cp2tform

Infer spatial transformation from control point pairs

cp2tform is not recommended. Use fitgeotrans instead.

Syntax

tform = cp2tform(movingPoints,fixedPoints,transformationType)

tform = cp2tform(movingPoints,fixedPoints,'polynomial',degree)

tform = cp2tform(movingPoints,fixedPoints,'lwm',n)

tform = cp2tform(movingPoints,fixedPoints,'piecewise linear')

[tform,usedMP,usedFP,badMP,badFP] = cp2tform(movingPoints,fixedPoints,'piecewise linear')

tform = cp2tform(cpstruct,transformationType,___)

[tform,usedMP,usedFP] = cp2tform(cpstruct,transformationType,___)

Description

tform = cp2tform(movingPoints,fixedPoints,transformationType) infers a spatial transformation from control point pairs and returns this transformation as a tform structure. Some of the transformation types have optional additional parameters, shown in the following syntaxes.

tform = cp2tform(movingPoints,fixedPoints,'polynomial',degree) lets you specify the order of the polynomials to use.

tform = cp2tform(movingPoints,fixedPoints,'lwm',n) creates a mapping by inferring a polynomial at each control point using neighboring control points. The mapping at any location depends on a weighted average of these polynomials. You can optionally specify the number of points, n, used to infer each polynomial. The n closest points are used to infer a polynomial of order 2 for each control point pair.

tform = cp2tform(movingPoints,fixedPoints,'piecewise linear') creates a Delaunay triangulation of the fixed control points, and maps corresponding moving control points to the fixed control points. The mapping is linear (affine) for each triangle and continuous across the control points but not continuously differentiable as each triangle has its own mapping.

[tform,usedMP,usedFP,badMP,badFP] = cp2tform(movingPoints,fixedPoints,'piecewise linear') returns in usedMP and usedFP the control points that were used for the piecewise linear transformation. This syntax also returns in badMP and badFP the control points that were eliminated because they were middle vertices of degenerate fold-over triangles.

tform = cp2tform(cpstruct,transformationType,___) uses a cpstruct structure to store the control point coordinates of the moving and fixed images.

[tform,usedMP,usedFP] = cp2tform(cpstruct,transformationType,___) also returns in usedMP and usedFP the control points that were used for the transformation. Unmatched and predicted points are not used. See cpstruct2pairs.

Examples

Use Control Points to Create Nonreflective Similarity Transformation Structure

Transform an image, use the cp2tform function to return the transformation, and compare the angle and scale of the tform to the angle and scale of the original transformation:

I = checkerboard;
J = imrotate(I,30);
fixedPoints = [11 11; 41 71];
movingPoints = [14 44; 70 81];
cpselect(J,I,movingPoints,fixedPoints);
 
t = cp2tform(movingPoints,fixedPoints,'nonreflective similarity');

Recover angle and scale by checking how a unit vector parallel to the x-axis is rotated and stretched.

u = [0 1]; 
v = [0 0]; 
[x, y] = tformfwd(t,u,v); 
dx = x(2) - x(1); 
dy = y(2) - y(1); 
angle = (180/pi) * atan2(dy, dx) 
scale = 1 / sqrt(dx^2 + dy^2)

Input Arguments

movingPoints — Control points in the moving image

m-by-2 matrix

Control points in the moving image, specified as an m-by-2 matrix. Each row specifies the [x y] coordinates of a control point.

Example: [11 11; 41 71]

Data Types: double

fixedPoints — Control points in the fixed image

m-by-2 matrix

Control points in the fixed image, specified as an m-by-2 matrix. Each row specifies the [x y] coordinates of a control point.

Example: [14 44; 70 81]

Data Types: double

transformationType — Type of transformation

'nonreflectivesimilarity' | 'similarity' | 'affine' | 'projective' | 'polynomial' | 'piecewise linear' | 'lwm'

Type of transformation, specified as one of the following, listed in order of increasing complexity. The cp2tform function requires a minimum number of control point pairs to infer a structure of each transform type.

Transformation Type	Description	Minimum Number of Control Point Pairs	Example
'nonreflective similarity'	Use this transformation when shapes in the moving image are unchanged, but the image is distorted by some combination of translation, rotation, and scaling. Straight lines remain straight, and parallel lines are still parallel.	2
'similarity'	Same as 'nonreflective similarity' with the addition of optional reflection.	3
'affine'	Use this transformation when shapes in the moving image exhibit shearing. Straight lines remain straight, and parallel lines remain parallel, but rectangles become parallelograms.	3
'projective'	Use this transformation when the scene appears tilted. Straight lines remain straight, but parallel lines converge toward vanishing points that might or might not fall within the image.	4
'polynomial'	Use this transformation when objects in the image are curved. The higher the order of the polynomial, the better the fit, but the result can contain more curves than the fixed image. You can specify the degree of the polynomial.	6 (order 2) 10 (order 3) 15 (order 4)
'piecewise linear'	Use this transformation when parts of the image appear distorted differently.	4
'lwm'	Use this transformation (local weighted mean), when the distortion varies locally and piecewise linear is not sufficient. You can specify the number n of points to use in the local weighed mean calculation.	6 (12 recommended)

Data Types: char

cpstruct — Preselected control points

structure

Preselected control points, specified as a structure. cpstruct contains information about the x- and y-coordinates of all control points in the moving and fixed images, including unpaired and predicted control points. cpstruct2pairs eliminates unmatched and predicted control points, and returns the set of valid control point pairs.

cpstruct is a structure produced by the Control Point Selection tool (cpselect) when you choose the Export Points to Workspace option. For more information, see Export Control Points to the Workspace.

Data Types: struct

degree — Degree of the polynomial

3 (default) | 2 | 4

Degree of the polynomial transformation, specified as the integer 2, 3, or 4.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

n — Number of points to use in local weighted mean calculation

12 (default) | positive integer

Number of points to use in local weighted mean calculation, specified as a positive integer. n can be as small as 6, but making n small risks generating ill-conditioned polynomials

Output Arguments

tform — Transformation

struct

Transformation, returned as a struct.

usedMP — Used moving points

n-by-2 matrix

Moving control points that were used to infer the spatial transformation, returned as an n-by-2 matrix. Unmatched and predicted points are not used.

usedFP — Used fixed points

n-by-2 matrix

Fixed control points that were used to infer the spatial transformation, returned as an n-by-2 matrix. Unmatched and predicted points are not used.

badMP — Eliminated moving points

p-by-2 matrix

Moving control points that were eliminated because they were determined to be outliers, returned as a p-by-2 matrix.

badFP — Eliminated fixed points

p-by-2 matrix

Fixed control points that were eliminated because they were determined to be outliers, returned as a p-by-2 matrix.

Tips

• When transformtype is 'nonreflective similarity', 'similarity', 'affine', 'projective', or 'polynomial', and movingPoints and fixedPoints (or cpstruct) have the minimum number of control points needed for a particular transformation, cp2tform finds the coefficients exactly.

• If movingPoints and fixedPoints have more than the minimum number of control points, a least-squares solution is found. See mldivide.

• When either movingPoints or fixedPoints has a large offset with respect to their origin (relative to range of values that it spans), cp2tform shifts the points to center their bounding box on the origin before fitting a tform structure. This enhances numerical stability and is handled transparently by wrapping the origin-centered tform within a custom tform that automatically applies and undoes the coordinate shift as needed. As a result, fields(T) can give different results for different coordinate inputs, even for the same transformation type.

Algorithms

cp2tform uses the following general procedure:

1. Use valid pairs of control points to infer a spatial transformation or an inverse mapping from output space (x,y) to input space (x,y) according to transformtype.

2. Return the tform structure containing spatial transformation.

The procedure varies depending on the transformtype.

Nonreflective Similarity

Nonreflective similarity transformations can include a rotation, a scaling, and a translation. Shapes and angles are preserved. Parallel lines remain parallel. Straight lines remain straight.

Let

sc = scale*cos(angle)
ss = scale*sin(angle)

[u v] = [x y 1] * [ sc -ss
                    ss  sc
                    tx  ty]

Solve for sc, ss, tx, and ty.

Similarity

Similarity transformations can include rotation, scaling, translation, and reflection. Shapes and angles are preserved. Parallel lines remain parallel. Straight lines remain straight.

Let

sc = s*cos(theta)
ss = s*sin(theta)

                   [ sc -a*-ss
 [u v] = [x y 1] *   ss  a*sc
                     tx  ty]

Solve for sc, ss, tx, ty, and a. If a = -1, reflection is included in the transformation. If a = 1, reflection is not included in the transformation.

Affine

In an affine transformation, the x and y dimensions can be scaled or sheared independently and there can be a translation. Parallel lines remain parallel. Straight lines remain straight. Nonreflective similarity transformations are a subset of affine transformations.

For an affine transformation,

[u v] = [x y 1] * Tinv

Tinv is a 3-by-2 matrix. Solve for the six elements of Tinv:

t_affine = cp2tform(movingPoints,fixedPoints,'affine');

The coefficients of the inverse mapping are stored in t_affine.tdata.Tinv.

At least three control-point pairs are needed to solve for the six unknown coefficients.

Projective

In a projective transformation, quadrilaterals map to quadrilaterals. Straight lines remain straight. Affine transformations are a subset of projective transformations.

For a projective transformation,

[up vp wp] = [x y w] * Tinv

where

u = up/wp 
v = vp/wp

Tinv is a 3-by-3 matrix.

Assuming

Tinv = [ A D G;
         B E H;
         C F I ];
u = (Ax + By + C)/(Gx + Hy + I)
v = (Dx + Ey + F)/(Gx + Hy + I)

Solve for the nine elements of Tinv:

t_proj = cp2tform(movingPoints,fixedPoints,'projective');

The coefficients of the inverse mapping are stored in t_proj.tdata.Tinv.

At least four control-point pairs are needed to solve for the nine unknown coefficients.

Note

An affine or projective transformation can also be expressed like this, for a 3-by-2 Tinv:

[u v]'  =  Tinv' * [x y 1]' 

Or, like this, for a 3-by-3 Tinv:

[u v 1]'  =  Tinv' * [x y 1]'

Polynomial

In a polynomial transformation, polynomial functions of x and y determine the mapping.

Second-Order Polynomials

For a second-order polynomial transformation, [u v] = [1 x y x*y x^2 y^2] * Tinv Both u and v are second-order polynomials of x and y. Each second-order polynomial has six terms. To specify all coefficients, Tinv has size 6-by-2. t_poly_ord2 = cp2tform(movingPoints,fixedPoints,'polynomial'); The coefficients of the inverse mapping are stored in t_poly_ord2.tdata. At least six control-point pairs are needed to solve for the 12 unknown coefficients.

Third-Order Polynomials

For a third-order polynomial transformation: [u v] = [1 x y x*y x^2 y^2 y*x^2 x*y^2 x^3 y^3] * Tinv Both u and v are third-order polynomials of x and y. Each third-order polynomial has 10 terms. To specify all coefficients, Tinv has size 10-by-2. t_poly_ord3 = cp2tform(movingPoints, fixedPoints,'polynomial',3); The coefficients of the inverse mapping are stored in t_poly_ord3.tdata. At least ten control-point pairs are needed to solve for the 20 unknown coefficients.

Fourth-Order Polynomials

For a fourth-order polynomial transformation: [u v] = [1 x y x*y x^2 y^2 y*x^2 x*y^2 x^3 y^3 x^3*y x^2*y^2 x*y^3 x^4 y^4] * Tinv Both u and v are fourth-order polynomials of x and y. Each fourth-order polynomial has 15 terms. To specify all coefficients, Tinv has size 15-by-2. t_poly_ord4 = cp2tform(movingPoints, fixedPoints,'polynomial',4); The coefficients of the inverse mapping are stored in t_poly_ord4.tdata. At least 15 control-point pairs are needed to solve for the 30 unknown coefficients.

Piecewise Linear

In a piecewise linear transformation, linear (affine) transformations are applied separately to each triangular region of the image[1].

1. Find a Delaunay triangulation of the fixed control points.

2. Using the three vertices of each triangle, infer an affine mapping from fixed to moving coordinates.

Note

At least four control-point pairs are needed. Four pairs result in two triangles with distinct mappings.

Local Weighted Mean

For each control point in fixedPoints:

1. Find the N closest control points.

2. Use these N points and their corresponding points in movingPoints to infer a second-order polynomial.

3. Calculate the radius of influence of this polynomial as the distance from the center control point to the farthest point used to infer the polynomial (using fixedPoints)[2].

Note

At least six control-point pairs are needed to solve for the second-order polynomial. Ill-conditioned polynomials might result if too few pairs are used.

References

[1] Goshtasby, Ardeshir, "Piecewise linear mapping functions for image registration," Pattern Recognition, Vol. 19, 1986, pp. 459-466.

[2] Goshtasby, Ardeshir, "Image registration by local approximation methods," Image and Vision Computing, Vol. 6, 1988, pp. 255-261.