Compose Transformations

Implement a function to compose multiple 2D transformations by multiplying their matrices, which is essential in computer vision for tasks like image formation and object recognition. This process involves combining various transformations such as rotation, scaling, and translation.

The concept of transformation matrices is crucial here, where each matrix $T_i$ represents a specific transformation. The combined transformation is obtained by multiplying these matrices in a specific order, which is important due to the non-commutative nature of matrix multiplication.

Here are the steps to achieve this composition:

Start with the identity matrix as the initial combined transformation matrix.
Multiply each given transformation matrix with the current combined transformation matrix from right to left. The key to this process is understanding how matrix multiplication works and how it applies to 2D transformations.

T_{combined} = T_n \cdot T_{n-1} \cdot... \cdot T_1

This technique is widely used in image processing and robotics.

Example:

Input:

compose([[[1,0,5],[0,1,0],[0,0,1]], [[2,0,0],[0,2,0],[0,0,1]]])

Output:

[[2,0,10],[0,2,0],[0,0,1]]

Reasoning:

We have two matrices: a translation $T_1 = \begin{bmatrix}1&0&5\\0&1&0\\0&0&1\end{bmatrix}$ and a scaling $T_2 = \begin{bmatrix}2&0&0\\0&2&0\\0&0&1\end{bmatrix}$ .
The combined transform is $T_{\text{combined}} = T_2 \cdot T_1$ :
- First row: $[2\cdot1 + 0\cdot0 + 0\cdot0,\; 2\cdot0 + 0\cdot1 + 0\cdot0,\; 2\cdot5 + 0\cdot0 + 0\cdot1] = [2, 0, 10]$
- Second row: $[0\cdot1 + 2\cdot0 + 0\cdot0,\; 0\cdot0 + 2\cdot1 + 0\cdot0,\; 0\cdot5 + 2\cdot0 + 0\cdot1] = [0, 2, 0]$
- Third row: $[0, 0, 1]$ (identity row stays the same)
So the final output matrix is $\begin{bmatrix}2&0&10\\0&2&0\\0&0&1\end{bmatrix}$ , which matches [[2,0,10],[0,2,0],[0,0,1]].

Constraints:

Input is a list of 3×3 matrices
Return the composed transformation matrix

Example:

Constraints:

Test Results