Laplacian of Gaussian Kernel Calculator

Calculator Input

Kernel Size

Use an odd size like 5, 7, 9, or 11.

Sigma

Sigma controls smoothing spread and edge scale.

Pixel Spacing

Spacing changes sample distance between kernel coordinates.

Amplitude

Amplitude scales all kernel values uniformly.

Center X Offset

Shift the kernel center horizontally.

Center Y Offset

Shift the kernel center vertically.

Display Precision

Choose decimal places for all displayed values.

Normalization

Pick a scaling rule for the final matrix.

Example Data Table

Item	Example Value	Meaning
Kernel Size	7	Creates a 7 × 7 sampled LoG matrix.
Sigma	1.20	Larger sigma increases smoothing and widens the kernel.
Pixel Spacing	1.00	Samples coordinates one unit apart.
Normalization	Exact zero-sum	Forces the discrete matrix sum to exactly zero.
Expected Pattern	Negative center, positive ring	Typical LoG shape highlights edge transitions clearly.

Formula Used

Continuous LoG Formula

LoG(x, y, σ) = A × [((x² + y²) - 2σ²) / σ⁴] × exp(-(x² + y²) / (2σ²))

Discrete Sample Coordinates

x = (column - center - offset_x) × spacing
y = (row - center - offset_y) × spacing

Zero-Crossing Radius

r = √2 × σ

The calculator samples the continuous LoG function on a discrete grid. Optional normalization can enforce zero-sum balance, unit absolute sum, unit energy, or unit peak magnitude.

How to Use This Calculator

Enter an odd kernel size. Common sizes are 5, 7, and 9.
Set sigma to control the width of the smoothing effect.
Choose pixel spacing if your sample coordinates need scaling.
Use amplitude to enlarge or shrink every matrix value.
Apply center offsets when you want a shifted kernel center.
Select a normalization mode for your filtering workflow.
Click Generate Kernel to show the result above the form.
Review the heatmap, matrix table, and summary metrics.
Use CSV or PDF buttons to export the output.

Frequently Asked Questions

1. What does a Laplacian of Gaussian kernel do?

It combines Gaussian smoothing with a second derivative operator. This helps detect edges and blob-like structures while reducing some noise before the derivative stage.

2. Why must the kernel size be odd?

An odd size gives the matrix a single center cell. That center is important because the LoG kernel is usually defined around a central origin point.

3. How does sigma affect the kernel?

Larger sigma values widen the kernel and smooth more strongly. Smaller sigma values create tighter kernels that respond to finer image details.

4. What is the benefit of zero-sum normalization?

A perfect LoG kernel should sum to zero. Discrete sampling can introduce slight imbalance, so zero-sum normalization corrects that numerical drift.

5. When should I use unit energy normalization?

Use unit energy when you want consistent filter strength comparisons across different kernel sizes or sigma values. It normalizes the matrix using its L2 norm.

6. What does the zero-crossing radius mean?

It is the radius where the continuous LoG response changes sign. In two dimensions, that radius equals √2 times sigma.

7. Why does the center cell often look negative?

For a centered LoG kernel with positive amplitude, the origin usually produces the strongest negative value. Positive values appear farther out in a surrounding ring.

8. Can I use this matrix in image filtering code?

Yes. The matrix can be copied directly into convolution workflows. The export options also make it easy to store, review, and reuse generated kernels.