On a very high level, my pose estimation pipeline looks somewhat like this:
Find features in image_1 and image_2 (let's say cv::ORB)
Match the features (let's say using the BruteForce-Hamming descriptor matcher)
Calculate Essential Matrix (using cv::findEssentialMat)
Decompose it to get the proper rotation matrix and translation unit vector (using cv::recoverPose)
Repeat
I noticed that at some point, the yaw angle (calculated using the output rotation matrix R of cv::recoverPose) suddenly jumps by more than 150 degrees. For that particular frame, the number of inliers is 0 (the return value of cv::recoverPose). So, to understand what exactly that means and what's going on, I asked this question on SO.
As per the answer to my question:
So, if the number of inliers is 0, then something went very wrong. Either your E is wrong, or the point matches are wrong, or both. In this case you simply cannot estimate the camera motion from those two images.
For that particular image pair, based on the visualization and from my understanding, matches look good:
The next step in the pipeline is finding the Essential Matrix. So, now, how can I check whether the calculated Essential Matrix is correct or not without decomposing it i.e. without calculating Roll Pitch Yaw angles (which can be done by finding the rotation matrix via cv::recoverPose)?
Basically, I want to double-check whether my Essential Matrix is correct or not before I move on to the next component (which is cv::recoverPose) in the pipeline!
The essential matrix maps each point p in image 1 to its epipolar line in image 2. The point q in image 2 that corresponds to p must be very close to the line. You can plot the epipolar lines and the matching points to see if they make sense. Remember that E is defined in normalized image coordinates, not in pixels. To get the epipolar lines in pixel coordinates, you would need to convert E to F (the fundamental matrix).
The epipolar lines must all intersect in one point, called the epipole. The epipole is the projection of the other camera's center in your image. You can find the epipole by computing the nullspace of F.
If you know something about the camera motion, then you can determine where the epipole should be. For example, if the camera is mounted on a vehicle that is moving directly forward, and you know the pitch and roll angles of the camera relative to the ground, then you can calculate exactly where the epipole will be. If the vehicle can turn in-plane, then you can find the horizontal line on which the epipole must lie.
You can get the epipole from F, and see if it is where it is supposed to be.
I need to calculate the minimum distance between a 3D point (latitude, longitude, elevation) and a line (defined as two points).
The elevation is not necessary on the ground, I need to consider flying objects.
I only found an article that explains how to do that on a generic space but my points are defined with lat/lon/altitude(meters).
Thank you for pointing in the right direction, in my case I need to do that in Javascript but couldn't find any library that takes into consideration the altitude.
Point-Line Distance--3-Dimensional
If you want to compare a 3d point to a 2d line, I suppose you mean a "line" on our earth, at elevation 0. Take a look at st_distance in postgis.
If I understand you correctly, that'll give you what you want.
https://postgis.net/docs/ST_Distance.html
I am trying to do normal mapping on flat surface but I can't get any noticeable result :(
My shader
http://pastebin.com/raEvBY92
For my eye, shader looks fine, but it doesn't render desired result( https://dl.dropbox.com/u/47585151/sss/final.png).
All values are passed.Normals,tengents and binormals are computed correctly when I create the grid,I have checked that!
Here are screens of ambient,diffuse,specular and bump map.
https://dl.dropbox.com/u/47585151/sss/ambient.png
https://dl.dropbox.com/u/47585151/sss/bumpMap.png
https://dl.dropbox.com/u/47585151/sss/diffuse.png
https://dl.dropbox.com/u/47585151/sss/specular.png
They seems to be legit...
The bump map,which is the result of (bump=normalize(mul(bump, input.WorldToTangentSpace)) definitely looks correct,but doesn't have any impact on end result.
Maybe I don't understand the different spaces idea or I changed the order of matrix multiplication.By world matrix I understand the position and orientation of the grid,which never changes and it is identity matrix.Only view matrix changes and represents camera position and orientation in own space.
Where is my mistake?
First of all, if you're having a problem, it's a good ideo to comment everything out, which doesn't belong to this. The whole lightcomputation with ambient, specular or even the diffusetexture isn't interesting at this moment. With
output.color=float4(diffuse ,1);
You can focus on your problem and see clearly what change, if you change something in you code.
If your quad lies in the xy-plane with z=0, you should change your lightvector, he wouldn't work. Generally I use for testing purpose a diagonal vector (like normalize(1,-1,1)) to prevent a parallel direction to my object.
When I look over your code it seems to me, that you didn't get the idea of the different spaces, as how you thought ;) The basic idea of normalmapping is to give additional information about the surface with additional normals. They are saved in a normalmap, so encoded to rgb, where b is usually the up-vector. Now you must fit them into your 3D-world, because they aren't in the world space, but in the tangent space (tangent space = surface space of the triangle). Because this transformation is more complex, the normal computation goes the other way round. You transform with the normal,binormal and tangent as a matrix your lightvector and viewvector from world space into tangent space (you are mapping the axis of world space xyz to tnb - tangent,normal,binormal, the order can be wrong I usually swap them until it works ;) ). With your line
bump = normalize(mul(bump, input.WorldToTangentSpace));
you try to transform you normal in tangent space to tangent space. Change this, so you transform the view and the lightvector in the vertexshader into tangent space and pass the transformed vectors to the pixelshader. There you can do the lightcomputation in tangent space. Maybe read an additional tutorial to normalmapping, then you will get this working! :)
PS: If youre finished with the basic lighting, your specular computation seems to have some errors, too.
float3 reflect = normalize(2*diffuse*bump-LightDirection);
This line seems to should compute the halfway-vector, but therefore you need the viewvector and shouldn't use a lightingstrength like diffuse. But a tutorial can explain this in more detail than me now.
I have a set of points to define a shape. These points are in order and essentially are my "selection".
I want to be able to contract this selection by an arbitrary amount to get a smaller version of my original shape.
In a basic example with a triangle, the points are simply moved along their normal which is defined by the points to the left and the right of the points in question.
Eventually all 3 points will meet and form one point but until that point they will make a smaller and smaller triangle.
For more complex shapes, when moving the individual points inward, they may pass through the outer edge of the shape resulting in weird artifacts. Obviously I'll need to cull these points and remove them from the array.
Any help in exactly how I can do that would be greatly appreciated.
Thanks!
This is just an idea but couldn't you find the center of mass of the object, create a vector from the center to each point, and move each point along this vector?
To find the center of mass would of course involve averaging each x and y coordinate. Getting a vector is as simple a subtracting the point in question with the center point. Normalizing and scaling are common vector operations that can be found with the Google.
EDIT
Another way to interpret what you're asking is you want to erode your collection of points. As in morphology erosion. This is typically applied to binary images but you can slightly modify the concept to work with a collection of points. Essentially, you need to write a function that, given a point, will return true (black) or false (white) depending on if that point is inside or outside the shape defined by your points. You'd have to look up how to do that for shapes that aren't always concave (it's harder but not impossible).
Now, obviously, every single one of your actual points will return false because they're all on the border (by definition). However, you now have a matrix of points around your point of interest that define where is "inside" and where is "outside". Average all of the "inside" points and move your actual point along the vector between itself and towards this average. You could play with different erosion kernels to see what works best.
You could even work with a kernel with floating point weights instead of either/or values which will affect your average calculation proportional to their weights. With this, you could approximate a circular kernel with a low number of points. Try the simpler method first.
Find the selection center (as suggested by colithium)
Map the selection points to the coordinate system with the selection center at (0,0). For example, if the selection center is at (150,150), and a given selection point is at (125,75), the mapped position of the point becomes (-25,-75).
Scale the mapped points (multiply X and Y by something in the range of 0.0..1.0)
Remap the points back to the original coordinate system
Only simple maths required, no need to muck about normalizing vectors.
I'm implementing a 2D game with ships in space.
In order to do it, I'm using LÖVE, which wraps Box2D with Lua. But I believe that my question can be answered by anyone with a greater understanding of physics than myself - so pseudo code is accepted as a response.
My problem is that I don't know how to move my spaceships properly on a 2D physics-enabled world. More concretely:
A ship of mass m is located at an initial position {x, y}. It has an initial velocity vector of {vx, vy} (can be {0,0}).
The objective is a point in {xo,yo}. The ship has to reach the objective having a velocity of {vxo, vyo} (or near it), following the shortest trajectory.
There's a function called update(dt) that is called frequently (i.e. 30 times per second). On this function, the ship can modify its position and trajectory, by applying "impulses" to itself. The magnitude of the impulses is binary: you can either apply it in a given direction, or not to apply it at all). In code, it looks like this:
function Ship:update(dt)
m = self:getMass()
x,y = self:getPosition()
vx,vy = self:getLinearVelocity()
xo,yo = self:getTargetPosition()
vxo,vyo = self:getTargetVelocity()
thrust = self:getThrust()
if(???)
angle = ???
self:applyImpulse(math.sin(angle)*thrust, math.cos(angle)*thrust))
end
end
The first ??? is there to indicate that in some occasions (I guess) it would be better to "not to impulse" and leave the ship "drift". The second ??? part consists on how to calculate the impulse angle on a given dt.
We are in space, so we can ignore things like air friction.
Although it would be very nice, I'm not looking for someone to code this for me; I put the code there so my problem is clearly understood.
What I need is an strategy - a way of attacking this. I know some basic physics, but I'm no expert. For example, does this problem have a name? That sort of thing.
Thanks a lot.
EDIT: Beta provided a valid strategy for this and Judge kindly implemented it directly in LÖVE, in the comments.
EDIT2: After more googling I also found openSteer. It's on C++, but it does what I pretended. It will probably be helpful to anyone reaching this question.
It's called motion planning, and it's not trivial.
Here's a simple way to get a non-optimal trajectory:
Stop. Apply thrust opposite to the direction of velocity until velocity is zero.
Calculate the last leg, which will be the opposite of the first, a steady thrust from a standing start that gets the ship to x0 and v0. The starting point will be at a distance of |v0|^2/(2*thrust) from x0.
Get to that starting point (and then make the last leg). Getting from one standing point to another is easy: thrust toward it until you're halfway there, then thrust backward until you stop.
If you want a quick and dirty approach to an optimal trajectory, you could use an iterative approach: Start with the non-optimal approach, above; that's just a time sequence of thrust angles. Now try doing little variations of that sequence, keeping a population of sequences that get close to the goal. reject the worst, experiment with the best -- if you're feeling bold you could make this a genetic algorithm -- and with luck it will start to round the corners.
If you want the exact answer, use the calculus of variations. I'll take a crack at that, and if I succeed I'll post the answer here.
EDIT: Here's the exact solution to a simpler problem.
Suppose instead of a thrust that we can point in any direction, we have four fixed thrusters pointing in the {+X, +Y, -X, -Y} directions. At any given time we will firing at most one of the +/-X and at most one of the +/-Y (there's no point in firing +x and -X at the same time). So now the X and Y problems are independent (they aren't in the original problem because thrust must be shared between X and Y). We must now solve the 1-D problem -- and apply it twice.
It turns out the best trajectory involves thrusting in one direction, then the other, and not going back to the first one again. (Coasting is useful only if the other axis's solution will take longer than yours so you have time to kill.) Solve the velocity problem first: suppose (WLOG) that your target velocity is greater than your initial velocity. To reach the target velocity you will need a period of thrust (+) of duration
T = (Vf - Vi)/a
(I'm using Vf: final velocity, Vi: initial velocity, a: magnitude of thrust.)
We notice that if that's all we do, the location won't come out right. The actual final location will be
X = (Vi + Vf)T/2
So we have to add a correction of
D = Xf - X = Xf -(Vi+Vf)T/2
Now to make the location come out right, we add a period of thrust in one direction before that, and an equal period in the opposite direction after. This will leave the final velocity undisturbed, but give us some displacement. If the duration of this first period (and the third) is t, then the displacement we get from it is
d = +/-(at^2 + atT)
The +/- depends on whether we thrust + then -, or - then +. Suppose it's +.
We solve the quadratic:
t = (-aT + sqrt(a^2 T^2 + 4 a D))/2a
And we're done.
In the absence of additional info, we can assume there are 3 forces acting upon the spaceship and eventually dictating its trajectory:
"impulses" : [user/program-controlled] force.
The user (or program) appear to have full control over this, i.e. it controls the direction of the impulse and its thrust (probably within a 0-to-max range)
some external force: call it gravity, whatever...
Such force could be driven by several sources but we're just interested in the resulting combined force: at a given time and space this external force acts upon the ship with a given strengh and direction. The user/program has no control over these.
inertia: this is related to the ship's current velocity and direction. This force generally causes the ship to continue in its current direction at its current speed. There may be other [space-age] parameters controlling the inertia but generally, it is proportional to both velocity and to the ship's mass (Intuitively, it will be easier to bring a ship to a stop if its current velocity is smaller and/or if its mass is smaller)
Apparently the user/program only controls (within limits) the first force.
It is unclear, from the question, whether the problem at hand is:
[Problem A] to write a program which discovers the dynamics of the system (and/or adapts to changes these dynamics).
or..
[Problem B] to suggest a model -a formula- which can be used to compute the combined force eventually applied to the ship: the "weighed" sum of the user-controlled impulse and the other two system/physics-driven forces.
The latter question, Problem B, is more readily and succinctly explained, so let's suggest the following model:
Constant Parameters:
ExternalForceX = strength of the external force in the X direction
ExternalForceY = id. Y direction
MassOfShip = coeficient controlling
Variable Parameters:
ImpulseAngle = direction of impulse
ImpulseThrust = force of thrust
Formula:
Vx[new] = (cos(ImpulseAngle) * ImpulseThrust) + ExternalForceX + (MassOfShip * Vx[current])
Vy[new] = (sin(ImpulseAngle) * ImpulseThrust) + ExternalForceY + (MassOfShip * Vy[current])
Note that the above model assumes a constant External force (constant both in terms of its strength and direction); that is: akin to that of a gravitational field relatively distant from the area displayed (just like say the Earth gravity, considered within the span of a football field). If the scale of the displayed area is big relative to the source(s) of external forces, the middle term of the formulas above should then be modified to include: a trigonometric factor based on the angle between the center of the source and the current position and/or a [reversely] proportional factor based on the distance between the center of the source and the current position.
Similarly, the Ship's mass is assumed to remain constant, it could well be a variable, based say on the mass of the Ship when empty, to which the weight of fuel is removed/added as the game progresses.
Now... All the above assume that the dynamics of the system are controlled by the game designer: essentially choosing a set of values for the parameter mentioned and possibly adding a bit of complexity in the math of the formula (and also ensuring proper scaling to generally "keep" the ship within the display area).
What if instead, the system dynamics were readily programmed into the game (and assumed to be hidden/random), and the task at hand is to write a program which will progressively decide the direction and thrust value of the impulses to drive the ship to its targeted destination, in a way that its velocity at the target be as close as possible to getTargetVelocity()? This is the "Problem A".
This type of problem can be tackled with a PID Controller. In a nuthell, such a controller "decides" which amount of action (in this game's case = which impulse angle and amount of thrust to apply), based on three, weighed, factors, loosely defined below:
how far-off we are the current values from "set point": this is the P=Proportional part of PID
how fast are we approaching the "set point": this is the D=Derivative part of PID
how long and how much have we been away from the "set point": this is the I=Intergral part of PID
A less sophisticated controller could for example only use the proportional factor. This would result in oscillating, sometimes with much amplitude on either side of the set point ("I'm X units away from where I'm supposed to be: let me yank the steering wheel and press on gas"). Such overshooting of the set point are tempered by the Derivative factor ("Yeah, I'm still not where I'm supposed to be but the progress I made since the last time I check is very big: better slow down a bit"). Finally the Integral part takes into account the fact that all things being equal with regards to the combined Proportional and Derivative part, a smaller or bigger action would be appropriate depending on whether we've been "off-track" for a long time or not and of much off track we've been all this time (eg. "Lately we've been tracking rather close to where we're supposed to be, no point in making rash moves")
We can discuss the details implementing PID controllers for the specific needs of the space ship game, if that is effectively what is required. The idea was to provide a flavor of what can be done.
To just get from the current position to the destination with an initial velocity, then apply thrust along the normalized difference between the shortest path and the current velocity. You don't actually need the angle.
-- shortest path minus initial velocity
dx,dy = x0 - x - vx, y0 - y - vy
-- normalize the direction vector
magnitude = sqrt(dx*dx + dy*dy)
dx,dy = dx/magnitude, dy/mangitude
-- apply the thrust in the direction we just calculated
self:applyImpulse(thrust*dx, thrust*dy)
Note that this does not take the target velocity into account because that gets extremely complicated.
I have a very small Lua module for handling 2D vectors in this paste bin. You are welcome to use it. The code above would reduce to:
d = destination - position - velocity
d:normalize()
d = d * thrust
self:applyImpulse(d.x, d.y)
Are you expelling fuel? Your mass will change with time if you are.
Thrust is a reactive force. It's the rate of change of mass, times the speed of the exhaust relative to the spaceship.
Do you have external forces? If you do, these need to enter into your impulse calculation.
Let's assume a magical thrust with no mass being expelled, and no external forces.
Impulse has units of momentum. It's the integral of a force over time.
First off, you'll need to figure out exactly what the API calls "thrust" and impulse. If you're feeding it a thrust multiplied by a scalar (number), then applyImpulse has to do something else to your input to be able to use it as an impulse, because the units don't match up.
Assuming your "thrust" is a force, then you multiply that thrust by the time interval (1/30 second) to get the impulse, and break out the components.
Don't know if I'm answering your question, but hopefully that helps you to understand the physics a bit.
It's easier to think about if you separate the ship's velocity into components, parallel and perpendicular to the target velocity vector.
Considering along the perpendicular axis, the ship wants to come in line with the target position as soon as possible, and then stay there.
Along the parallel axis, it should be accelerating in whatever direction will bring it close to the target velocity. (Obviously if that acceleration takes it away from the target point, you'll need to decide what to do. Fly past the point and then double-back?)
I would deal with the two of them separately, and probably perpendicular first. Once it's working, and if that doesn't prove nice enough, you can start to think about if there are ways to get the ship to fire intelligent angles between perpendicular and parallel.
(EDIT: also, I forgot to mention, this will take some adjustment to deal with the scenario where you are offset a lot in the perpendicular direction but not much in the parallel direction. The important lesson here is to take the components, which gives you useful numbers on which to base a decision.)
Your angle is the Inverse Tangent the Opposite/Adjacent
So angle = InvTan(VY/VX)
No sure what your are talking about concerning wanting to drift??