Uniform map of R3 colour space onto a discrete number line












0












$begingroup$


I'm thinking about theoretical colour palette formation.



I have to start with a locus of $1000^3$ points in $R^3$, with each colour component axis being a discrete variable $0 le r, g, b < 1000$. (The resolution of 1000 is due to software constraints). This represents an RGB colour space - the cubic solid shown in the lower left of this diagram:



color solids



I have to decide on a scheme to choose up to $2^8$ of those points in a manner that attempts to get approximate uniform representation from each axis, and then map those points to an $R^1$ discrete space $0 le p < 2^8$. The mapping doesn't need to be linear but that would make things easier.



Uniform method



If we choose $n$ values on each axis, then



$n = lfloor 2^{8/3} rfloor = 6$ values per axis



$6^3 = 216$ values



$ frac {100 cdot 6^3} {2^8} approx 84% $ p space usage



For each channel value $c_i$, for $0 le i < 3$,



$c_i = frac {1000} 6 modleft( lfloor frac p {6^i} rfloor, 6 right) $



That method has perfect channel uniformity but wastes a lot (16%) of the p space.



The following is a generated visual representation of the uniform palette. The x-axis is red, the "short" y-axis is green, and the "long" y-axis is blue.



uniform-palette



Bitfield method



Another method is to divide the colour channels into bit fields: for field bit count $f_i, 0 le i < 3$:



$sum_i f_i = 8$



$f = (3, 3, 2)$



$c_i = frac {1000} {f_i} modleft( lfloor frac p {f_{i-1}^i} rfloor, f_i right) $



$f_{i-1} = 1$ for $i = 0$.



This method has perfect p space usage but poor uniformity. Two of the channels have $2^3=8$ values per axis but the last only has $2^2=4$ values.



The following is a generated visual representation of the bitfield palette, with the same axes as the previous one.



uniform palette



I'd like to learn of another method to map p to the $R^3$ space that achieves better uniformity than the bit field method while still using all of the p space.










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    0












    $begingroup$


    I'm thinking about theoretical colour palette formation.



    I have to start with a locus of $1000^3$ points in $R^3$, with each colour component axis being a discrete variable $0 le r, g, b < 1000$. (The resolution of 1000 is due to software constraints). This represents an RGB colour space - the cubic solid shown in the lower left of this diagram:



    color solids



    I have to decide on a scheme to choose up to $2^8$ of those points in a manner that attempts to get approximate uniform representation from each axis, and then map those points to an $R^1$ discrete space $0 le p < 2^8$. The mapping doesn't need to be linear but that would make things easier.



    Uniform method



    If we choose $n$ values on each axis, then



    $n = lfloor 2^{8/3} rfloor = 6$ values per axis



    $6^3 = 216$ values



    $ frac {100 cdot 6^3} {2^8} approx 84% $ p space usage



    For each channel value $c_i$, for $0 le i < 3$,



    $c_i = frac {1000} 6 modleft( lfloor frac p {6^i} rfloor, 6 right) $



    That method has perfect channel uniformity but wastes a lot (16%) of the p space.



    The following is a generated visual representation of the uniform palette. The x-axis is red, the "short" y-axis is green, and the "long" y-axis is blue.



    uniform-palette



    Bitfield method



    Another method is to divide the colour channels into bit fields: for field bit count $f_i, 0 le i < 3$:



    $sum_i f_i = 8$



    $f = (3, 3, 2)$



    $c_i = frac {1000} {f_i} modleft( lfloor frac p {f_{i-1}^i} rfloor, f_i right) $



    $f_{i-1} = 1$ for $i = 0$.



    This method has perfect p space usage but poor uniformity. Two of the channels have $2^3=8$ values per axis but the last only has $2^2=4$ values.



    The following is a generated visual representation of the bitfield palette, with the same axes as the previous one.



    uniform palette



    I'd like to learn of another method to map p to the $R^3$ space that achieves better uniformity than the bit field method while still using all of the p space.










    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I'm thinking about theoretical colour palette formation.



      I have to start with a locus of $1000^3$ points in $R^3$, with each colour component axis being a discrete variable $0 le r, g, b < 1000$. (The resolution of 1000 is due to software constraints). This represents an RGB colour space - the cubic solid shown in the lower left of this diagram:



      color solids



      I have to decide on a scheme to choose up to $2^8$ of those points in a manner that attempts to get approximate uniform representation from each axis, and then map those points to an $R^1$ discrete space $0 le p < 2^8$. The mapping doesn't need to be linear but that would make things easier.



      Uniform method



      If we choose $n$ values on each axis, then



      $n = lfloor 2^{8/3} rfloor = 6$ values per axis



      $6^3 = 216$ values



      $ frac {100 cdot 6^3} {2^8} approx 84% $ p space usage



      For each channel value $c_i$, for $0 le i < 3$,



      $c_i = frac {1000} 6 modleft( lfloor frac p {6^i} rfloor, 6 right) $



      That method has perfect channel uniformity but wastes a lot (16%) of the p space.



      The following is a generated visual representation of the uniform palette. The x-axis is red, the "short" y-axis is green, and the "long" y-axis is blue.



      uniform-palette



      Bitfield method



      Another method is to divide the colour channels into bit fields: for field bit count $f_i, 0 le i < 3$:



      $sum_i f_i = 8$



      $f = (3, 3, 2)$



      $c_i = frac {1000} {f_i} modleft( lfloor frac p {f_{i-1}^i} rfloor, f_i right) $



      $f_{i-1} = 1$ for $i = 0$.



      This method has perfect p space usage but poor uniformity. Two of the channels have $2^3=8$ values per axis but the last only has $2^2=4$ values.



      The following is a generated visual representation of the bitfield palette, with the same axes as the previous one.



      uniform palette



      I'd like to learn of another method to map p to the $R^3$ space that achieves better uniformity than the bit field method while still using all of the p space.










      share|cite|improve this question











      $endgroup$




      I'm thinking about theoretical colour palette formation.



      I have to start with a locus of $1000^3$ points in $R^3$, with each colour component axis being a discrete variable $0 le r, g, b < 1000$. (The resolution of 1000 is due to software constraints). This represents an RGB colour space - the cubic solid shown in the lower left of this diagram:



      color solids



      I have to decide on a scheme to choose up to $2^8$ of those points in a manner that attempts to get approximate uniform representation from each axis, and then map those points to an $R^1$ discrete space $0 le p < 2^8$. The mapping doesn't need to be linear but that would make things easier.



      Uniform method



      If we choose $n$ values on each axis, then



      $n = lfloor 2^{8/3} rfloor = 6$ values per axis



      $6^3 = 216$ values



      $ frac {100 cdot 6^3} {2^8} approx 84% $ p space usage



      For each channel value $c_i$, for $0 le i < 3$,



      $c_i = frac {1000} 6 modleft( lfloor frac p {6^i} rfloor, 6 right) $



      That method has perfect channel uniformity but wastes a lot (16%) of the p space.



      The following is a generated visual representation of the uniform palette. The x-axis is red, the "short" y-axis is green, and the "long" y-axis is blue.



      uniform-palette



      Bitfield method



      Another method is to divide the colour channels into bit fields: for field bit count $f_i, 0 le i < 3$:



      $sum_i f_i = 8$



      $f = (3, 3, 2)$



      $c_i = frac {1000} {f_i} modleft( lfloor frac p {f_{i-1}^i} rfloor, f_i right) $



      $f_{i-1} = 1$ for $i = 0$.



      This method has perfect p space usage but poor uniformity. Two of the channels have $2^3=8$ values per axis but the last only has $2^2=4$ values.



      The following is a generated visual representation of the bitfield palette, with the same axes as the previous one.



      uniform palette



      I'd like to learn of another method to map p to the $R^3$ space that achieves better uniformity than the bit field method while still using all of the p space.







      computer-science closed-map






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      edited Jan 2 at 23:05







      Reinderien

















      asked Jan 2 at 22:47









      ReinderienReinderien

      1135




      1135






















          1 Answer
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          $begingroup$

          6*6*7 = 252, which is much closer to 256. If $R in [0, 6)$, $G in [0, 7)$, and $B in [0, 6)$, the 8-bit indexed colour could be



          $C = R*7*6 + G*6 + B$



          That's pretty good.



          I also tried a spherical packing. For a face-centred packing into a cube with points at all corners, I get that a pattern $2x + 1$ layers deep has $4x^3 + 6x^2 + 3x + 1$ points in it. When $x = 3$, the pattern has 172 points. When $x = 4$, the pattern has 365 points. So neither of those is any good... On the other hand, I just found a totally awesome way of visually representing non-leap-years.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            6,7,6 yields about 1.6% waste. Awesome!
            $endgroup$
            – Reinderien
            Jan 3 at 4:17











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          active

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          active

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          1












          $begingroup$

          6*6*7 = 252, which is much closer to 256. If $R in [0, 6)$, $G in [0, 7)$, and $B in [0, 6)$, the 8-bit indexed colour could be



          $C = R*7*6 + G*6 + B$



          That's pretty good.



          I also tried a spherical packing. For a face-centred packing into a cube with points at all corners, I get that a pattern $2x + 1$ layers deep has $4x^3 + 6x^2 + 3x + 1$ points in it. When $x = 3$, the pattern has 172 points. When $x = 4$, the pattern has 365 points. So neither of those is any good... On the other hand, I just found a totally awesome way of visually representing non-leap-years.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            6,7,6 yields about 1.6% waste. Awesome!
            $endgroup$
            – Reinderien
            Jan 3 at 4:17
















          1












          $begingroup$

          6*6*7 = 252, which is much closer to 256. If $R in [0, 6)$, $G in [0, 7)$, and $B in [0, 6)$, the 8-bit indexed colour could be



          $C = R*7*6 + G*6 + B$



          That's pretty good.



          I also tried a spherical packing. For a face-centred packing into a cube with points at all corners, I get that a pattern $2x + 1$ layers deep has $4x^3 + 6x^2 + 3x + 1$ points in it. When $x = 3$, the pattern has 172 points. When $x = 4$, the pattern has 365 points. So neither of those is any good... On the other hand, I just found a totally awesome way of visually representing non-leap-years.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            6,7,6 yields about 1.6% waste. Awesome!
            $endgroup$
            – Reinderien
            Jan 3 at 4:17














          1












          1








          1





          $begingroup$

          6*6*7 = 252, which is much closer to 256. If $R in [0, 6)$, $G in [0, 7)$, and $B in [0, 6)$, the 8-bit indexed colour could be



          $C = R*7*6 + G*6 + B$



          That's pretty good.



          I also tried a spherical packing. For a face-centred packing into a cube with points at all corners, I get that a pattern $2x + 1$ layers deep has $4x^3 + 6x^2 + 3x + 1$ points in it. When $x = 3$, the pattern has 172 points. When $x = 4$, the pattern has 365 points. So neither of those is any good... On the other hand, I just found a totally awesome way of visually representing non-leap-years.






          share|cite|improve this answer









          $endgroup$



          6*6*7 = 252, which is much closer to 256. If $R in [0, 6)$, $G in [0, 7)$, and $B in [0, 6)$, the 8-bit indexed colour could be



          $C = R*7*6 + G*6 + B$



          That's pretty good.



          I also tried a spherical packing. For a face-centred packing into a cube with points at all corners, I get that a pattern $2x + 1$ layers deep has $4x^3 + 6x^2 + 3x + 1$ points in it. When $x = 3$, the pattern has 172 points. When $x = 4$, the pattern has 365 points. So neither of those is any good... On the other hand, I just found a totally awesome way of visually representing non-leap-years.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 3 at 3:58









          enigmaticPhysicistenigmaticPhysicist

          1985




          1985












          • $begingroup$
            6,7,6 yields about 1.6% waste. Awesome!
            $endgroup$
            – Reinderien
            Jan 3 at 4:17


















          • $begingroup$
            6,7,6 yields about 1.6% waste. Awesome!
            $endgroup$
            – Reinderien
            Jan 3 at 4:17
















          $begingroup$
          6,7,6 yields about 1.6% waste. Awesome!
          $endgroup$
          – Reinderien
          Jan 3 at 4:17




          $begingroup$
          6,7,6 yields about 1.6% waste. Awesome!
          $endgroup$
          – Reinderien
          Jan 3 at 4:17


















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