What is the range of parameters rho and theta in normal equation of a straight line?











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I'm implementing Hough Transform in MatLab to detect straight lines in an Image. It uses normal form of a straight line equation
$$x cos(theta) + y sin(theta) = rho$$



I'm interested in the lines which pass through the 1st quadrant only, that too in the rectangle with vertices $(0,0), (M,0), (0,N), (M,N)$ i.e. any line that doesn't pass through this rectangle, I'm not interested in finding the equation of that line. Orientation of the line can be anything. What is the range of values for the parameters $rho$ and $theta$?



If I assume $theta$ varies from $-90^text{o}$ to $90^text{o}$, what is the range of values for $rho$? Can $rho$ be negative?










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    up vote
    2
    down vote

    favorite












    I'm implementing Hough Transform in MatLab to detect straight lines in an Image. It uses normal form of a straight line equation
    $$x cos(theta) + y sin(theta) = rho$$



    I'm interested in the lines which pass through the 1st quadrant only, that too in the rectangle with vertices $(0,0), (M,0), (0,N), (M,N)$ i.e. any line that doesn't pass through this rectangle, I'm not interested in finding the equation of that line. Orientation of the line can be anything. What is the range of values for the parameters $rho$ and $theta$?



    If I assume $theta$ varies from $-90^text{o}$ to $90^text{o}$, what is the range of values for $rho$? Can $rho$ be negative?










    share|cite|improve this question


























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I'm implementing Hough Transform in MatLab to detect straight lines in an Image. It uses normal form of a straight line equation
      $$x cos(theta) + y sin(theta) = rho$$



      I'm interested in the lines which pass through the 1st quadrant only, that too in the rectangle with vertices $(0,0), (M,0), (0,N), (M,N)$ i.e. any line that doesn't pass through this rectangle, I'm not interested in finding the equation of that line. Orientation of the line can be anything. What is the range of values for the parameters $rho$ and $theta$?



      If I assume $theta$ varies from $-90^text{o}$ to $90^text{o}$, what is the range of values for $rho$? Can $rho$ be negative?










      share|cite|improve this question















      I'm implementing Hough Transform in MatLab to detect straight lines in an Image. It uses normal form of a straight line equation
      $$x cos(theta) + y sin(theta) = rho$$



      I'm interested in the lines which pass through the 1st quadrant only, that too in the rectangle with vertices $(0,0), (M,0), (0,N), (M,N)$ i.e. any line that doesn't pass through this rectangle, I'm not interested in finding the equation of that line. Orientation of the line can be anything. What is the range of values for the parameters $rho$ and $theta$?



      If I assume $theta$ varies from $-90^text{o}$ to $90^text{o}$, what is the range of values for $rho$? Can $rho$ be negative?







      geometry






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      edited Nov 18 at 18:21









      learner

      226




      226










      asked Nov 18 at 18:17









      Nagabhushan S N

      22814




      22814






















          1 Answer
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          retta_r_t_1



          As depicted in the graph we have better split for negative anf for positive values of $theta$.

          Thus we must have
          $$
          left{ {matrix{
          {left{ matrix{
          - rho /cos theta le M hfill cr
          - rho /sin theta le N hfill cr} right.} hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
          {left{ matrix{
          rho /cos theta - Ntan theta le M hfill cr
          0 le rho hfill cr} right.} hfill & {left| {;0 < theta < pi /2} right.} hfill cr
          } } right.
          $$

          that is
          $$
          left{ {matrix{
          { - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
          {0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
          } } right.
          $$






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            1 Answer
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            1 Answer
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            active

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            active

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            active

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            up vote
            1
            down vote



            accepted










            retta_r_t_1



            As depicted in the graph we have better split for negative anf for positive values of $theta$.

            Thus we must have
            $$
            left{ {matrix{
            {left{ matrix{
            - rho /cos theta le M hfill cr
            - rho /sin theta le N hfill cr} right.} hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
            {left{ matrix{
            rho /cos theta - Ntan theta le M hfill cr
            0 le rho hfill cr} right.} hfill & {left| {;0 < theta < pi /2} right.} hfill cr
            } } right.
            $$

            that is
            $$
            left{ {matrix{
            { - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
            {0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
            } } right.
            $$






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              retta_r_t_1



              As depicted in the graph we have better split for negative anf for positive values of $theta$.

              Thus we must have
              $$
              left{ {matrix{
              {left{ matrix{
              - rho /cos theta le M hfill cr
              - rho /sin theta le N hfill cr} right.} hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
              {left{ matrix{
              rho /cos theta - Ntan theta le M hfill cr
              0 le rho hfill cr} right.} hfill & {left| {;0 < theta < pi /2} right.} hfill cr
              } } right.
              $$

              that is
              $$
              left{ {matrix{
              { - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
              {0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
              } } right.
              $$






              share|cite|improve this answer























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                retta_r_t_1



                As depicted in the graph we have better split for negative anf for positive values of $theta$.

                Thus we must have
                $$
                left{ {matrix{
                {left{ matrix{
                - rho /cos theta le M hfill cr
                - rho /sin theta le N hfill cr} right.} hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
                {left{ matrix{
                rho /cos theta - Ntan theta le M hfill cr
                0 le rho hfill cr} right.} hfill & {left| {;0 < theta < pi /2} right.} hfill cr
                } } right.
                $$

                that is
                $$
                left{ {matrix{
                { - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
                {0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
                } } right.
                $$






                share|cite|improve this answer












                retta_r_t_1



                As depicted in the graph we have better split for negative anf for positive values of $theta$.

                Thus we must have
                $$
                left{ {matrix{
                {left{ matrix{
                - rho /cos theta le M hfill cr
                - rho /sin theta le N hfill cr} right.} hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
                {left{ matrix{
                rho /cos theta - Ntan theta le M hfill cr
                0 le rho hfill cr} right.} hfill & {left| {;0 < theta < pi /2} right.} hfill cr
                } } right.
                $$

                that is
                $$
                left{ {matrix{
                { - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
                {0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
                } } right.
                $$







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                answered Nov 19 at 1:21









                G Cab

                16.9k31237




                16.9k31237






























                     

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