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?
geometry
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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?
geometry
add a comment |
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?
geometry
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
geometry
edited Nov 18 at 18:21
learner
226
226
asked Nov 18 at 18:17
Nagabhushan S N
22814
22814
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1 Answer
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1
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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
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
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
$$
left{ {matrix{
{ - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
{0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
} } right.
$$
add a comment |
up vote
1
down vote
accepted
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
$$
left{ {matrix{
{ - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
{0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
} } right.
$$
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
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
$$
left{ {matrix{
{ - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
{0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
} } right.
$$
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
$$
left{ {matrix{
{ - Mcos theta le rho le - Nsin theta } hfill & {left| {; - pi /2 < theta < 0} right.} hfill cr
{0 le rho le Mcos theta + Nsin theta } hfill & {left| {;0 < theta < pi /2} right.} hfill cr
} } right.
$$
answered Nov 19 at 1:21
G Cab
16.9k31237
16.9k31237
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