i9482: Improve documentation concerning norm functionality.

Added forkfour Latex command to math js support.

Split cv::norm documentation between the cv::norm and its overload, to make things clearer

Corrected some typos and cleaned up grammar.

Result is clearer documentation for the norms.

Work pending...

doc/mymath.js

@@ -6,6 +6,7 @@ MathJax.Hub.Config(
           matTT: [ "\\[ \\left|\\begin{array}{ccc} #1 & #2 & #3\\\\ #4 & #5 & #6\\\\ #7 & #8 & #9 \\end{array}\\right| \\]", 9],
           fork: ["\\left\\{ \\begin{array}{l l} #1 & \\mbox{#2}\\\\ #3 & \\mbox{#4}\\\\ \\end{array} \\right.", 4],
           forkthree: ["\\left\\{ \\begin{array}{l l} #1 & \\mbox{#2}\\\\ #3 & \\mbox{#4}\\\\ #5 & \\mbox{#6}\\\\ \\end{array} \\right.", 6],
+          forkfour: ["\\left\\{ \\begin{array}{l l} #1 & \\mbox{#2}\\\\ #3 & \\mbox{#4}\\\\ #5 & \\mbox{#6}\\\\ #7 & \\mbox{#8}\\\\ \\end{array} \\right.", 8],
           vecthree: ["\\begin{bmatrix} #1\\\\ #2\\\\ #3 \\end{bmatrix}", 3],
           vecthreethree: ["\\begin{bmatrix} #1 & #2 & #3\\\\ #4 & #5 & #6\\\\ #7 & #8 & #9 \\end{bmatrix}", 9],
           hdotsfor: ["\\dots", 1],

doc/mymath.sty

@@ -28,7 +28,14 @@
   #3 & \mbox{#4}\\
   #5 & \mbox{#6}\\
   \end{array} \right.}
-
+\newcommand{\forkthree}[8]{
+  \left\{
+  \begin{array}{l l}
+  #1 & \mbox{#2}\\
+  #3 & \mbox{#4}\\
+  #5 & \mbox{#6}\\
+  #7 & \mbox{#8}\\
+  \end{array} \right.}
 \newcommand{\vecthree}[3]{
 \begin{bmatrix}
  #1\\

modules/core/include/opencv2/core.hpp

@@ -640,34 +640,44 @@ Scalar_ 's.
 CV_EXPORTS_W void meanStdDev(InputArray src, OutputArray mean, OutputArray stddev,
                              InputArray mask=noArray());
 
-/** @brief Calculates an absolute array norm, an absolute difference norm, or a
-relative difference norm.
+/** @brief Calculates an absolute array norm.
 
-The function cv::norm calculates an absolute norm of src1 (when there is no
-src2 ):
+This version of cv::norm calculates the absolute norm of src1. The type of norm to calculate is specified using cv::NormTypes.
 
-\f[norm =  \forkthree{\|\texttt{src1}\|_{L_{\infty}} =  \max _I | \texttt{src1} (I)|}{if  \(\texttt{normType} = \texttt{NORM_INF}\) }
+\f[norm =  \forkfour{\|\texttt{src1}\|_{L_{\infty}} =  \max _I | \texttt{src1} (I)|}{if  \(\texttt{normType} = \texttt{NORM_INF}\) }
 { \| \texttt{src1} \| _{L_1} =  \sum _I | \texttt{src1} (I)|}{if  \(\texttt{normType} = \texttt{NORM_L1}\) }
-{ \| \texttt{src1} \| _{L_2} =  \sqrt{\sum_I \texttt{src1}(I)^2} }{if  \(\texttt{normType} = \texttt{NORM_L2}\) }\f]
+{ \| \texttt{src1} \| _{L_2} =  \sqrt{\sum_I \texttt{src1}(I)^2} }{if  \(\texttt{normType} = \texttt{NORM_L2}\) }
+{ \| \texttt{src1} \| _{L_2} ^{2} = \sum_I \texttt{src1}(I)^2} {if  \(\texttt{normType} = \texttt{NORM_L2SQR}\)}\f]
+
+If normType is not specified, NORM_L2 is used.
 
 or an absolute or relative difference norm if src2 is there:
 
-\f[norm =  \forkthree{\|\texttt{src1}-\texttt{src2}\|_{L_{\infty}} =  \max _I | \texttt{src1} (I) -  \texttt{src2} (I)|}{if  \(\texttt{normType} = \texttt{NORM_INF}\) }
-{ \| \texttt{src1} - \texttt{src2} \| _{L_1} =  \sum _I | \texttt{src1} (I) -  \texttt{src2} (I)|}{if  \(\texttt{normType} = \texttt{NORM_L1}\) }
-{ \| \texttt{src1} - \texttt{src2} \| _{L_2} =  \sqrt{\sum_I (\texttt{src1}(I) - \texttt{src2}(I))^2} }{if  \(\texttt{normType} = \texttt{NORM_L2}\) }\f]
 
-or
-
-\f[norm =  \forkthree{\frac{\|\texttt{src1}-\texttt{src2}\|_{L_{\infty}}    }{\|\texttt{src2}\|_{L_{\infty}} }}{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_INF}\) }
-{ \frac{\|\texttt{src1}-\texttt{src2}\|_{L_1} }{\|\texttt{src2}\|_{L_1}} }{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_L1}\) }
-{ \frac{\|\texttt{src1}-\texttt{src2}\|_{L_2} }{\|\texttt{src2}\|_{L_2}} }{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_L2}\) }\f]
+As example for one array consider the function \f$r(x)= \begin{pmatrix} x \\ 1-x \end{pmatrix}, x \in [-1;1]\f$.
+The \f$ L_{1}, L_{2} \f$ and \f$ L_{\infty} \f$ norm for the sample value \f$r(-1) = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\f$
+is calculated as follows
+\f{align*}
+    \| r(-1) \|_{L_1} &= |-1| + |2| = 3 \\
+    \| r(-1) \|_{L_2} &= \sqrt{(-1)^{2} + (2)^{2}} = \sqrt{5} \\
+    \| r(-1) \|_{L_\infty} &= \max(|-1|,|2|) = 2
+\f}
+and for \f$r(0.5) = \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}\f$ the calculation is
+\f{align*}
+    \| r(0.5) \|_{L_1} &= |0.5| + |0.5| = 1 \\
+    \| r(0.5) \|_{L_2} &= \sqrt{(0.5)^{2} + (0.5)^{2}} = \sqrt{0.5} \\
+    \| r(0.5) \|_{L_\infty} &= \max(|0.5|,|0.5|) = 0.5.
+\f}
+The following graphic shows all values for the three norm functions \f$\| r(x) \|_{L_1}, \| r(x) \|_{L_2}\f$ and \f$\| r(x) \|_{L_\infty}\f$.
+It is notable that the \f$ L_{1} \f$ norm forms the upper and the \f$ L_{\infty} \f$ norm forms the lower border for the example function \f$ r(x) \f$.
+![Graphs for the different norm functions from the above example](pics/NormTypes_OneArray_1-2-INF.png)
 
 The function cv::norm returns the calculated norm.
 
 When the mask parameter is specified and it is not empty, the norm is
 calculated only over the region specified by the mask.
 
-A multi-channel input arrays are treated as a single-channel, that is,
+Multi-channel input arrays are treated as single-channel arrays, that is,
 the results for all channels are combined.
 
 @param src1 first input array.
@@ -676,7 +686,24 @@ the results for all channels are combined.
 */
 CV_EXPORTS_W double norm(InputArray src1, int normType = NORM_L2, InputArray mask = noArray());
 
-/** @overload
+/** @brief Calculates an absolute difference norm or a relative difference norm.
+
+This version of cv::norm calculates the absolute difference norm 
+or the relative difference norm of arrays src1 and src2.
+The type of norm to calculate is specified using cv::NormTypes.
+
+\f[norm =  \forkfour{\|\texttt{src1}-\texttt{src2}\|_{L_{\infty}} =  \max _I | \texttt{src1} (I) -  \texttt{src2} (I)|}{if  \(\texttt{normType} = \texttt{NORM_INF}\) }
+{ \| \texttt{src1} - \texttt{src2} \| _{L_1} =  \sum _I | \texttt{src1} (I) -  \texttt{src2} (I)|}{if  \(\texttt{normType} = \texttt{NORM_L1}\) }
+{ \| \texttt{src1} - \texttt{src2} \| _{L_2} =  \sqrt{\sum_I (\texttt{src1}(I) - \texttt{src2}(I))^2} }{if  \(\texttt{normType} = \texttt{NORM_L2}\) }
+{ \| \texttt{src1} - \texttt{src2} \| _{L_2} ^{2} =  \sum_I (\texttt{src1}(I) - \texttt{src2}(I))^2 }{if  \(\texttt{normType} = \texttt{NORM_L2SQR}\) }
+\f]
+
+or
+
+\f[norm =  \forkthree{\frac{\|\texttt{src1}-\texttt{src2}\|_{L_{\infty}}    }{\|\texttt{src2}\|_{L_{\infty}} }}{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_INF}\) }
+{ \frac{\|\texttt{src1}-\texttt{src2}\|_{L_1} }{\|\texttt{src2}\|_{L_1}} }{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_L1}\) }
+{ \frac{\|\texttt{src1}-\texttt{src2}\|_{L_2} }{\|\texttt{src2}\|_{L_2}} }{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_L2}\) }\f]
+
 @param src1 first input array.
 @param src2 second input array of the same size and the same type as src1.
 @param normType type of the norm (cv::NormTypes).

modules/core/include/opencv2/core/base.hpp

@@ -167,23 +167,6 @@ enum DecompTypes {
 { \frac{\|\texttt{src1}-\texttt{src2}\|_{L_1} }{\|\texttt{src2}\|_{L_1}} }{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_L1}\) }
 { \frac{\|\texttt{src1}-\texttt{src2}\|_{L_2} }{\|\texttt{src2}\|_{L_2}} }{if  \(\texttt{normType} = \texttt{NORM_RELATIVE | NORM_L2}\) }\f]
 
-As example for one array consider the function \f$r(x)= \begin{pmatrix} x \\ 1-x \end{pmatrix}, x \in [-1;1]\f$.
-The \f$ L_{1}, L_{2} \f$ and \f$ L_{\infty} \f$ norm for the sample value \f$r(-1) = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\f$
-is calculated as follows
-\f{align*}
-    \| r(-1) \|_{L_1} &= |-1| + |2| = 3 \\
-    \| r(-1) \|_{L_2} &= \sqrt{(-1)^{2} + (2)^{2}} = \sqrt{5} \\
-    \| r(-1) \|_{L_\infty} &= \max(|-1|,|2|) = 2
-\f}
-and for \f$r(0.5) = \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}\f$ the calculation is
-\f{align*}
-    \| r(0.5) \|_{L_1} &= |0.5| + |0.5| = 1 \\
-    \| r(0.5) \|_{L_2} &= \sqrt{(0.5)^{2} + (0.5)^{2}} = \sqrt{0.5} \\
-    \| r(0.5) \|_{L_\infty} &= \max(|0.5|,|0.5|) = 0.5.
-\f}
-The following graphic shows all values for the three norm functions \f$\| r(x) \|_{L_1}, \| r(x) \|_{L_2}\f$ and \f$\| r(x) \|_{L_\infty}\f$.
-It is notable that the \f$ L_{1} \f$ norm forms the upper and the \f$ L_{\infty} \f$ norm forms the lower border for the example function \f$ r(x) \f$.
-![Graphs for the different norm functions from the above example](pics/NormTypes_OneArray_1-2-INF.png)
  */
 enum NormTypes { NORM_INF       = 1,
                  NORM_L1        = 2,