transformer3d

transformer3d.js

A node.js library for transforming between a 3d coordinate systems. Check it out here!

Installation

npm install transformer3d

Basic Usage

The core object for this library is the CoordinateNetwork, which allows to user to build up a graph, where the nodes are various coordinate systems, and the edges are transformations between the two node pairs. Nodes are identified with some unique name, and edges are instances of the transform.Transform object (or it's many subclasses).

Vector Transformations

Euclidean Transformations

Basic example

Take for example, a network with two coordinate systems "A" and "B", where the origin of B is at the vector [ 1, 0, 0 ] in A's coordinate frame. Then the network can be built up as:

const { CoordinateNetwork, transform } = require('transformer3d');

// Initialize the network
const net = new CoordinateNetwork();

// Attach the A and B coordinate system
net.connect_systems(
  "A", 
  new transform.ShiftStaticTransform([ 1, 0, 0]), // B is at [1,0,0] in A's frame
  "B"
);

// Tell the network that you are done adding coordinate systems
net.compile();

// Transform [1,0,0] in A's frame to B's frame
net.transform_vec([1,0,0], "A", "B"); // returns [2,0,0]

// Tranform [0, 2, 0] in B's frame to A's frame
net.tranform_vec([0,2,0], "B", "A"); // returns [-1,2,0]

If instead of being shifted relative to one another, B was rotated 90 degrees around the x-axis relative to A:

// Initialize the network
const net = new CoordinateNetwork();

// Attach the A and B coordinate system
net.connect_systems(
  "A", 
  //                                  +90 deg     around x
  new transform.RotateStaticTransform(Math.PI/2, [ 1, 0, 0]),
  "B"
);

// Tell the network that you are done adding coordinate systems
net.compile();

// Transform [0,1,0] in A's frame to B's frame
net.transform_vec([0,1,0], "A", "B"); // returns [0,0,1]

// Tranform [0, 2, 0] in B's frame to A's frame
net.tranform_vec([0,2,0], "B", "A"); // returns [0,0,-2]

As a rule, methods which mutate a transformer3d object will return a reference to the object itself, so many set up methods can be changed together:

// Same as the previous example
const net = (new CoordinateNetwork())
  .connect_systems("A", new transform.RotateStaticTransform(Math.PI/2, [ 1, 0, 0]), "B")
  .compile();

net.transform_vec([0,1,0], "A", "B"); // returns [0,0,1]

Many connections

Coordinate systems need to only specify the minimal number of connections, and the network can compile the rest of the connections. So, if you have coordinate systems A, B, C and D, you don't need to explicitly specify the connection between A and D, if you have specified the connection between A to B, B to C and C to D.

const net = (new CoordinateNetwork())
  .connect_systems("A", new transform.ShiftStaticTransform([1,0,0]), "B")
  .connect_systems("B", new transform.ShiftStaticTransform([1,0,0]), "C")
  .connect_systems("C", new transform.ShiftStaticTransform([1,0,0]), "D")
  .compile();

net.transform_vec([0,0,0], "A", "D"); // returns [3,0,0]

That being said, if you happen to know the explicit connection between A and D, and tell the network, it will be able to evaluate that transformation much more quickly (becuase it will not need to route through B and C).

const net = (new CoordinateNetwork())
  .connect_systems("A", new transform.ShiftStaticTransform([1,0,0]), "B")
  .connect_systems("B", new transform.ShiftStaticTransform([1,0,0]), "C")
  .connect_systems("C", new transform.ShiftStaticTransform([1,0,0]), "D")
  .connect_systems("A", new transform.ShiftStaticTransform([3,0,0]), "D") // provide shortcut
  .compile();

net.transform_vec([0,0,0], "A", "D"); // returns [3,0,0] much faster

The more connections you provide, the faster the network can evaluate transformations

Dynamic Connections

Sometimes a coordinate transformation is dependent upon some external data which is either not available at the time of instantiation, or will periodically change. This can be handled by using 'Dynamic' transformation objects, together with the CoordinateNetwork.update method

const net = (new CoordinateNetwork())
  .connect_systems(
    "A", 
    //                            object key for data
    new transform.ShiftDynamicTransform("A_to_B_shift"), 
    "B"
  ).compile();

// some time later
net.update({ A_to_B_shift : [1,0,0] })
net.transform_vec([0,0,0], "A", "B"); // returns [1,0,0]

// some time even later
net.update({ A_to_B_shift : [3,0,0] })
net.transform_vec([0,0,0], "A", "B"); // returns [3,0,0]

Check out the full API for details on using dynamic connections

Connection to Affine Transformations

A common alternative to the quaternion approach used in this library is to implement coordinate transformations using matrix multiplication. This is often construed as a subset of affine transformations, where the matrix part is required to be determinant 1. The basic form of an affine transformation is:

v'^T = A * v^T + b^T

where v and v' are row vectors for before and after the affine transformation, A is the matrix part (the rotation) and b is the vector part (the shift). These rotate and shift operations are accomplished in this library using the quaternion.UnitQuaternion and shift.Shift classes, but it is possible to convert between the two representations. For example:

const { Euclidean } = require('transformer3d').euclidean;
const { UnitQuaternion } = require('transformer3d').quaternion;
const { Shift } = require('transformer3d').shift;
euc = new Euclidean();
q = UnitQuaternion.from_axis( Math.PI/4, [0,0,1] ); // rotate 45 deg about z axis
s = Shift.from_vec([0,0,1]);                        // shift 1 unit in z
euc.set_objects( q, s );

euc.affine; // returns { A : [[1/sqrt(2),1/sqrt(2),0],[-1/sqrt(2),1/sqrt(2),0],[0,0,1]], B : [0,0,1] }

Note that network.CoordinateNetwork instances are also aware of affine transformations too, so that you can get the total equivalent affine transformation for any two coordinate systems which are connected by Euclidean-type transformations:

const { transform, CoordinateNetwork } = require('transformer3d');

let net = new CoordinateNetwork();
net.connect_systems("A", transform.ShiftStaticTransform([0,1,0]), "B") // B is 1 unit shifted from A in y dir
net.connect_systems("B", transform.RotateStaticTransform(Math.PI/2, [1,0,0]), "C") // C is rotated 90 about x from B
net.compile();

net.get_affine("A","B"); // returns { A : [[1,0,0],[0,1,0],[0,0,1]], b : [0,1,0] }
net.get_affine("A","C"); // returns { A : [[0,1,0],[-1,0,0],[0,0,1]], b : [1,0,0] }

Notice in that last case that the affine shift b : [1,0,0] is in the coordinate system of 'C', since affine transformations always apply their shifts at the end, even though the network would apply the shift from "A" to "B" before rotating from "B" to "C".

As a final note, the net.get_affine method will only work if all the internal transformations between the two coordinate systems are Euclidean-type. If any non-Euclidean transformations are present in the chain between the two endpoints, the method will throw an error. This is because generic transformation objects are not guaranteed to have an effect affine transformation equivalent (e.g. the transform.PinholeCameraTransform is distinctly not affine in nature, since disparity and distance are inversely related to one another).

Pinhole Camera Transformations

This library also has limited support for transforming between pixel-space (i, j and disparity) and world-space (X, Y, Z) for a pair of stereocalibrated cameras, following the conventions of OpenCV. You can see here for a discussion of stereorectifying a pair of cameras, and see here and here for a futher discussion of the details of disparity imaging.

const net = (new CoordinateNetwork())
  .connect_systems("world", new transform.PinholeCameraTransform("Q" /* data key */), "image")
  .compile()
  .update({ Q : [[1,0,0,-50],[0,1,0,-50],[0,0,1,100],[0,0,0.1,0]] }); // set Q matrix

net.transform_vec([0, 0, 10], "world", "image"); // projects vector into an image

Homogeneous Transformations

Just like with affine transformations, this library also exposes transformations for homogeneous coordinates . Briefly, homogeneous coordinates are a coordinate representation in n+1 dimensions, in our case this means coordinates are expressed as [ x, y, z, w ], where the 3d vector can be recovered as [ x/w, y/w, z/w ]. A homogeneous (also called projective) transformation is then any 4x4 matrix which operates on the homogeneous coordinates. All affine transformations can be expressed as a homogeneous transformation, but the pinhole camera transformation (discussed above) can also be expressed as a homogeneous transformation.

CoordinateNetwork instances expose a method .get_homogeneous( start, end ) which will return the homogeneous transformation which is equivalent to the transformation between start and end. Note that just like for affine transformations, this method will only work if all the subtransformations between start and end are homogeneous-compatible (this includes transform.PinholeCameraTransform and all transform.EuclideanTransformMixin derivatives). The method will return the 4x4 matrix for which the following are equivalent:

const net = (new CoordinateNetwork())
  .connect_systems("camera", new transform.PinholeCameraTransform("Q" /* data key */), "image")
  .connect_systems("world", new transform.ShiftStaticTransform([0,0,1]), "camera") // the world is 1m below the camera
  .compile()
  .update({ Q : [[1,0,0,-50],[0,1,0,-50],[0,0,1,100],[0,0,0.1,0]] }); // set Q matrix

// Use the network to transform a vector
let [i,j,d] = net.transform_vec([0, 0, 10], "world", "image");

// Get the homogeneous matrix and transform the vector yourself
let mat = net.get_homogeneous( "world", "image" ); // 4x4 matrix
let world_vec = [0,0,10,1]; // same world vector in homogeneous coordinates
// compute matrix product: img_vec^T = mat * world_vec^T
let img_vec = [0,0,0,0]; 
for ( let _i = 0; _i < 4; _i ++ ) {
  for ( let _j = 0; _j < 4; _j ++) {
      img_vec[_i] += mat[_i][_j] * world_vec[_j]
    }
}
let [ ip, jp, dp, wp ] = img_vec;

i === ip / wp; // returns True
j === jp / wp; // returns True
d === dp / wp; // returns True

Quaternion Transformations

This library, in addition to transforming 3d vectors, allows supports transforming rigid body orientations in the form of unit quaternions (UnitQuaternion class). In this construction, the unit quaternion is imagined to represent the rotation required to transform a reference object orientation into a new orientation. For example, if an objects orientation is "rotated 90 degrees around the x-axis from the starting point", this is represented as q = 0.707 + 0.707i + 0j + 0k, and can be generated using:

const { UnitQuaternion } = require('transformer3d');

let q = UnitQuaternion.from_axis(
  Math.PI/2, // Angle
  [1,0,0]    // Axis
)

Suppose that there is another coordinate reference frame B which is reached from A by rotating around A's z-axis by +90 degrees. One can transform the quaternion q to express it's orientation relative to the B axis via the following:

const net = (new CoordinateNetwork())
  .connect_systems("A", new transform.RotateStaticTransform( Math.PI/2, [0,0,1] ), "B")
  .compile();

net.orient_quat(q); // returns q', which is the same orientation as q, but referenced against B

Advanced Usage

See the full API for more details on advanced usage.

Development notes

The author welcomes any feedback, pull requests, feature request, and forks!

Known Bugs

• If the real part of a UnitQuaternion is negative, that might produce some strange behavoir with some of the composite getter functions, like .angle

TODO

• Finish commenting the pinhole camera module
• Add methods to quaternion module for construction based on vectors: i.e. 'quat which rotates vec1 onto vec2 about a perpendicular', or 'quat which maps a pair of perpendicular unit vectors onto a different pair of perpendicular unit vectors'
• Consider removing .transform_quat from the transform module, since is is pretty niche
• Should in place transformations return the mutated object, or undefined? c.f. Euclidean.transform_vec_ip
• Should the quaternion module just be a separate npm pacakge?