Affine Transformation: Unlocking Visual Perspectives: Exploring Affine Transformation in Computer Vision

By Fouad Sabry

What is Affine Transformation

In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Affine Transformation

Chapter 2: Linear Map

Chapter 3: Translation (Geometry)

Chapter 4: Affine Group

Chapter 5: Affine Space

Chapter 6: Transformation Matrix

Chapter 7: Barycentric Coordinate System

Chapter 8: Real Coordinate Space

Chapter 9: Eigenvalues and Eigenvectors

Chapter 10: Eigendecomposition of a Matrix

(II) Answering the public top questions about affine transformation.

(III) Real world examples for the usage of affine transformation in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Affine Transformation.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Apr 28, 2024

Book preview

Chapter 1: Affine transformation

An affine transformation (from the Latin affinis, connected with) is a geometric transformation in Euclidean geometry that maintains straight lines and parallelism but changes the lengths and directions of the angles and distances involved.

A more general definition of an affine transformation is an automorphism of an affine space (Euclidean spaces are special cases of affine spaces), that is, a function which maps an affine space onto itself while maintaining the ratio of the lengths of parallel line segments. Therefore, after an affine transformation, sets of parallel affine subspaces retain their parallelism. Distances and angles between lines are not always preserved by an affine transformation, but distance ratios along a straight line are preserved.

Assuming X is the point set of some affine space, we can write every affine transformation on X as the combination of a linear transformation on X and a translation of X. The starting point of the affine space is not required to be kept the same during an affine transformation, unlike a linear one. Accordingly, every affine transformation is linear, but not every linear transformation is affine.

Affine transformations include translation, enlargement, reduction, homology, similarity, reflection, rotation, shear mapping, and any combination or sequence of these.

Affine transformations are those projective transformations of a projective space that preserve the invariance of the hyperplane at infinity, defining the affine space as the complement of the hyperplane at infinity.

An affine map is a more general form of an affine transformation.

Imagine a field k and an affine space X, Let V denote the vector space to which it belongs.

A bijection f from X onto itself is called an affine transformation; this means that a linear map g from V to V is well defined by the equation {\displaystyle g(y-x)=f(y)-f(x);} here, as usual, The free vector from point 2 to point 1 is denoted by the difference of these two points, and well-defined means that {\displaystyle y-x=y'-x'} implies that

{\displaystyle f(y)-f(x)=f(y')-f(x').}

If X has at least two dimensions, then there exists a bijection from X onto itself, denoted by f, such that:

If S is an affine subspace of X in dimension d, then f (S) is also an affine subspace of X in dimension d.

It follows that f (S) and f (T) are parallel if and only if S and T are parallel affine subspaces of X.

Affine transformations satisfy these two conditions, which precisely express what is meant by the phrase f preserves parallelism..

The second condition follows logically from the first, so they cannot be considered separate.

An affine space, by definition,, V acts on X, so that, for each set of two (x, v) in X × V there is associated a point y in X.

We can denote this action by v→(x) = y.

Here we use the convention that v→ = v are two interchangeable notations for an element of V.

By fixing a point c in X one can define a function mc : X → V by mc(x) = cx→.

Assuming a c, It's a one-to-one mapping with this function, and so, has an inverse function mc−1 : V → X given by mc−1(v) = v→(c).

By defining these operations, we can transform X into a vector space (with respect to c)::

{\displaystyle x+y=m_{c}^{-1}\left(m_{c}(x)+m_{c}(y)\right),{\text{ for all }}x,y{\text{ in }}X,}

and

{\displaystyle rx=m_{c}^{-1}\left(rm_{c}(x)\right),{\text{ for all }}r{\text{ in }}k{\text{ and }}x{\text{ in }}X.}

Despite the fact that this vector space with origin c should be formally distinguished from the affine space X, in practice it is usually denoted by the same symbol and only after an origin has been specified is it mentioned that it is a vector space. This recognition enables the transformation from vector to point representation and back again.

For any linear transformation λ of V, L(c) is a function that can be defined, λ) : X → X by

{\displaystyle L(c,\lambda )(x)=m_{c}^{-1}\left(\lambda (m_{c}(x))\right)=c+\lambda ({\vec {cx}}).}

If so, L(c), λ) is an affine transformation of X which leaves the point c fixed.

It's a linear map from X to another variable, represented by a vector space with c as its center.

Let σ be any affine transformation of X.

Pick a point c in X and consider the translation of X by the vector {\displaystyle {\mathbf {w}}={\overrightarrow {c\sigma (c)}}} , denoted by Tw.

Affine transformations include translations, and affine transformations include their composition.

In light of this specific c, there exists a unique linear transformation λ of V such that

{\displaystyle \sigma (x)=T_{\mathbf {w}}\left(L(c,\lambda )(x)\right).}

In other words, if we consider X to be a vector space, then any arbitrary affine transformation of X can be written as the composition of a linear transformation of X and a translation of X.

Affine transformations are typically defined in terms of this representation (with the choice of origin being implicit).

Given the foregoing, Affine maps are constructed by combining a translation function with a linear map.

Multiplication of matrices is used to represent linear maps in standard vector algebra, to represent translations via vector addition.

Formally, in the limit of finite dimensions, if the linear map is represented as a multiplication by an invertible matrix A and the translation as the addition of a vector \mathbf {b} , an affine map f acting on a vector \mathbf {x} can be represented as

{\displaystyle \mathbf {y} =f(\mathbf {x} )=A\mathbf {x} +\mathbf {b} .}

With the help of an enhanced matrix and enhanced vector, Multiple matrix multiplications are not required to represent the translation and linear map.

The method necessitates adding a final 1 to all vectors, and that the bottom row of all matrices be filled in with zeros, added right-most column (translation vector), plus a single number in the bottom right corner.

If A is a matrix,

{\displaystyle {\begin{bmatrix}\mathbf {y} \\1\end{bmatrix}}=\left[{\begin{array}{ccc|c}&A&&\mathbf {b} \\0&\cdots &0&1\end{array}}\right]{\begin{bmatrix}\mathbf {x} \\1\end{bmatrix}}}

means the same thing as

{\displaystyle \mathbf {y} =A\mathbf {x} +\mathbf {b} .}

Affine transformation matrix is another name for the augmented matrix shown above.

In most situations, when the last row vector is not restricted to be {\displaystyle \left[{\begin{array}{ccc|c}0&\cdots &0&1\end{array}}\right]} , The matrix is converted into a matrix of projective transformations (as it can also be used to perform projective transformations).

This representation exhibits the set of all invertible affine transformations as the semidirect product of K^{n} and {\displaystyle \operatorname {GL} (n,K)} .

The law of the composition of functions defines this group, referred to as the affine group.

When multiplying matrices and vectors, the origin is always transferred to the origin, and thus never stand in for a translation, where the starting point must be moved to another location.

Adding the extra digit 1 to the end of each vector, This additional dimension can be thought of as a subset of the space being mapped.

At that point, When the extra coordinate is 1, the original space is contained in that smaller region.

Thus the origin of the original space can be found at {\displaystyle (0,0,\dotsc ,0,1)} .

By applying a linear transformation to the higher-dimensional space, we can perform a translation within the original space (more precisely,, a deformation in shear).

Homogeneous coordinates include, for instance, the ones used to describe the higher-dimensional space.

Assuming a Euclidean starting point, True projective space exists in the higher dimensions.

By multiplying the corresponding matrices, any number of affine transformations can be combined into a single one when working with homogeneous coordinates. Numerous applications in robotics, computer vision, and computer graphics rely on this property.

If the vectors {\displaystyle \mathbf {x} _{1},\dotsc ,\mathbf {x} _{n+1}} are a basis of the domain's projective vector space and if {\displaystyle \mathbf {y} _{1},\dotsc ,\mathbf {y} _{n+1}} are the corresponding vectors in the codomain vector space then the augmented matrix M that achieves this affine transformation

{\displaystyle {\begin{bmatrix}\mathbf {y} \\1\end{bmatrix}}=M{\begin{bmatrix}\mathbf {x} \\1\end{bmatrix}}}

is

{\displaystyle M={\begin{bmatrix}\mathbf {y} _{1}&\cdots &\mathbf {y} _{n+1}\\1&\cdots &1\end{bmatrix}}{\begin{bmatrix}\mathbf {x} _{1}&\cdots &\mathbf {x} _{n+1}\\1&\cdots &1\end{bmatrix}}^{-1}.}

Whether or not the domain, codomain, and image vector spaces all have the same number of dimensions, this formulation still applies.

For example, the affine transformation of a vector plane is uniquely determined from the knowledge of where the three vertices ( {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\mathbf {x} _{3}} ) of a non-degenerate triangle are mapped to ( {\displaystyle \mathbf {y} _{1},\mathbf {y} _{2},\mathbf {y} _{3}} ), regardless of whether or not the triangle is non-degenerate in the codomain and the number of dimensions of the codomain.

Preserves affine structure during:

When three or more points lie along the same line, they are said to be collinear, and this property survives transformation.

When two or more lines undergo a transformation, their parallelism is preserved.

A set that is convex before a transformation remains convex after the transformation. In addition, the transformed set's extreme points correspond to the original set's extreme points.

ratios of lengths of parallel line segments: for distinct parallel segments defined by points p_{1} and p_{2} , p_{3} and p_4 , the ratio of {\overrightarrow {p_{1}p_{2}}} and {\displaystyle {\overrightarrow {p_{3}p_{4}}}} is the same as that of {\overrightarrow {f(p_{1})f(p_{2})}} and {\displaystyle {\overrightarrow {f(p_{3})f(p_{4})}}} .

barycenters for sets of points with different weights.

Considering that affine transformations can be inverted, the square matrix A appearing in its matrix representation is invertible.

Therefore, the inverse transformation's matrix representation is

{\displaystyle \left[{\begin{array}{ccc|c}&A^{-1}&&-A^{-1}{\vec {b}}\ \\0&\ldots &0&1\end{array}}\right].}

The affine group is the set of invertible affine transformations (from one affine space to another), which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n+1 .

The similarity transformations form the subgroup where A is a scalar times an orthogonal matrix.

For example, if the affine transformation acts on the plane and if the determinant of A is 1 or −1 then the transformation is an equiareal mapping.

The group formed by such transformations is known as the equi-affine group.

An isometry of the plane in terms of Euclidean distance is a transformation that is both equi-affine and a similarity.

Each of these groups has a subgroup of orientation-preserving or positive affine transformations: those where the determinant of A is positive.

Last but not least, this is the class of rigid transformations in three dimensions (proper rotations and pure translations).

In the presence of a fixed point, the affine transformation simplifies to a linear one. This could improve our ability to categorize and comprehend the change. For instance, it may be easier to visualize the overall behavior of a transformation if it is described as a rotation by a certain angle with respect to a certain axis, rather than as a combination of a translation and a rotation. However, this is situationally and contextually dependent.

An affine map {\displaystyle f\colon {\mathcal {A}}\to {\mathcal {B}}} between two affine spaces is a map on the points that acts linearly on the vectors (that is, the space's connecting