MATLAB Implementation of Affine Transformation Based on Least Squares Method

In the fields of image processing and computer vision, affine transformation is a fundamental geometric transformation method used for operations such as translation, rotation, scaling, and shearing on images. The least squares method is a classic mathematical optimization technique employed to fit data points and find optimal solutions. By combining these two approaches, we can achieve efficient affine transformation calculations in MATLAB. ### Fundamental Concepts of Affine Transformation Affine transformation combines linear transformation and translation, typically represented by a 2x3 matrix. This matrix describes how to map coordinates from the input image to the output image. Affine transformations preserve the parallelism of lines but do not necessarily maintain lengths and angles. ### Application of Least Squares Method The least squares method fits data points by minimizing the sum of squared errors. In affine transformation, we utilize least squares to solve for the transformation matrix that minimizes the error between transformed points and target points. Specifically, given a set of source points and their corresponding target points, we can construct a system of linear equations and use least squares to solve for optimal affine transformation parameters. The mathematical formulation involves solving Ax = b where A contains source point coordinates and b contains target point coordinates. ### MATLAB Implementation Approach In MATLAB, the implementation of least squares-based affine transformation follows these key steps: Data Preparation: Collect key points from the source image and their corresponding points in the target image. This typically involves using feature detection algorithms like SIFT or SURF. Equation Construction: Based on the mathematical model of affine transformation, convert the problem into a system of linear equations. The transformation matrix can be represented as [a11 a12 a13; a21 a22 a23] where the last column represents translation parameters. Parameter Solving: Utilize MATLAB's matrix computation capabilities (such as the backslash operator `` or the `pinv` function for pseudo-inverse) to compute the least squares solution. The core implementation uses: T = [x_source, y_source, ones(n,1)] \ [x_target, y_target]; where T is the 3x2 transformation matrix. Transformation Application: Apply the solved affine transformation matrix to the original image using MATLAB's `imwarp` function or manual coordinate transformation to complete the geometric transformation. ### Advantages and Application Scenarios The least squares method offers advantages in mathematical rigor and computational efficiency, particularly suitable for situations with noise or partially imprecise matching points. This approach finds wide applications in image registration, feature matching in computer vision, and medical image analysis. Through MATLAB's powerful matrix computation capabilities, we can easily implement this algorithm and validate its effectiveness in practical projects. The implementation leverages MATLAB's built-in optimization functions and image processing toolbox for robust performance.