Finding an Affine Transform using three 2D point correspondences using Simplex Affine Mapping (SAM) in Swift

(2021–01–01: for a more traditional method to find an affine transform from three point pairs, refer to this post: https://rethunk.medium.com/finding-an-affine-transform-the-traditional-way-with-three-2d-point-correspondences-in-swift-7c602682bfbc)

During the end-of-year holidays I was looking for a Swift implementation of a workhorse algorithm for point set registration that I could run in an iOS app. A StackOverflow post entitled “How to apply affine transformation according to 3 points with Swift” prompted me to look for a solution that didn’t require 3rd party libraries.

After a bit of googling, I stumbled onto a minimalist algorithm called Simplex Affine Mapping (SAM) that should be better known: it can be calculated by hand, it’s simple to implement from scratch, and it’s perfectly suitable for use in prototypes and lab exercises for image processing applications. The 2019 papers about SAM can be found on ResearchGate:

Beginner’s guide to mapping simplexes affinely” by Tymchyshyn and Khlevniuk

Workbook on mapping simplexes affinely” by Tymchyshyn and Khlevniuk

Point set registration is a common problem in computer vision. First you accumulate a list of (x,y) points for features identified in a reference image, and then you find the corresponding (x,y,) points for the same features in a sample image. Finally, you find the transform — translation, rotation, and (possibly) scaling — that maps one set of points to the other.

For industrial applications this technique can be used for robot guidance. For phone-based applications this technique can be used for image stitching or 3D point cloud stitching. The Wikipedia article on point set registration describes the applications and methods to register one set of 2D or 3D points to another.

Image processing libraries such as OpenCV include functions to find transforms for point set registration. Algorithms such as SIFT (Scale Invariant Feature Transform), SURF (Speeded Up Robust Features), and their descendant algorithms can automatically identify key points and then find the transform mapping those keypoints from one image to another image.

OpenCV and its point registration functions can be used in iOS applications written in Swift, as described in posts by Tim Poulsen, Yiwei Ni, and Anurag Ajwani.

But what if you want a minimalist point registration method to get started?

Arbitrary Transform using Simplex Affine Mapping (SAM) in 2D

To use SAM for 2D image processing, we need to find three point correspondences — points a, b, c in a reference image, and then matching points d, e, f in a sample image. The SAM formula then calculates the affine transform from points a, b, c to points d, e, f.

The SAM formula for 2D requires calculation of the determinant of a 4x4 matrix divided by the determinant of a 3x3 matrix. You can refer to the paper for the full form. For this article, I’m simply going to provide the worked example the authors include in the “Arbitrary Transform” section of their workbook PDF.

With apologies to the authors for copying and pasting their work directly, and to those for whom the screenshot will be inaccessible, here is an image showing the calculation of the affine transfrom from the points (1,1), (2, 3), (3,2) to the points (3,2), (1,5), (-2,1):

I’ve included this example from the authors in Swift sample code intended for use in an XCode playground.

Swift code

The Swift 5 code below defines a Triangle struct for the three points. The function sam(from:to:) returns a CGAffineTransform. With the exception of the Foundation framework, used only for NumberFormatter, the code relies entirely on CoreGraphics and some simple extensions I wrote.

To see the code run, copy and paste all of it into an XCode 12 playground. The code includes test functions that print messages to the console. There are a few colinearity tests and simple examples of translation, rotation, and scaling that can be checked by hand.

import CoreGraphics
import Foundation //for NumberFormatter, used in debug functions

// MARK: - Simple Affine Map function for 2D and supporting types

/// Calculate the affine transform from one set of three 2D points (the "from" triangle) to another set of three 2D points (the "to" triangle).
/// Throws an error if one of the sets of points is colinear.
/// The transform is calculated using the 2D expression of the Simplex Affine Map [SAM] by V. B. Tymchyshyn and A. V. Khlevniuk.
/// https://www.researchgate.net/publication/332410209_Beginner%27s_guide_to_mapping_simplexes_affinely
///
/// The workbook from the authors provides numerous examples showing step-by-step calculation for simplexes in N dimensions.
/// https://www.researchgate.net/publication/332971934_Workbook_on_mapping_simplexes_affinely\
///
/// A triangle is a simplex in 2D space, and hence works using the SAM scheme.
/// https://en.wikipedia.org/wiki/Simplex
///
/// The matrix elements include a mix of vectors and scalars. For this code, use letter names corresponding to the vectors and scalars in the SAM paper.
/// a, b, c are 2D points in the "from" triangle
/// d, e, f are 2D points in the "to" triangle
///
/// |0 d e f |
/// |p ax bx cx | // p called "x1" in the SAM paper
/// det |q ay by cy | // q called "x2" in the SAM paper
/// |1 1 1 1 |
/// (-1) ----------------------
/// |ax bx cx | // ax = point A, x component
/// det |ay by cy |
/// |1 1 1 |
///
/// For image processing applications this technique may not be appropriate if point finding is not sufficiently repeatable.
/// This mapping will be sensitive to noise / inaccuracy in finding any of the "from" or "to" points: any triangle can map to any other triangle!
/// For point registration, typically more than 3 point correspondences are used. The transform may be found
/// using a least squares fitting, and often by eliminating outlier points. Related topics include Singular Value Decomposition (SVD) and
/// Random Sampling and Consensus (RANSAC).
func sam(from: Triangle, to: Triangle) throws -> CGAffineTransform {
if from.colinear() {
throw SamError.colinearPoints(description: "Points in 'from' triangle are colinear. Can not calculate SAM transform.")
}

if to.colinear() {
throw SamError.colinearPoints(description: "Points in 'to' triangle are colinear. Can not calculate SAM transform.")
}

let a = from.vector1
let b = from.vector2
let c = from.vector3

let d = to.vector1
let e = to.vector2
let f = to.vector3

// Determinant of 4x4 in numerator of SAM formula
// Reduce to p, q, and translation. (p, q) here are (x1, x2) in the SAM paper.
// We can find the determinant starting with any row or any column, so we make it easy to isolate p ("x1") and q ("x2")

// +0 * (...)

// -p ( [d] (by - cy) - [e] (ay - cy) + [f] (ay - by) )
let p = CGFloat(-1.0) * (d * (b.dy - c.dy) - e * (a.dy - c.dy) + f * (a.dy - b.dy))

// +q ( [d] (bx - cx) - [e] (ax - cx) + [f] (ax - bx) )
let q = d * (b.dx - c.dx) - e * (a.dx - c.dx) + f * (a.dx - b.dx)

// -1 ( [d] (bx * cy - by * cx) - [e] (ax * cy - ay * cx) + [f] * (ax * by - ay * bx) )
let translation = CGFloat(-1.0) * (d * (b.dx * c.dy - b.dy * c.dx) - e * (a.dx * c.dy - a.dy * c.dx) + f * (a.dx * b.dy - a.dy * b.dx))

// Determinant of 3x3 used in denominator of SAM formula
let denominator = determinant(from)

let pd = CGFloat(-1.0) * p / denominator
let qd = CGFloat(-1.0) * q / denominator
let td = CGFloat(-1.0) * translation / denominator

// plug values into CoreGraphics affine transform
// |a b 0| = |pd.x pd.y 0|
// |c d 0| |qd.x qd.y 0|
// |tx ty 1| |td.x td.y 1|

return CGAffineTransform(a: pd.dx, b: pd.dy, c: qd.dx, d: qd.dy, tx: td.dx, ty: td.dy)
}

enum SamError: Error {
case colinearPoints(description: String)
}

/// Three points nominally defining a triangle, but possibly colinear.
struct Triangle {
var point1: CGPoint
var point2: CGPoint
var point3: CGPoint

var x1: CGFloat { point1.x }
var y1: CGFloat { point1.y }
var x2: CGFloat { point2.x }
var y2: CGFloat { point2.y }
var x3: CGFloat { point3.x }
var y3: CGFloat { point3.y }

/// Point1 as a 2D vector
var vector1: CGVector { CGVector(point1) }

/// Point2 as a 2D vector
var vector2: CGVector { CGVector(point2) }

/// Point3 as a 2D vector
var vector3: CGVector { CGVector(point3) }

/// Return a Triangle after applying an affine transform to self.
func applying(_ t: CGAffineTransform) -> Triangle {
Triangle(
point1: self.point1.applying(t),
point2: self.point2.applying(t),
point3: self.point3.applying(t)
)
}

init(point1: CGPoint, point2: CGPoint, point3: CGPoint) {
self.point1 = point1
self.point2 = point2
self.point3 = point3
}

init(x1: CGFloat, y1: CGFloat, x2: CGFloat, y2: CGFloat, x3: CGFloat, y3: CGFloat) {
point1 = CGPoint(x: x1, y: y1)
point2 = CGPoint(x: x2, y: y2)
point3 = CGPoint(x: x3, y: y3)
}

/// Returns a (Bool, CGFloat) tuple indicating whether the points in the Triangle are colinear, and the angle between vectors tested.
func colinear(degreesTolerance: CGFloat = 0.5) -> Bool {
let v1 = vector2 - vector1
let v2 = vector3 - vector2
let radians = CGVector.angleBetweenVectors(v1: v1, v2: v2)

if radians.isNaN {
return true
}

var degrees = radiansToDegrees(radians)

if degrees > 90 {
degrees = 180 - degrees
}

// code to get around parsing error
return 0 > degrees - degreesTolerance
}
}

func degreesToRadians(_ degrees: CGFloat) -> CGFloat {
degrees * CGFloat.pi / 180.0
}

func radiansToDegrees(_ radians: CGFloat) -> CGFloat {
180.0 * radians / CGFloat.pi
}

extension CGVector {
init(_ point: CGPoint) {
self.init(dx: point.x, dy: point.y)
}

static func + (lhs: CGVector, rhs: CGVector) -> CGVector {
CGVector(dx: lhs.dx + rhs.dx, dy: lhs.dy + rhs.dy)
}

static func - (lhs: CGVector, rhs: CGVector) -> CGVector {
CGVector(dx: lhs.dx - rhs.dx, dy: lhs.dy - rhs.dy)
}

static func * (_ vector: CGVector, _ scalar: CGFloat) -> CGVector {
CGVector(dx: vector.dx * scalar, dy: vector.dy * scalar)
}

static func * (_ scalar: CGFloat, _ vector: CGVector) -> CGVector {
CGVector(dx: vector.dx * scalar, dy: vector.dy * scalar)
}

static func * (lhs: CGVector, rhs: CGVector) -> CGFloat {
lhs.dx * rhs.dx + lhs.dy * rhs.dy
}

static func dotProduct(v1: CGVector, v2: CGVector) -> CGFloat {
v1.dx * v2.dx + v1.dy * v2.dy
}

static func / (_ vector: CGVector, _ scalar: CGFloat) -> CGVector {
CGVector(dx: vector.dx / scalar, dy: vector.dy / scalar)
}

static func / (_ scalar: CGFloat, _ vector: CGVector) -> CGVector {
CGVector(dx: vector.dx / scalar, dy: vector.dy / scalar)
}

// Returns again between vectors in range 0 to 2 * pi [positive]
// a * b = ||a|| ||b|| cos(theta)
// theta = arc cos (a * b / ||a|| ||b||)
static func angleBetweenVectors(v1: CGVector, v2: CGVector) -> CGFloat {
acos( v1 * v2 / (v1.length() * v2.length()) )
}

func length() -> CGFloat {
sqrt(self.dx * self.dx + self.dy * self.dy)
}
}

/// Determinant of three points used in the denominator of the SAM formula.
/// x1 x2 x3
/// y1 y2 y3
/// 1 1 1
func determinant(_ t: Triangle) -> CGFloat {
t.x1 * (t.y2 - t.y3) - t.x2 * (t.y1 - t.y3) + t.x3 * (t.y1 - t.y2)
}

//MARK: - Debug
extension NumberFormatter {
func string(_ value: CGFloat, digits: Int, failText: String = "[?]") -> String {
minimumFractionDigits = max(0, digits)
maximumFractionDigits = minimumFractionDigits

guard let s = string(from: NSNumber(value: Double(value))) else {
return failText
}

return s
}

func string(_ point: CGPoint, digits: Int = 1, failText: String = "[?]") -> String {
let sx = string(point.x, digits: digits, failText: failText)
let sy = string(point.y, digits: digits, failText: failText)
return "(\(sx), \(sy))"
}

func string(_ vector: CGVector, digits: Int = 1, failText: String = "[?]") -> String {
let sdx = string(vector.dx, digits: digits, failText: failText)
let sdy = string(vector.dy, digits: digits, failText: failText)
return "(\(sdx), \(sdy))"
}

func string(_ transform: CGAffineTransform, rotationDigits: Int = 2, translationDigits: Int = 1, failText: String = "[?]") -> String {
let sa = string(transform.a, digits: rotationDigits)
let sb = string(transform.b, digits: rotationDigits)
let sc = string(transform.c, digits: rotationDigits)
let sd = string(transform.d, digits: rotationDigits)
let stx = string(transform.tx, digits: translationDigits)
let sty = string(transform.ty, digits: translationDigits)

var s = "a: \(sa) b: \(sb) 0"
s += "\nc: \(sc) d: \(sd) 0"
s += "\ntx: \(stx) ty: \(sty) 1"
return s
}
}

let formatter = NumberFormatter()

/// Checks whether the three points in the Triangle are colinear.
/// Returns a test message spanning multiple lines.
func testColinearity(_ t: Triangle) -> String {
let co = t.colinear()
let st1 = formatter.string(t.point1, digits: 2)
let st2 = formatter.string(t.point2, digits: 2)
let st3 = formatter.string(t.point3, digits: 2)

var s = "\(st1), \(st2), \(st3): points in triangle"
s += "\n\(co): colinear"

return s
}

/// Checks how close two points are. Used to test whether a transformed point (actual) is equivalent to the expected point.
/// Returns a test message spanning multiple lines.
func testCoincidence(actual: CGPoint, expected: CGPoint, pointTolerance: CGFloat = 0.1) -> String {
let vector = CGVector(actual) - CGVector(expected)
let distance = vector.length()
let sd = formatter.string(distance, digits: 2)
let sa = formatter.string(actual, digits: 2)
let se = formatter.string(expected, digits: 2)

return "\(sd) offset from actual \(sa) to expected \(se)"
}

/// Calculates the affine transform and checks its validity.
/// Returns a test message spanning multiple lines.
func testSAM(from t1: Triangle, to t2: Triangle, pointTolerance: CGFloat = 0.1) -> String {
var s = "from \(t1) to \(t2)"

do {
let transform = try sam(from: t1, to: t2)
s += "\n" + formatter.string(transform)

// apply transform to points in t1
let a = t1.applying(transform)

// get difference between transformed points (actual) and t2 points (expected)
s += "\n" + testCoincidence(actual: a.point1, expected: t2.point1, pointTolerance: pointTolerance)
s += "\n" + testCoincidence(actual: a.point2, expected: t2.point2, pointTolerance: pointTolerance)
s += "\n" + testCoincidence(actual: a.point3, expected: t2.point3, pointTolerance: pointTolerance)

return s
}
catch SamError.colinearPoints(let description) {
return description
}
catch {
return error.localizedDescription
}
}

//MARK: - Tests for Playground
print()
print("** Colinearity tests **")
let c1 = Triangle(x1: 1, y1: 1, x2: 2, y2: 2, x3: 3, y3: 3)
let sc1 = testColinearity(c1)
print(sc1)

print()
let c2 = Triangle(x1: 0, y1: 0, x2: 5, y2: 5, x3: 0.01, y3: 0.02)
let sc2 = testColinearity(c2)
print(sc2)

print()
let c3 = Triangle(x1: 1, y1: 1, x2: 2, y2: 3, x3: 3, y3: 2)
let sc3 = testColinearity(c3)
print(sc3)

print()
print("** Arbitrary translation example from SAM workbook **")
let a1 = Triangle(x1: 1, y1: 1, x2: 2, y2: 3, x3: 3, y3: 2)
let a2 = Triangle(x1: 3, y1: 2, x2: 1, y2: 5, x3: -2, y3: 1)
print(testColinearity(a1))
print()
print(testColinearity(a2))
print()
print(testSAM(from: a1, to: a2))

print()
print("** Pure Translation **")
let t1 = Triangle(x1: 0, y1: 0, x2: -2, y2: 3, x3: -5, y3: 3)
let t2 = Triangle(x1: 3, y1: -2, x2: 1, y2: 1, x3: -2, y3: 1)
print(testSAM(from: t1, to: t2))

print()
print("** Rotation and Translation **")
let r1 = Triangle(x1: 0, y1: 0, x2: 1, y2: 0, x3: 0, y3: 1)
let r2 = Triangle(x1: 6, y1: 5, x2: 5, y2: 5, x3: 6, y3: 4)
print(testSAM(from: r1, to: r2))

print()
print("** Pure Scaling **")
let s1 = Triangle(x1: 0, y1: 0, x2: 1, y2: 0, x3: 0, y3: 1)
let s2 = Triangle(x1: 0, y1: 0, x2: 5, y2: 0, x3: 0, y3: 5)
print(testSAM(from: s1, to: s2))

print()
print("** Test in which a triangle has colinear points")
let k1 = Triangle(x1: 1, y1: 1, x2: -2, y2: -2, x3: 5, y3: 5)
let k2 = Triangle(x1: -5, y1: 3, x2: 2, y2: 8, x3: -3, y3: 1)
print()
print(testColinearity(k1))
print()
print(testColinearity(k2))
print()
print(testSAM(from: k1, to: k2))

Pasting the original code into the StackOverflow post caused a weird parsing error that corrupted the code inside the colinearity check function. I modified the code until I could paste it into StackOverflow, copy it back from StackOverflow into a fresh playground, and then have it run again.

Caveats about Use of SAM in Real-World Applications

Real-world applications would typically use least-squares fits and other statistical methods to process as many as thousands of point correspondences. Using a large number of correspondences and rejecting outliers yields an accurate transform despite the presence of image noise, occlusion (i.e. objects partly blocking the view), changes in lighting, and other conditions that can affect the accuracy of detecting the (x,y) position of particular features.

So in general I would suggest being careful using SAM for real-world applications delivered to end users. Once you integrate a 3rd party library such as OpenCV and learn the ins and outs of using more complicated point registration algorithms, you’ll have a general technique for point set registration.

That said, there are times when it’s handy and even preferred to employ a minimalist technique to yield a quick answer. If you’re building an app that will rely on point set registration or image registration, and if you need to account for translation, rotation, and scaling, then I recommend using the Simplex Affine Mapping formula during your prototyping stage.

There may be another technique that is mathematically equivalent and similarly simple, but SAM makes sense to me, and it was fun to implement.

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