Basic space carving algorithm

I have the following problem as shown in the figure. I have point cloud and a mesh generated by a tetrahedral algorithm. How would I carve the mesh using the that algorithm ? Are landmarks are the point cloud ?

Pseudo code of the algorithm:

for every 3D feature point
 convert it 2D projected coordinates

for every 2D feature point
cast a ray toward the polygons of the mesh 
get intersection point
if zintersection < z of 3D feature point
for ( every triangle vertices )
cull that triangle.

Here is a follow up implementation of the algorithm mentioned by the Guru Spektre :)

Update code for the algorithm:

 int i;
        for (i = 0; i < out.numberofpoints; i++)
        {
            Ogre::Vector3 ray_pos = pos; // camera position);
            Ogre::Vector3 ray_dir = (Ogre::Vector3 (out.pointlist[(i*3)], out.pointlist[(3*i)+1], out.pointlist[(3*i)+2]) - pos).normalisedCopy();  // vertex - camea pos ;

            Ogre::Ray ray;
            ray.setOrigin(Ogre::Vector3( ray_pos.x, ray_pos.y, ray_pos.z));
            ray.setDirection(Ogre::Vector3(ray_dir.x, ray_dir.y, ray_dir.z));
            Ogre::Vector3 result;
            unsigned int u1;
            unsigned int u2;
            unsigned int u3;
            bool rayCastResult = RaycastFromPoint(ray.getOrigin(), ray.getDirection(), result, u1, u2, u3);

            if ( rayCastResult )
            {
                Ogre::Vector3 targetVertex(out.pointlist[(i*3)], out.pointlist[(3*i)+1], out.pointlist[(3*i)+2]);
                float distanceTargetFocus = targetVertex.squaredDistance(pos);
                float distanceIntersectionFocus = result.squaredDistance(pos);
                if(abs(distanceTargetFocus) >= abs(distanceIntersectionFocus))
                {
                    if ( u1 != -1 && u2 != -1 && u3 != -1)
                    {
                        std::cout << "Remove index "<< "u1 ==> " <<u1 << "u2 ==>"<<u2<<"u3 ==> "<<u3<< std::endl;
                        updatedIndices.erase(updatedIndices.begin()+ u1);
                        updatedIndices.erase(updatedIndices.begin()+ u2);
                        updatedIndices.erase(updatedIndices.begin()+ u3);

                    }
                }
            }

            }


            if ( updatedIndices.size() <= out.numberoftrifaces)
            {
                std::cout << "current face list===> "<< out.numberoftrifaces << std::endl;

                std::cout << "deleted face list===> "<< updatedIndices.size() << std::endl;
                manual->begin("Pointcloud", Ogre::RenderOperation::OT_TRIANGLE_LIST);

                for (int n = 0; n < out.numberofpoints; n++)
                {
                    Ogre::Vector3 vertexTransformed = Ogre::Vector3( out.pointlist[3*n+0], out.pointlist[3*n+1], out.pointlist[3*n+2]) - mReferencePoint;
                    vertexTransformed *=1000.0 ;
                    vertexTransformed = mDeltaYaw * vertexTransformed;

                    manual->position(vertexTransformed);

                }

                for (int n = 0 ; n < updatedIndices.size(); n++)
                {
                     int n0 = updatedIndices[n+0];
                     int n1 = updatedIndices[n+1];
                     int n2 = updatedIndices[n+2];

                    if ( n0 < 0 || n1 <0 || n2 <0 )
                    {
                        std::cout<<"negative indices"<<std::endl;
                        break;
                    }
                    manual->triangle(n0, n1, n2);

                }

                manual->end();

Follow up with the algorithm:

I have now two versions one is the triangulated one and the other is the carved version.

It's not not a surface mesh. Here are the two files http://www.mediafire.com/file/cczw49ja257mnzr/ahmed_non_triangulated.obj http://www.mediafire.com/file/cczw49ja257mnzr/ahmed_triangulated.obj

I see it like this:

So you got image from camera with known matrix and FOV and focal length.

From that you know where exactly the focal point is and where the image is proected onto the camera chip (Z_near plane). So any vertex, its corresponding pixel and focal point lies on the same line.

So for each view cas ray from focal point to each visible vertex of the pointcloud. and test if any face of the mesh hits before hitting face containing target vertex. If yes remove it as it would block the visibility.

Landmark in this context is just feature point corresponding to vertex from pointcloud. It can be anything detectable (change of intensity, color, pattern whatever) usually SIFT/SURF is used for this. You should have them located already as that is the input for pointcloud generation. If not you can peek pixel corresponding to each vertex and test for background color.

Not sure how you want to do this without the input images. For that you need to decide which vertex is visible from which side/view. May be it is doable form nearby vertexes somehow (like using vertex density points or corespondence to planar face...) or the algo is changed somehow for finding unused vertexes inside mesh.

To cast a ray do this:

ray_pos=tm_eye*vec4(imgx/aspect,imgy,0.0,1.0);
ray_dir=ray_pos-tm_eye*vec4(0.0,0.0,-focal_length,1.0);

where tm_eye is camera direct transform matrix, imgx,imgy is the 2D pixel position in image normalized to <-1,+1> where (0,0) is the middle of image. The focal_length determines the FOV of camera and aspect ratio is ratio of image resolution image_ys/image_xs

Ray triangle intersection equation can be found here:

Reflection and refraction impossible without recursive ray tracing?

If I extract it:

vec3 v0,v1,v2; // input triangle vertexes
vec3 e1,e2,n,p,q,r;
float t,u,v,det,idet;
//compute ray triangle intersection
e1=v1-v0;
e2=v2-v0;
// Calculate planes normal vector
p=cross(ray[i0].dir,e2);
det=dot(e1,p);
// Ray is parallel to plane
if (abs(det)<1e-8) no intersection;
idet=1.0/det;
r=ray[i0].pos-v0;
u=dot(r,p)*idet;
if ((u<0.0)||(u>1.0)) no intersection;
q=cross(r,e1);
v=dot(ray[i0].dir,q)*idet;
if ((v<0.0)||(u+v>1.0)) no intersection;
t=dot(e2,q)*idet;
if ((t>_zero)&&((t<=tt))  // tt is distance to target vertex
    {
    // intersection
    }

Follow ups:

To move between normalized image (imgx,imgy) and raw image (rawx,rawy) coordinates for image of size (imgxs,imgys) where (0,0) is top left corner and (imgxs-1,imgys-1) is bottom right corner you need:

imgx = (2.0*rawx / (imgxs-1)) - 1.0
imgy = 1.0 - (2.0*rawy / (imgys-1))

rawx = (imgx + 1.0)*(imgxs-1)/2.0
rawy = (1.0 - imgy)*(imgys-1)/2.0

[progress update 1]

I finally got to the point I can compile sample test input data for this to get even started (as you are unable to share valid data at all):

I created small app with hard-coded table mesh (gray) and pointcloud (aqua) and simple camera control. Where I can save any number of views (screenshot + camera direct matrix). When loaded back it aligns with the mesh itself (yellow ray goes through aqua dot in image and goes through the table mesh too). The blue lines are casted from camera focal point to its corners. This will emulate the input you got. The second part of the app will use only these images and matrices with the point cloud (no mesh surface anymore) tetragonize it (already finished) now just cast ray through each landmark in each view (aqua dot) and remove all tetragonals before target vertex in pointcloud is hit (this stuff is not even started yet may be in weekend)... And lastly store only surface triangles (easy just use all triangles which are used just once also already finished except the save part but to write wavefront obj from it is easy ...).

[Progress update 2]

I added landmark detection and matching with the point cloud

as you can see only valid rays are cast (those that are visible on image) so some points on point cloud does not cast rays (singular aqua dots)). So now just the ray/triangle intersection and tetrahedron removal from list is what is missing...