regress.md

Polynomial Regression for Ground Control Points (GCP)

Overview

This script implements polynomial regression to transform Ground Control Points (GCPs) between image and real-world coordinates. It performs bidirectional mapping:

• Forward Transformation: Maps real-world coordinates ((X, Y)) to image coordinates ((x, y)).
• Backward Transformation: Maps image coordinates ((x, y)) to real-world coordinates ((X, Y)).

The model utilizes polynomial regression of degree (d) and normalizes data before fitting.

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Mathematical Formulation

1. Normalization

To improve numerical stability, data is normalized using mean and standard deviation:

$$\hat{x} = \frac{x - \mu_x}{\sigma_x}, \quad \hat{y} = \frac{y - \mu_y}{\sigma_y}$$ $$\hat{X} = \frac{X - \mu_X}{\sigma_X}, \quad \hat{Y} = \frac{Y - \mu_Y}{\sigma_Y}$$

where:

• ( \mu ) is the mean.
• ( \sigma ) is the standard deviation.

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2. Polynomial Basis Expansion

The design matrix (A) consists of polynomial basis functions up to a given degree (d):

$$A[i, j] = X^p Y^q, \quad \text{where} \quad p+q \leq d$$

The number of terms in the polynomial:

$$N = \frac{(d+1)(d+2)}{2}$$

For degree (d=2), the design matrix takes the form:

$$A = \begin{bmatrix} 1 & X & Y & X^2 & XY & Y^2 \end{bmatrix}$$

3. Polynomial Regression

The transformation is modeled as:

$$x = A \cdot c_x, \quad y = A \cdot c_y$$ $$X = A' \cdot c_X, \quad Y = A' \cdot c_Y$$

where:

• ( A ) and ( A' ) are the design matrices.
• ( c_x, c_y, c_X, c_Y ) are the regression coefficients computed using least squares estimation:

$$c = (A^T A)^{-1} A^T b$$

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4. Evaluation

Given a new input ((X, Y)) or ((x, y)), the estimated output is:

$$\hat{x} = A \cdot c_x, \quad \hat{y} = A \cdot c_y$$ $$\hat{X} = A' \cdot c_X, \quad \hat{Y} = A' \cdot c_Y$$

Denormalization is applied to restore original values:

$$x = \hat{x} \cdot \sigma_x + \mu_x, \quad y = \hat{y} \cdot \sigma_y + \mu_y$$ $$X = \hat{X} \cdot \sigma_X + \mu_X, \quad Y = \hat{Y} \cdot \sigma_Y + \mu_Y$$

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5. Root Mean Square Error (RMSE)

The error is measured using Root Mean Square Error (RMSE):

$$\text{RMSE} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2}$$