Rath, T., Credner, J., Fadami, M., Rehrmann, P.: Blue Apple – Transformation of histograms to Gaussian distributions. BLab Code No. 20223, http://www.blab-osnabrueck.de/blue-apple3.txt, 2022.

Title: Blue Apple - Transformation of histograms to Gaussian distributions

Authors: Thomas Rath, Jonas Credner, Maryam Fadami, Peter Rehrmann, 

Year:2022

Keywords: Image pre-processing, histogramm transformation, color selection, identification biological objects, binarisation, Halcon

Abstract:
Image processing of color images often requires to analyse histograms of color chanels. Theses histograms are normally not normal distributed. With this algorithm one can transform a histogramm, which consists of any data, to a pseudo normal distributed histogram. Pseudo normal distributed means, that the histogram is as much as possible (or similar as) normal distributed. It is calculated by separating the histogram into a left side and a right side histogram. The code for the transformation is given. Additionally the code for transfomation of an absolute histogram to a relative histogram is given.  

Cite as:
t.Rath, J. Credner, M. Fadami, P. Rehrmann (2022): 
Blue Apple - Transformation of histograms to Gaussian distributions.
BLab Code No. 20223, http://www.blab-osnabrueck.de/blue-apple3.txt

Literature:
Halcon (2022): Halcon - Die Leistungsstarke Software fuer Ihre Bildverarbeitungsanwendung. MVTec Software GmbH, Munich, Germany

Affiliation:
T.Rath, J. Credner, M. Fadami, P. Rehrmann
Biosystem Engineering Laboratory (BLab), Osnabrueck University of Applied sciences
email: t.rath@hs-osnabrueck.de


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# File generated by HDevelop for text file Version 13.0.3
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procedure gauss_classification_histogram (: : Rel_Histogramm: Klassifikations_Histogramm, 
    L1, L2, L3, L4, R1, R2, R3, R4, Mue, MueY)
  * ***********************************************************************************************
  * *Berechnung des gauss-normierten Histgogrammes und Ausgabe von Stuetzstellen und Mittelwerten
  * ***********************************************************************************************
  
    
  
  * ***********************************************************************************************************************
  *  Bestimmung von MueX bei beliebigem Histogramm, durch Berechnung der Schwerpunktlinie der Histogrammfläche, 2.3.2021 *
  * ***********************************************************************************************************************
  
  HistoSum := 0
  for i := 1 to 256 by 1
      tuple_select (Rel_Histogramm, i-1, Wert)
      HistoSum := HistoSum + Wert
  endfor
  
  * *********
  * MueX-Suche von links
  * *********
  Sum_von_links := 0.0
  il := -1
  repeat
      il := il + 1
      tuple_select (Rel_Histogramm, il, Wert)
      Sum_von_links := Sum_von_links + Wert
  until ((Sum_von_links >= HistoSum/2) or (il = 255))
  
  
  * *********
  * MueX-Suche von rechts
  * *********
  Sum_von_rechts := 0.0
  ir := 256
  repeat
      ir := ir - 1
      tuple_select (Rel_Histogramm, ir, Wert)
      Sum_von_rechts := Sum_von_rechts + Wert
  until ((Sum_von_rechts >= HistoSum/2) or (ir = 0))
  
  * *********
  * MueX-Ermittlung als Mittelwert ungerundet
  * *********
  
  MueX := (il + ir)/2.0
  
  Mue2 := int(MueX)
  
  
  MueLinks := Mue2
  MueRechts := Mue2 + 1
  
  
  * *************************************   Ende Bestimmung MueX, also mittlerer X-Wert des Histogramms aufgrund der Histogramm-Y-W
  * ***************************************
  
  
  
  F4 := 0.00135 * HistoSum
  F3 := 0.02140 * HistoSum
  F2 := 0.13590 * HistoSum
  F1 := 0.34135 * HistoSum
  
  
  * ********Werte um 1 hochsetzen*****************************
  * for i := 0 to 255 by 1
  *     tuple_select (Rel_Histogramm, i, Wert)
  *     Wert := Wert + 1.0
  *     tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
  
  * endfor
  
  
  
  * tuple_round (Mue, Mue)
  
  tuple_gen_const (301, 0, LinkerZusatz)
  tuple_gen_const (300, 0, RechterZusatz)
  
  tuple_concat (LinkerZusatz, Rel_Histogramm, Rel_Histogramm)
  tuple_concat (Rel_Histogramm, RechterZusatz, Rel_Histogramm)
  
  MueLinks := MueLinks + 301
  MueRechts := MueRechts + 301
  
  
  tuple_select (Rel_Histogramm, MueLinks, MueY1)
  tuple_select (Rel_Histogramm, MueRechts, MueY2)
  
  MueY := (MueY1 + MueY2) / 2.0
  
  FlaecheLinks := 0.5 * MueY
  
  
  i := MueLinks + 1
  
  
  repeat
  
      i := i - 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheLinks := FlaecheLinks + Auswahl
  
  until ((FlaecheLinks >= F1) or (i = 300))
  
  L1 := i
  
  FlaecheRechts := 0.5 * MueY
  
  i := MueRechts - 1
  repeat
  
      i := i + 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheRechts := FlaecheRechts + Auswahl
  
  until ((FlaecheRechts >= F1) or (i = 557))
  
  R1 := i
  * ************************************************
  
  FlaecheLinks := 0.5 * MueY
  i := MueLinks + 1
  repeat
  
      i := i - 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheLinks := FlaecheLinks + Auswahl
  
  until ((FlaecheLinks >= (F1+F2)) or (i=300))
  
  L2 := i
  
  if (L2 >= L1)
  *     L2 := L1 - 1
      L2 := L1
  endif
  
  FlaecheRechts := 0.5 * MueY
  
  i := MueRechts -1
  repeat
  
      i := i + 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheRechts := FlaecheRechts + Auswahl
  
  until ((FlaecheRechts >= (F1+F2)) or (i=557))
  
  R2 := i
  
  if (R2 <= R1)
  *     R2 := R1 + 1
      R2 := R1
  endif
  * *******************************************
  FlaecheLinks := 0.5 * MueY
  i := MueLinks + 1
  repeat
  
      i := i - 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheLinks := FlaecheLinks + Auswahl
  
  until ((FlaecheLinks >= (F1+F2+F3)) or (i=300))
  
  L3 := i
  
  if (L3 >= L2)
  *     L3 := L2 - 1
      L3 := L2
  endif
  
  FlaecheRechts := 0.5 * MueY
  
  i := MueRechts - 1
  repeat
  
      i := i + 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheRechts := FlaecheRechts + Auswahl
  
  until ((FlaecheRechts >= (F1+F2+F3)) or (i = 557 ))
  
  R3 := i
  
  if (R3 <= R2)
  *     R3 := R2 + 1
      R3 := R2
  endif
  * **************************************************
  FlaecheLinks := 0.5 * MueY
  i := MueLinks + 1
  repeat
  
      i := i - 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheLinks := FlaecheLinks + Auswahl
  
  until ((FlaecheLinks >= (F1+F2+F3+F4)) or (i=300))
  
  L4 := i
  
  if (L4 >= L3)
  *     L4 := L3 -1
      L4 := L3
  
  endif
  
  FlaecheRechts := 0.5 * MueY
  
  i := MueRechts - 1
  repeat
  
      i := i + 1
      tuple_select (Rel_Histogramm, i, Auswahl)
      FlaecheRechts := FlaecheRechts + Auswahl
  
  until ((FlaecheRechts >= (F1+F2+F3+F4)) or (i=557))
  
  R4 := i
  
  if (R4 <= R3)
  *     R4 := R3 + 1
      R4 := R3
  endif
  * *****************************************************
  DurchschnittlichL := ((L1-L2) + (L2-L3) + (L3-L4)) / 3
  DurchschnittlichR := ((R2-R1) + (R3-R2) + (R4-R3)) / 3
  
  
  LNix := L4 - DurchschnittlichL
  
  if (LNix >= 301)
      LNix := 300
  endif
  
  RNix := R4 + DurchschnittlichR
  
  if (RNix <= 556)
      RNix := 557
  endif
  
  * **************************************************
  
  if (MueLinks != L1)
  
      L1Y := F1 /(MueLinks - L1)
  else
  
      L1Y := F1 / 1.0
  
  endif
  
  
  if (L1 != L2)
      L2Y := F2 /(L1 - L2)
  else
      L2Y := L1Y
  endif
  
  if (L2 != L3)
      L3Y := F3 /(L2 - L3)
  else
      L3Y := L2Y
  endif
  
  if (L3 != L4)
      L4Y := F4 /(L3 - L4)
  else
      L4Y := L3Y
  endif
  
  if (MueRechts != R1)
  
      R1Y := F1 /(R1 - MueRechts)
  
  else
  
      R1Y := F1/1.0
  
  endif
  
  
  
  if (R1 != R2)
      R2Y := F2 /(R2 - R1)
  else
      R2Y := R1Y
  endif
  
  if (R2 != R3)
      R3Y := F3 /(R3 - R2)
  else
      R3Y := R2Y
  endif
  
  if (R3 != R4)
      R4Y := F4 /(R4 - R3)
  else
      R4Y := R3Y
  endif
  
  
  
  * RNixY := 0
  * LNixY := 0
  * **************************************************
  * if (L2Y > L1Y)
  
  *     L2Y := L1Y - (MueY/256)
  
  * endif
  
  * if (L3Y > L2Y)
  
  *     L3Y := L2Y - (MueY/256)
  
  * endif
  
  * if (L4Y > L3Y)
  
  *     L4Y := L3Y - (MueY/256)
  
  * endif
  
  * LNixY := L4Y  - (MueY/256)
  LNixY := 0
  
  
  * if (R2Y > R1Y)
  
  *     R2Y := R1Y - (MueY/256)
  
  * endif
  
  * if (R3Y > R2Y)
  
  *     R3Y := R2Y - (MueY/256)
  
  * endif
  
  * if (R4Y > R3Y)
  
  *     R4Y := R3Y - (MueY/256)
  
  * endif
  
  * RNixY := R4Y - (MueY/256)
  RNixY := 0
  * **************************************************
  for i := L1 to R1 by 1
      Wert := ((R1Y-L1Y)/(R1-L1))*i+((L1Y*R1-R1Y*L1)/(R1-L1))
      tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
  endfor
  
  if (R1 != R2)
      for i := R1+1 to R2 by 1
          Wert := ((R2Y-R1Y)/(R2-R1))*i+((R1Y*R2-R2Y*R1)/(R2-R1))
          tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
      endfor
  endif
  
  if (L1 != L2)
      for i := L1-1 to L2 by -1
          Wert := ((L2Y-L1Y)/(L2-L1))*i+((L1Y*L2-L2Y*L1)/(L2-L1))
          tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
      endfor
  endif
  
  if (R2 != R3)
      for i := R2+1 to R3 by 1
          Wert := ((R3Y-R2Y)/(R3-R2))*i+((R2Y*R3-R3Y*R2)/(R3-R2))
          tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
      endfor
  endif
  
  if (L2 != L3)
      for i := L2-1 to L3 by -1
          Wert := ((L3Y-L2Y)/(L3-L2))*i+((L2Y*L3-L3Y*L2)/(L3-L2))
          tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
      endfor
  endif
  
  if (R3 != R4)
      for i := R3+1 to R4 by 1
          Wert := ((R4Y-R3Y)/(R4-R3))*i+((R3Y*R4-R4Y*R3)/(R4-R3))
          tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
      endfor
  endif
  
  if (L3 != L4)
      for i := L3-1 to L4 by -1
          Wert := ((L4Y-L3Y)/(L4-L3))*i+((L3Y*L4-L4Y*L3)/(L4-L3))
          tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
      endfor
  endif
  
  for i := R4+1 to RNix by 1
      Wert := ((RNixY-R4Y)/(RNix-R4))*i+((R4Y*RNix-RNixY*R4)/(RNix-R4))
      tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
  endfor
  
  for i := L4-1 to LNix by -1
      Wert := ((LNixY-L4Y)/(LNix-L4))*i+((L4Y*LNix-LNixY*L4)/(LNix-L4))
      tuple_replace (Rel_Histogramm, i, Wert, Rel_Histogramm)
  endfor
  
  
  * ******************************************************************
  tuple_select_range (Rel_Histogramm, 301, 556, Rel_Histogramm)
  
  Klassifikations_Histogramm := Rel_Histogramm
  
  * if (HistogrammRelative = true)
  
  *     histogram_norm (Klassifikations_Histogramm, Klassifikations_Histogramm)
  
  * endif
  
  L1 := L1 - 301
  L2 := L2 - 301
  L3 := L3 - 301
  L4 := L4 - 301
  
  R1 := R1 - 301
  R2 := R2 - 301
  R3 := R3 - 301
  R4 := R4 - 301
  Mue := MueX - 301
  * MueY := MueY/2
  
  
  
  return ()


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# File generated by HDevelop for text file Version 13.0.3
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procedure histogram_norm (: : Lineares_Gauss_Histogramm: Lineares_Gauss_Histogramm_Normiert)
  * *******************************************************
  * *Normierung eines Histogrammes auf den Summenwert 1
  * *******************************************************
  
  tuple_length (Lineares_Gauss_Histogramm, Length)
  
  tuple_sum (Lineares_Gauss_Histogramm, Sum)
  
  Faktor := 1.0/Sum
  
  for i := 0 to Length-1 by 1
  
      tuple_select (Lineares_Gauss_Histogramm, i, Selected)
      NormierterWert := Selected * Faktor
      tuple_replace (Lineares_Gauss_Histogramm, i, NormierterWert, Lineares_Gauss_Histogramm)
  
  
  endfor
  
  Lineares_Gauss_Histogramm_Normiert := Lineares_Gauss_Histogramm
  
  
  
  return ()