteright put
 dup 40 /parenleft put
 dup 41 /parenright put
 dup 44 /comma put
@@ -1694,29 +714,23 @@ dup 48 /zero put
 dup 49 /one put
 dup 50 /two put
 dup 51 /three put
-dup 52 /four put
-dup 53 /five put
-dup 54 /six put
-dup 55 /seven put
-dup 56 /eight put
 dup 57 /nine put
 dup 58 /colon put
-dup 63 /question put
 dup 65 /A put
 dup 66 /B put
 dup 67 /C put
-dup 68 /D put
+dup 69 /E put
+dup 70 /F put
+dup 71 /G put
 dup 73 /I put
+dup 74 /J put
+dup 77 /M put
 dup 78 /N put
 dup 79 /O put
 dup 80 /P put
 dup 82 /R put
+dup 83 /S put
 dup 84 /T put
-dup 87 /W put
-dup 88 /X put
-dup 90 /Z put
-dup 91 /bracketleft put
-dup 93 /bracketright put
 dup 97 /a put
 dup 98 /b put
 dup 99 /c put
@@ -1726,6 +740,7 @@ dup 102 /f put
 dup 103 /g put
 dup 104 /h put
 dup 105 /i put
+dup 106 /j put
 dup 107 /k put
 dup 108 /l put
 dup 109 /m put
@@ -1741,509 +756,835 @@ dup 119 /w put
 dup 120 /x put
 dup 121 /y put
 dup 122 /z put
-dup 127 /dieresis put
 readonly def
 currentdict end
 currentfile eexec
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+%!PS-AdobeFont-1.0: CMSY10 003.002
+%%Title: CMSY10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
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@@ -2290,20 +1631,44 @@ FontDirectory/CMMI10 known{/CMMI10 findfont dup/UniqueID known{dup
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
+dup 11 /alpha put
+dup 22 /mu put
+dup 25 /pi put
+dup 26 /rho put
+dup 27 /sigma put
 dup 58 /period put
 dup 61 /slash put
 dup 65 /A put
+dup 66 /B put
+dup 67 /C put
+dup 69 /E put
 dup 72 /H put
+dup 73 /I put
+dup 74 /J put
+dup 76 /L put
 dup 77 /M put
+dup 78 /N put
+dup 80 /P put
 dup 83 /S put
-dup 84 /T put
+dup 97 /a put
+dup 98 /b put
 dup 99 /c put
+dup 100 /d put
+dup 101 /e put
+dup 102 /f put
 dup 105 /i put
 dup 107 /k put
+dup 108 /l put
 dup 109 /m put
 dup 110 /n put
+dup 111 /o put
 dup 115 /s put
+dup 116 /t put
+dup 117 /u put
+dup 120 /x put
+dup 121 /y put
 dup 122 /z put
+dup 126 /vector put
 readonly def
 currentdict end
 currentfile eexec
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-%%EndFont 
-%%BeginFont: CMSY10
-%!PS-AdobeFont-1.0: CMSY10 003.002
-%%Title: CMSY10
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY10.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
-FontDirectory/CMSY10 known{/CMSY10 findfont dup/UniqueID known{dup
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-{save true}{false}ifelse}{false}ifelse
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-/FontName /CMSY10 def
-/FontBBox {-29 -960 1116 775 }readonly def
-/PaintType 0 def
-/FontInfo 9 dict dup begin
-/version (003.002) readonly def
-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY10.) readonly def
-/FullName (CMSY10) readonly def
-/FamilyName (Computer Modern) readonly def
-/Weight (Medium) readonly def
-/ItalicAngle -14.04 def
-/isFixedPitch false def
-/UnderlinePosition -100 def
-/UnderlineThickness 50 def
-end readonly def
-/Encoding 256 array
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@@ -2880,14 +2138,17 @@ FontDirectory/CMR10 known{/CMR10 findfont dup/UniqueID known{dup
 end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
+dup 8 /Phi put
 dup 11 /ff put
 dup 12 /fi put
 dup 13 /fl put
 dup 14 /ffi put
-dup 34 /quotedblright put
+dup 19 /acute put
+dup 33 /exclam put
 dup 39 /quoteright put
 dup 40 /parenleft put
 dup 41 /parenright put
+dup 43 /plus put
 dup 44 /comma put
 dup 45 /hyphen put
 dup 46 /period put
@@ -2904,6 +2165,7 @@ dup 56 /eight put
 dup 57 /nine put
 dup 58 /colon put
 dup 59 /semicolon put
+dup 61 /equal put
 dup 63 /question put
 dup 65 /A put
 dup 66 /B put
@@ -2921,6 +2183,7 @@ dup 77 /M put
 dup 78 /N put
 dup 79 /O put
 dup 80 /P put
+dup 81 /Q put
 dup 82 /R put
 dup 83 /S put
 dup 84 /T put
@@ -2931,9 +2194,8 @@ dup 88 /X put
 dup 89 /Y put
 dup 90 /Z put
 dup 91 /bracketleft put
-dup 92 /quotedblleft put
 dup 93 /bracketright put
-dup 96 /quoteleft put
+dup 95 /dotaccent put
 dup 97 /a put
 dup 98 /b put
 dup 99 /c put
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 0000000000000000000000000000000000000000000000000000000000000000
 0000000000000000000000000000000000000000000000000000000000000000
 0000000000000000000000000000000000000000000000000000000000000000
@@ -3738,11 +3021,8 @@ end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
 dup 12 /fi put
-dup 40 /parenleft put
-dup 41 /parenright put
 dup 44 /comma put
-dup 45 /hyphen put
-dup 46 /period put
+dup 48 /zero put
 dup 49 /one put
 dup 50 /two put
 dup 51 /three put
@@ -3751,6 +3031,7 @@ dup 53 /five put
 dup 54 /six put
 dup 55 /seven put
 dup 56 /eight put
+dup 57 /nine put
 dup 65 /A put
 dup 67 /C put
 dup 68 /D put
@@ -3774,7 +3055,6 @@ dup 98 /b put
 dup 99 /c put
 dup 100 /d put
 dup 101 /e put
-dup 102 /f put
 dup 103 /g put
 dup 104 /h put
 dup 105 /i put
@@ -3783,14 +3063,11 @@ dup 108 /l put
 dup 109 /m put
 dup 110 /n put
 dup 111 /o put
-dup 112 /p put
 dup 114 /r put
 dup 115 /s put
 dup 116 /t put
 dup 117 /u put
 dup 118 /v put
-dup 119 /w put
-dup 120 /x put
 dup 121 /y put
 dup 122 /z put
 dup 123 /endash put
@@ -4001,367 +3278,2238 @@ D4FFED9073979E1F49C984D0D8731F93868E8F137226777551D4DE437AE5E27D
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+%%EndFont 
+%%BeginFont: CMSY9
+%!PS-AdobeFont-1.0: CMSY9 003.002
+%%Title: CMSY9
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSY9.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMSY9 known{/CMSY9 findfont dup/UniqueID known{dup
+/UniqueID get 5096650 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+11 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /CMSY9 def
+/FontBBox {-29 -958 1146 777 }readonly def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMSY9.) readonly def
+/FullName (CMSY9) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /minus put
+readonly def
+currentdict end
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+%%BeginFont: CMMI9
+%!PS-AdobeFont-1.0: CMMI9 003.002
+%%Title: CMMI9
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI9.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+%%EndFont 
+%%BeginFont: CMR6
+%!PS-AdobeFont-1.0: CMR6 003.002
+%%Title: CMR6
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR6.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+FontDirectory/CMR6 known{/CMR6 findfont dup/UniqueID known{dup
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+{save true}{false}ifelse}{false}ifelse
+11 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /CMR6 def
+/FontBBox {-20 -250 1193 750 }readonly def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR6.) readonly def
+/FullName (CMR6) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
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+readonly def
+currentdict end
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+%%Title: CMMI8
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI8.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+%%BeginFont: CMMI6
+%!PS-AdobeFont-1.0: CMMI6 003.002
+%%Title: CMMI6
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI6.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+%%EndFont 
+%%BeginFont: CMR8
+%!PS-AdobeFont-1.0: CMR8 003.002
+%%Title: CMR8
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR8.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+/FontName /CMR8 def
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+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR8.) readonly def
+/FullName (CMR8) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+end readonly def
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+%!PS-AdobeFont-1.0: CMEX10 003.002
+%%Title: CMEX10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMEX10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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 0000000000000000000000000000000000000000000000000000000000000000
 0000000000000000000000000000000000000000000000000000000000000000
@@ -4408,17 +5556,24 @@ end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
 dup 12 /fi put
+dup 13 /fl put
 dup 40 /parenleft put
 dup 41 /parenright put
 dup 44 /comma put
 dup 45 /hyphen put
 dup 46 /period put
+dup 49 /one put
+dup 50 /two put
 dup 51 /three put
 dup 65 /A put
+dup 67 /C put
 dup 68 /D put
+dup 72 /H put
 dup 73 /I put
 dup 76 /L put
+dup 77 /M put
 dup 84 /T put
+dup 86 /V put
 dup 97 /a put
 dup 98 /b put
 dup 99 /c put
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 0000000000000000000000000000000000000000000000000000000000000000
 0000000000000000000000000000000000000000000000000000000000000000
 0000000000000000000000000000000000000000000000000000000000000000
@@ -4912,9 +6112,6 @@ end readonly def
 /Encoding 256 array
 0 1 255 {1 index exch /.notdef put} for
 dup 12 /fi put
-dup 39 /quoteright put
-dup 40 /parenleft put
-dup 41 /parenright put
 dup 44 /comma put
 dup 45 /hyphen put
 dup 46 /period put
@@ -4938,12 +6135,15 @@ dup 76 /L put
 dup 77 /M put
 dup 78 /N put
 dup 79 /O put
+dup 80 /P put
 dup 82 /R put
 dup 83 /S put
 dup 84 /T put
 dup 85 /U put
 dup 87 /W put
 dup 89 /Y put
+dup 91 /bracketleft put
+dup 93 /bracketright put
 dup 97 /a put
 dup 98 /b put
 dup 99 /c put
@@ -4969,7 +6169,6 @@ dup 119 /w put
 dup 120 /x put
 dup 121 /y put
 dup 122 /z put
-dup 127 /dieresis put
 readonly def
 currentdict end
 currentfile eexec
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-%!PS-AdobeFont-1.0: CMTI10 003.002
-%%Title: CMTI10
-%Version: 003.002
-%%CreationDate: Mon Jul 13 16:17:00 2009
-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
-%Copyright: (<http://www.ams.org>), with Reserved Font Name CMTI10.
-% This Font Software is licensed under the SIL Open Font License, Version 1.1.
-% This license is in the accompanying file OFL.txt, and is also
-% available with a FAQ at: http://scripts.sil.org/OFL.
-%%EndComments
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-/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMTI10.) readonly def
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-/isFixedPitch false def
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 %%EndFont 
+%%BeginFont: CMSS12
+%!PS-AdobeFont-1.0: CMSS12 003.002
+%%Title: CMSS12
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMSS12.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
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+{save true}{false}ifelse}{false}ifelse
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+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
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+/FullName (CMSS12) readonly def
+/FamilyName (Computer Modern) readonly def
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+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
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diff --git a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.synctex.gz b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.synctex.gz
index e852b39..0dfffe6 100644
Binary files a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.synctex.gz and b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.synctex.gz differ
diff --git a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.tex b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.tex
index 15195e0..1a974c5 100644
--- a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.tex	
+++ b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.tex	
@@ -286,7 +286,7 @@ The magnetic flux in loop 2, that results from the current of wire loop 1 that r
 
 The total magnetic flux in wire loop 2 resulting from the currents in wire loop 1, can be described with the following formula.
 
-\begin{eqnarray}
+\begin{eqnarray}\label{eq:localization_function}
 \Phi_{21} = \Phi_{21_{r}} + \Phi_{21_{l}} =  \frac{\mu_0 l}{2\pi}i_{1}(ln\frac{b}{a}-ln\frac{d}{c}) = \frac{\mu_0 l}{2\pi}i_{1}ln\frac{bc}{ad}
 \end{eqnarray}
 
@@ -459,12 +459,18 @@ The receiving coils without ferrit core were self-made wooden cubes with an edge
 	\label{fig:infrared_camera_1}
 \end{figure}
 
-In order to verify the objects position, a reference measurement was performed. For this an infrared based localization system of the company ART (Advanced Realtime Tracking GmbH, Am Oeferl 6, 82362, Weilheim i.OB, Germany) was used. Six infrared cameras of the series ARTTRACK5  (Figure \ref{fig:infrared_camera_1}) were used. These cameras act as both infrared light emitter and receiver. The ARTTRACK5 have a tracking range of up to 7.5\metre, a field of view of around 80 degrees and offer up to 1.3 \Mega pix at up to 300\hertz. The cameras were connected to the ART Controller (ATC), which then again was connected to the host PC. The host PC ran the software (DTrack2). The object was mounted onto the target CT11 (Figure \ref{fig:loc_object_on_art_frame}) with four infrared marker spheres. The spheres reflect the infrared light back to the cameras enabling them to track the object. After calibration (Chapter \ref{sec:Setting up the localization systems}) the system had an accuracy of 0.33\milli\metre , according to the DTrack2 software.
+In order to verify the objects position, a reference measurement was performed. For this an infrared based localization system of the company ART (Advanced Realtime Tracking GmbH, Am Oeferl 6, 82362, Weilheim i.OB, Germany) was used. This system was chosen because of its sub millimetre precise localization and its adequate localization area of around $20\metre^2$. An alternative available system would have been the Qualisys system located in the LINK testing hall of the Fraunhofer IIS. During this project the Qualisys system was not adequate as its localization area of around $1400\metre^2$ is too large for this project. Also its availability is lower, since the LINK testing hall gets closed on several occasions for other Fraunhofer projects. The ART system was always available during this project. Because of these reasons the decision fell on the ART system. \\
+The following components of the ART system were used. Six infrared cameras of the series ARTTRACK5  (Figure \ref{fig:infrared_camera_1}) were used. These cameras act as both infrared light emitter and receiver. The ARTTRACK5 have a tracking range of up to 7.5\metre, a field of view of around 80 degrees and offer up to 1.3 \Mega pix at up to 300\hertz frame rate. The cameras were connected to the ART Controller (ATC), which then again was connected to the host PC. The host PC ran the software (DTrack2). The object was mounted onto the target CT11 (Figure \ref{fig:loc_object_on_art_frame}) with four infrared marker spheres. The spheres reflect the infrared light back to the cameras enabling them to track the object. After calibration (Chapter \ref{sec:Setting up the localization systems}) the system had an accuracy of 0.33\milli\metre , according to the DTrack2 software.
 
 
 
 \newpage
 \section{Methods}
+The following chapter is structured as follows. The first section (Chapter \ref{sec:Setting up the localization systems}) describes the necessary steps to set up the testing systems (Inductive localization system and infrared reference localization system).
+\\
+The second section (Chapter \ref{sec:precision_measurements}) describes how the accuracy measurement data was created. The following subsections (Chapter \ref{sec:scaling_factor_test}, \ref{sec:K-nearest interpolation test}, \ref{sec:remove_highest_frame_receiving_coil_from_the_localization_test}) explain how the data was then interpreted using different software paramters. These parameters were then altered in an attempt to optimize the systems localization accuracy.
+
+
 \subsection{Setting up the localization systems}\label{sec:Setting up the localization systems}
 In order to determine the precision of the IndLoc setup a series of tests were performed. The localization object was placed in different positions and the position was then measured by the IndLoc system. These measured positions are then compared to an infrared based camera system.
 
@@ -499,22 +505,29 @@ As the system is heating up the ART system can be calibrated. How to place the c
 \\
 Now that both localization systems are calibrated correctly a few test measurements should be performed. In order to do this place the localization object on the localization plane and compare the position which both systems display. If they are reasonably close to each other (around 1-2\centi\metre) the systems are working correctly. Confirm this by repeating this process for two or three positions which are at least 10\centi\metre $\:$ apart.
 
-\subsection{Precision measurements}
-All of the following precision measurements were performed by placing the localization object on specific positions in the localization area. It was positioned by eye using the millimetre sheet as a first rough positioning assistance. The exact position was then determined by recording three seconds with the ART system. Then the position was measured with the IndLoc system by recording 2000 samples with it. These recordings contain the receiving coils voltages. Thus saving a 2D array of 2000 by 16. The file names of each measurement were documented. Then the object was placed 1 \centi\metre 
+\subsection{Precision measurements}\label{sec:precision_measurements}
+All of the following precision measurements were performed by placing the localization object on specific positions in the localization area and measuring the resulting receiving coil voltages. It was positioned by eye using the millimetre sheet as a first rough positioning assistance. The exact position was then determined by recording three seconds with the ART system. As the frame rate of the ART system can differentiate, at least three seconds were recorded to make sure enough samples for further analysis were available. As the ART system is sub millimetre (0.33\milli\metre) precise this position was used as the correct reference position. 
+\\
+Afterwards the position was measured with the IndLoc system by recording 2000 samples per position. 
+
+2000 samples were recorded for each point so later on the possibility of averaging and analyzing the signal to noise ratio was available. At a sampling rate of 1MHz recording 2000 samples took 2\milli s. These recordings contain the receiving coils voltages. Thus saving a 2D array of 2000 by 16. The file names of each measurement were documented. Then the object was placed 1 \centi\metre 
 $\:$ to the side on the mm sheet and the process was repeated until a measurement set of points came together. The data of the precision measurements were then evaluated with different hard- or software parameters in order to optimize them.
 
-\subsubsection{Scaling factor test}
+\subsection{Movement measurements}
+
+\subsection{Precision measurements}  
+\subsubsection{Scaling factor test}\label{sec:scaling_factor_test}
 The scaling factor is a factor that is multiplied onto each receiving coil voltage. Its purpose is to diminish the difference between the calculated receiving coil voltages in the fingerprinting table and the real measured voltages. As the fingerprinting table is created certain parameters are required as input, such as for example: Exciter wire resistance, Inductance of the receiving coils, Resistance of the receiving coils, Windings of the localization object, etc. As measuring all of these parameters would take too much effort the scaling factor has been implemented. Usually the scaling factor is tested empirically for each IndLoc setup or prototype that is built.
 
 
 \begin{figure}[h]
 	\includegraphics[scale=0.4]{estimation_scaling_factor_1}
 	\centering
-	\caption{Before being able to localize precisely the correct scaling factor, which multiplies each receiving coil voltages, has to be determined. For this the scaling factor from $10^{2}$ to $ 10^{-4}$ were tested.}
+	\caption{Before being able to localize precisely the correct scaling factor, which multiplies each receiving coil voltages, has to be determined. For this the scaling factors $10^{2}$ to $ 10^{-4}$ were tested in logarithmic steps.}
 	\label{fig:estimation_scaling_factor_1}
 \end{figure}
 
-A first rough estimation of the scaling factor was done by testing $10^{2}$ to $ 10^{-3}$ in logarithmic steps (Figure \ref{fig:estimation_scaling_factor_1}, \ref{fig:estimation_scaling_factor_2}). Figure \ref{fig:estimation_scaling_factor_1} shows the entire localization area. The blue outside rectangle is the Exciter, which creates the primary magnetic field (Chapter \ref{sec:The_exciter}). The black rectangles along the exciter are the receiving coils, measuring the secondary magnetic field of the localization object (Chapter \ref{sec:The_receiving_coils}). The axis represent the localization area (Figure \ref{fig:Testing_setup}). The green cross markers in the middle of plot are the positions estimated by the ART system. As the precision of the ART system is 0.33\milli\metre $\:$ the green markers are used as the position the localization object is really in. The differently shaped black markers are the position the IndLoc system calculated with the different scaling factor, which can be looked up in the legend. It is to be noted that only the first sample of the ART and the IndLoc recordings were used during this test.
+A first rough estimation of the scaling factor was done by testing $10^{2}$ to $ 10^{-3}$ in logarithmic steps (Figure \ref{fig:estimation_scaling_factor_1}, \ref{fig:estimation_scaling_factor_2}). Figure \ref{fig:estimation_scaling_factor_1} shows the entire localization area. The blue outside rectangle represents the Exciter, which creates the primary magnetic field (Chapter \ref{sec:The_exciter}). The black rectangles along the exciter represent the receiving coils, measuring the secondary magnetic field of the localization object (Chapter \ref{sec:The_receiving_coils}). The axis span the localization area (Figure \ref{fig:Testing_setup}). The green cross markers in the middle of plot are the positions determined by the ART system. As the precision of the ART system is 0.33\milli\metre $\:$ the green markers are used as the correct reference position of the localization object. The different shaped black markers are the position the IndLoc system calculated with the different scaling factor. The scaling factor for each marker shape can be seen in the legend. It is to be noted that only the first sample of the ART and the IndLoc recordings were used during this test. How the noise affects the localization during the 2000 samples of one recording will be investigated later (Chapter \ref{sec:Noise_investigation}).
 
 
 \begin{figure}[h]
@@ -528,19 +541,19 @@ Figure \ref{fig:estimation_scaling_factor_2} shows a magnified view of Figure \r
 
 As the evaluation of the differences in localization precision by eye became impossible a statistical method was attempted. For this the distance between each pair of points was calculated with the following formula. 
 
-\begin{equation}
-d_{Pi} = \sqrt{(x_{ART_{Pi}} - x_{IndLoc_{Pi}})^2 + y_{ART_{Pi}} - y_{IndLoc_{Pi}})^2 }
+\begin{equation}\label{eq:distances}
+d_{Pi} = \sqrt{(x_{ART_{Pi}} - x_{IndLoc_{Pi}})^2 + (y_{ART_{Pi}} - y_{IndLoc_{Pi}})^2 }
 \end{equation}
 
-Afterwards the mean distance for each scaling factor was calculated with.
+Afterwards the mean distance for each scaling factor was calculated with the formula:
 
-\begin{equation}
+\begin{equation}\label{eq:mean}
 \mu ={\frac {1}{n}}\sum _{i=1}^{n}d_{i}={\frac {d_{1}+d_{2}+\cdots +d_{n}}{n}}
 \end{equation}
 
 The standard deviation for each scaling factor was calculated with the formula:
 
-\begin{equation}
+\begin{equation}\label{eq:std_dev}
 \sigma = \sqrt{\frac{\sum |P_{i} - \mu|^2}{N}}
 \end{equation}
 
@@ -552,9 +565,6 @@ The standard deviation for each scaling factor was calculated with the formula:
 	\label{fig:estimation_scaling_factor_3}
 \end{figure}
 
-\newpage
-Each equation was applied to a total of 500 different scaling factors ((0.0015, 0.002, 0.000001 - start, stop, step)). An optimum was found at a scaling factor of 0.001755 with $\mu = 3.76\milli\metre \:$ and $\sigma = 3.28\milli\metre \:$ resulting in the localization which can be seen in Figure \ref{fig:estimation_scaling_factor_3}. In order to verify these results another set of positions was used, which was positioned more to the edge of the localization area.
-
 \begin{comment}
 avg distance 3.757373465139104
 median distance 2.6566228652376584
@@ -562,19 +572,31 @@ mean distance 3.757373465139104
 std dev distance 3.2792325092288714
 \end{comment}
 
+\clearpage
+Each equation was applied to a total of 500 different scaling factors ((0.0015, 0.002, 0.000001 - start, stop, step)). An optimum was found at a scaling factor of 0.001755 with $\mu = 3.76\milli\metre \:$ and $\sigma = 3.28\milli\metre \:$ resulting in the localization which can be seen in Figure \ref{fig:estimation_scaling_factor_3}. In order to verify these results another set of positions was used, which was positioned near the corner of the localization area (Figure \ref{fig:estimation_scaling_factor_4}).
+
+
+
 \begin{figure}[h]
 	\includegraphics[scale=0.4]{estimation_scaling_factor_4}
 	\centering
-	\caption{Placing the object in 25 positions near the corner of the localization area with a scaling factor of 0.001755, , resulting in mean error $\mu = 6.18\milli\metre$ and standard deviation $\sigma = 7.71\milli\metre$.}
+	\caption{Placing the object in 25 positions near the corner of the localization area with a scaling factor of 0.002100, resulting in mean error $\mu = 10.39\milli\metre$ and standard deviation $\sigma = 7.52\milli\metre$.}
 	\label{fig:estimation_scaling_factor_4}
 \end{figure}
 
-Statistically optimizing the scaling factor for the off center set of points resulted in a scaling factor of 0.002105. Resulting in a mean error $\mu = 6.18\milli\metre \:$ and a standard deviation $\sigma = 7.71\milli\metre$.
+\begin{comment}
+ scaling_factor =  0.0021000000000000016 
+ mean_error =  10.39 
+ standard_deviation =  7.52
+Plotting positions for: sf= 0.0021000000000000016 , k= 3
+\end{comment}
+
+Statistically optimizing the scaling factor for the off centre set of points resulted in a scaling factor of 0.002105. Resulting in a mean error $\mu = 6.18\milli\metre \:$ and a standard deviation $\sigma = 7.71\milli\metre$.
 
 \begin{comment}
 Best scaling factor =    mean =  6.180273611932338 std_dev =  7.715220033729418 ; min_both =  13.895493645661755
 \end{comment}
-As the localization accuracy declined it suggested that the optimal scaling factor is positional dependant. For this reason the whole localization system had to be taken into account (Figure \ref{fig:estimation_scaling_factor_5}).
+As the localization accuracy declined it suggested that the optimal scaling factor is positional dependant. For this reason the whole localization system had to be taken into account.
 
 \begin{figure}[h]
 	\includegraphics[scale=0.4]{estimation_scaling_factor_5}
@@ -583,19 +605,129 @@ As the localization accuracy declined it suggested that the optimal scaling fact
 	\label{fig:estimation_scaling_factor_5}
 \end{figure}
 
+In order to look at the entire localization area the object was placed 42 positions running in a horizontal and a diagonal line from the centre to the edge of the localization area (Figure \ref{fig:estimation_scaling_factor_5}). Due to the symmetric setup these lines can be mirrored and as such give a rough estimation of the entire localization area. 
+
 \begin{comment}
 best sf = 0.00204  mean =  7.421424662824991 std dev =  6.481103235160666 ; min both =  13.902527897985657
 \end{comment}
 
+\clearpage
+\subsubsection{K-nearest interpolation test}\label{sec:K-nearest interpolation test}
+During localization a k-nearest neighbour interpolation is performed. The 16 measured receiving coil voltages are compared to the fingerprinting table (Chapter \ref{sec:The_host system}). The k best matching entries of the fingerprinting table are selected. The X-,Y-,Z-positions of the selected entries are then used for the k-nearest neighbour interpolation resulting in a final X-,Y-,Z-position.
+\\
+If the k-nearest neighbour interpolation would not be performed the final position could only have \centi\metre $\:$ values without decimal places. The reason being that the fingerprinting table is created using a 1\centi\metre $\:$ X-,Y-,Z-step width (Chapter \ref{sec:The_host system}). That is why the k-nearest neighbour interpolation is a crucial step to achieve sub \centi\metre$\:$ precision. During all previous tests k=2 was used, as this is the lowest possible value. If k=1 would have been used the previously problem of achieving sub \centi\metre $\:$ precision would have occurred and may have distorted previous results.
+
+During this chapter multiple k values will be tested and the resulting localization precision will be determined. As previously mentioned (Chapter \ref{sec:precision_measurements}) the data, which was analysed during the scaling factor test (Chapter \ref{sec:scaling_factor_test}) will be analysed during this test as well. Namely the set of points in the centre (Figure \ref{fig:estimation_scaling_factor_3}), in the corner (Figure \ref{fig:estimation_scaling_factor_4}) and the set of points which cover the entirety of the localization area(Figure \ref{fig:estimation_scaling_factor_5}). In order to receive comparable results, the individual optimized scaling factor for each set of points were used (Chapter \ref{sec:scaling_factor_test}).
 
 \begin{figure}[h]
-	\includegraphics[scale=0.4]{estimation_scaling_factor_6}
+	\includegraphics[scale=0.4]{estimation_k_nearest_1}
 	\centering
-	\caption{With averaged IndLoc feature vectors 0.00204, , resulting in mean error $\mu = 7.34\milli\metre$ and standard deviation $\sigma = 6.49\milli\metre$.}
-	\label{fig:estimation_scaling_factor_6}
+	\caption{placing the localization object in 25 positions near the centre of the localization are with a k-nearest value of 19, resulting in mean error $\mu = 3.21\milli\metre$ and standard deviation $\sigma = 1.84\milli\metre$.}
+	\label{fig:estimation_k_nearest_1}
 \end{figure}
 
-Averaging the 2000 samples before localizing Figure \ref{fig:estimation_scaling_factor_6}.
+\begin{comment}
+
+[BEST MEAN]: k_nearest: 9  --> mean= 3.17 , std_dev= 2.04
+[BEST MEAN+STD]: current k_nearest: 19  --> mean= 3.21 , std_dev= 1.84
+
+current k_nearest: 2  --> mean= 7.44 , std_dev= 4.87
+current k_nearest: 100  --> mean= 5.29 , std_dev= 2.19
+
+Trend: mean and std start high, both decrease as k rises. Lowest mean at k=9. Lowest both at k=19 (statistical optimum). Then after that it gets worse until k=100 the mean rises until 5 and std_dev until 2.
+\end{comment}
+
+A total of 100 k values were tested (start:1, stop:100, step:1) and were each evaluated using the formulas (\ref{eq:distances}), (\ref{eq:mean}) and (\ref{eq:std_dev}). An optimum was determined at k=19 with $\mu = 3.21\milli\metre$ and $\sigma = 1.84\milli\metre$ (Figure \ref{fig:estimation_k_nearest_1}). 
+
+
+\begin{figure}[h]
+	\includegraphics[scale=0.4]{estimation_k_nearest_2}
+	\centering
+	\caption{placing the localization object in 25 positions near the corner of the localization are with a k-nearest value of 12, resulting in mean error $\mu = 10.39\milli\metre$ and standard deviation $\sigma = 7.52\milli\metre$.}
+	\label{fig:estimation_k_nearest_2}
+\end{figure}
+
+\begin{comment}
+ scaling_factor =  0.0021 
+ mean_error =  10.39 
+ standard_deviation =  7.52
+Plotting positions for: sf= 0.0021 , k= 3
+\end{comment}
+
+\begin{figure}[h]
+	\includegraphics[scale=0.4]{estimation_k_nearest_3}
+	\centering
+	\caption{Placing the object in 42 positions running from the centre to the edges of the localization area with a k-nearest value of 4, resulting in mean error $\mu = 7.57\milli\metre$ and standard deviation $\sigma = 5.34\milli\metre$.}
+	\label{fig:estimation_k_nearest_3}
+\end{figure}
+
+\begin{comment}
+scaling_factor =  0.00204 
+ mean_error =  7.57 
+ standard_deviation =  5.34
+Plotting positions for: sf= 0.00204 , k= 4
+\end{comment}
+
+\clearpage
+\subsubsection{Remove highest frame receiving coil from the localization test}\label{sec:remove_highest_frame_receiving_coil_from_the_localization_test}
+It is known in the team that the localization near the receiving coils  declines. For this reason the frame receiving coil which shows the highest voltage, is removed from the localization calculations.
+
+\begin{figure}[h]
+	\includegraphics[scale=0.4]{estimation_removing_highest_frame_1}
+	\centering
+	\caption{(sf= 0.001755 , k= 19) Placing the localization object in 25 positions near the centre of the localization area with removing the highest frame receiving coil, resulting in mean error $\mu = 5.56\milli\metre$ and standard deviation $\sigma = 3.2\milli\metre$.}
+	\label{fig:estimation_removing_highest_frame_1}
+\end{figure}
+
+\begin{comment}
+ scaling_factor =  0.001755 
+ mean_error =  5.56 
+ standard_deviation =  3.2
+Plotting positions for: sf= 0.001755 , k= 19
+\end{comment}
+
+
+\begin{figure}[h]
+	\includegraphics[scale=0.4]{estimation_removing_highest_frame_2}
+	\centering
+	\caption{(sf= 0.0021 , k= 12) Placing the localization object in 25 positions near the corner of the localization area with a k-nearest value of 4, resulting in mean error $\mu = 13.32\milli\metre$ and standard deviation $\sigma = 8.24\milli\metre$.}
+	\label{fig:estimation_k_nearest_2}
+\end{figure}
+
+\begin{comment}
+ scaling_factor =  0.0021 
+ mean_error =  13.32 
+ standard_deviation =  8.24
+Plotting positions for: sf= 0.0021 , k= 12
+
+So far my method was: Determine the best scaling_factor, then for the next tests keep using this sf.
+Then after that with this sf determine the best k_nearest value. 
+
+Then in this test I used the best sf from sf_test for this set of data (horiz+diag) and the best k from k-test. Both of these parameters were used in this "take out highest frame" test and the localization was fairly shitty.
+I tested a bit around and the best I could get was at totally different sf and k. 
+!!!----------!!!------------!!!-------------!!!--------------!!!---
+In general I feel like that when changing the sf, the k nearest needs to be readjusted. And vice versa. So the two are dependant.
+!---------!!!!-------------!!!--------------!!!--------!!!---------
+\end{comment}
+
+
+
+\begin{figure}[h]
+	\includegraphics[scale=0.4]{estimation_removing_highest_frame_3}
+	\centering
+	\caption{(sf= 0.00204 , k= 4) Placing the object in 42 positions running from the centre to the edges of the localization area with a k-nearest value of 4, resulting in mean error $\mu = 15.21\milli\metre$ and standard deviation $\sigma = 14.33 \milli\metre$.}
+	\label{fig:estimation_k_nearest_3}
+\end{figure}
+
+\begin{comment}
+ scaling_factor =  0.00204 
+ mean_error =  15.21 
+ standard_deviation =  14.33
+Plotting positions for: sf= 0.00204 , k= 4
+\end{comment}
+
+\clearpage
+\subsubsection{Noise investigation}\label{sec:Noise_investigation}
 
 \clearpage
 \section{RESULTS}
@@ -609,6 +741,7 @@ Averaging the 2000 samples before localizing Figure \ref{fig:estimation_scaling_
 
 
 \clearpage
+
 \bibliography{Literatur_BA_2}
 \end{document}
 
diff --git a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.toc b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.toc
index 61dcd25..f29b65d 100644
--- a/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.toc	
+++ b/BachelorThesis/Writing/2021_Seyffer_Investigating the precision of an induction-based localization system for medical applications.toc	
@@ -24,8 +24,13 @@
 \contentsline {subsection}{\numberline {4.2}ART system}{17}%
 \contentsline {section}{\numberline {5}Methods}{19}%
 \contentsline {subsection}{\numberline {5.1}Setting up the localization systems}{19}%
-\contentsline {subsection}{\numberline {5.2}Precision measurements}{20}%
-\contentsline {subsubsection}{\numberline {5.2.1}Scaling factor test}{21}%
-\contentsline {section}{\numberline {6}RESULTS}{26}%
-\contentsline {section}{\numberline {7}DISCUSSION}{27}%
-\contentsline {section}{\numberline {8}SUMMARY, OUTLOOK INTO THE FUTURE}{28}%
+\contentsline {subsection}{\numberline {5.2}Precision measurements}{21}%
+\contentsline {subsection}{\numberline {5.3}Movement measurements}{21}%
+\contentsline {subsection}{\numberline {5.4}Precision measurements}{21}%
+\contentsline {subsubsection}{\numberline {5.4.1}Scaling factor test}{21}%
+\contentsline {subsubsection}{\numberline {5.4.2}K-nearest interpolation test}{27}%
+\contentsline {subsubsection}{\numberline {5.4.3}Remove highest frame receiving coil from the localization test}{30}%
+\contentsline {subsubsection}{\numberline {5.4.4}Noise investigation}{32}%
+\contentsline {section}{\numberline {6}RESULTS}{33}%
+\contentsline {section}{\numberline {7}DISCUSSION}{34}%
+\contentsline {section}{\numberline {8}SUMMARY, OUTLOOK INTO THE FUTURE}{35}%
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