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Gradient Descent Update

MediumOptimization 📚 CV: Image Alignment and Stitching

Implement a gradient descent update step for camera parameters in the context of Bundle Adjustment, a crucial process in Image Alignment and Stitching. You are given the current camera parameters and their gradient, and need to perform one update step to minimize a cost function.

The goal of gradient descent is to iteratively update parameters to minimize the cost function, which measures the difference between observed and predicted values. The learning rate controls the step size of each update. The gradient of the cost function points in the direction of the steepest ascent, so subtracting it from the current parameters moves them downhill toward the minimum.

Here are the steps to update the parameters:

Compute the gradient of the cost function with respect to the current parameters.
Multiply the gradient by the learning rate.
Subtract the result from the current parameters to obtain the updated parameters.

\theta_{new} = \theta_{old} - \eta \cdot \nabla C

This technique is widely used in computer vision applications, such as image stitching and 3D reconstruction.

Example:

Input:

translation = [1, 1, 1]
gradient = [0.1, 0.2, 0.3]
learning_rate = 0.1

Output:

[0.99, 0.98, 0.97]

Reasoning:

Applying gradient descent update:

θ_new = θ_old - η × gradient

For each component:

tx_new = 1 - 0.1 × 0.1 = 1 - 0.01 = 0.99
ty_new = 1 - 0.1 × 0.2 = 1 - 0.02 = 0.98
tz_new = 1 - 0.1 × 0.3 = 1 - 0.03 = 0.97

Result: [0.99, 0.98, 0.97]

The translation moves in the opposite direction of the gradient.

Constraints:

translation: current [tx, ty, tz]
gradient: cost gradient [∂C/∂tx, ∂C/∂ty, ∂C/∂tz]
learning_rate: step size η
Return updated translation
Round to 4 decimal places

More from CV: Image Alignment and Stitching

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Gradient descent update in this problem is just applying the formula $\theta_{$ \text{new} $} =$ \theta $_{$ \text{old} $} - \eta$ \nabla $C$ to the camera translation parameters using the provided gradient and learning rate.

1. Background Knowledge

In bundle adjustment, we jointly refine camera parameters (poses, intrinsics) and 3D points to minimize a reprojection error cost: how far projected 3D points are from observed 2D image points. This is typically solved by iterative optimization (Gauss–Newton, Levenberg–Marquardt, or gradient descent). Each iteration updates parameters in the direction that reduces the cost.

Gradient descent is a first-order optimization method: at each step, we compute the gradient of the cost with respect to parameters and move against this gradient. For a parameter vector $\theta \in \mathbb{R}^d$ , and cost $C($ \theta $)$ , the update is:

\theta_{\text{new}} = \theta_{\text{old}} - \eta \nabla C(\theta_{\text{old}})

Here, $\theta$ corresponds to the 3D camera translation (e.g., $[t_x, t_y, t_z]$ ), $\nabla C$ is a vector of partial derivatives of the cost with respect to each translation component, and $\eta$ is a small scalar controlling step size.

2. Algorithm / General Approach

The pattern for this type of problem is:

Treat the camera translation as a vector parameter.
Treat the provided gradient as the same-shape vector of derivatives of cost with respect to translation.
Apply the gradient descent formula elementwise:

new_translation = old_translation − learning_rate * grad_translation.

Return or store the updated translation.

This is a simple vector arithmetic operation, no loops over pixels or points are required if translation and gradient are already aggregated.

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