MATLAB - Laplacian of Gaussian Filter

MATLAB - Laplacian of Gaussian (LoG) for Edge Detection

The Laplacian of Gaussian (LoG) is a powerful filter used in image processing for detecting edges. It combines two steps into one: smoothing the image with a Gaussian filter to reduce noise, and then applying a Laplacian filter to find regions of rapid intensity change (i.e., edges).

The LoG is known as a second-derivative operator. Edges in an image correspond to zero-crossings in the filtered output (where pixel values change from positive to negative). By adjusting the sigma (standard deviation) of the Gaussian component, you can detect edges at different scales—a small sigma finds fine details, while a large sigma finds broader, more prominent edges.

Modern Approach: Creating the LoG Filter Manually

While MATLAB has a legacy fspecial('log') function, the modern and more instructive approach is to create the filter kernel yourself. This gives you full control and a better understanding of the process.

Example 1: Manual LoG Filter Creation and Application

%% 1. Load and Prepare the Image
originalImage = imread('peppers.png');
grayImage = rgb2gray(originalImage);

% Convert to double for accurate calculations
doubleImage = double(grayImage);

%% 2. Define LoG Filter Parameters
% Sigma controls the scale of edges to be detected
sigma = 2.0;

% Determine filter size based on sigma (a common rule of thumb)
filterSize = 2 * ceil(3 * sigma) + 1;

%% 3. Create the LoG Filter Kernel
% Create a grid of (x,y) coordinates
[X, Y] = meshgrid(-floor(filterSize/2):floor(filterSize/2));

% The Laplacian of Gaussian formula
logKernel = -1 / (pi * sigma^4) * (1 - (X.^2 + Y.^2) / (2 * sigma^2)) .* exp(-(X.^2 + Y.^2) / (2 * sigma^2));

% Best practice: Normalize the kernel to sum to zero for a Laplacian operator
logKernel = logKernel - mean(logKernel(:));

%% 4. Apply the Filter
% 'replicate' handles border pixels by copying edge pixel values
filteredImage = imfilter(doubleImage, logKernel, 'replicate');

%% 5. Display the Results
figure;
subplot(1, 2, 1);
imshow(grayImage);
title('Original Grayscale Image');

% The LoG output highlights edges. We use an empty colormap range `[]`
% to autoscale the display to show both positive and negative values.
subplot(1, 2, 2);
imshow(filteredImage, []);
title(sprintf('LoG Filtered (Sigma = %.1f)', sigma));

Explanation

• Filter Creation: We first create a coordinate grid (meshgrid). Then, we apply the mathematical formula for the 2D LoG function to this grid. This creates a kernel (a small matrix) that represents the filter.
• Normalization: A key step for second-derivative filters is ensuring the kernel’s elements sum to zero. This guarantees that the filter has no response in flat, uniform areas of the image, only at changes in intensity.
• Filtering: imfilter convolves the image with our custom logKernel. We use the 'replicate' option to avoid dark borders caused by zero-padding.
• Displaying: The output of the LoG filter contains positive, negative, and zero values. To visualize this correctly, imshow(filteredImage, []) scales the display range to fit the data, typically showing positive values as white, negative values as black, and zero-crossings (the edges) as gray.

Expected Output

The output window will show the original image on the left. On the right, the filtered image will be mostly gray. Strong edges in the original image will appear as sharp, thin double lines (one black, one white). This black-to-white transition represents the zero-crossing, which precisely locates the edge.

Legacy Method: Using fspecial

For quick tests or working with older code, you can still use fspecial. It’s less flexible but more concise.

Example 2: Comparing a Simple Laplacian with LoG

This example highlights why the Gaussian smoothing in LoG is so important. A pure Laplacian filter is very sensitive to noise.

I = imread('cameraman.tif');

% 1. Simple Laplacian Filter (noise-sensitive)
h_lap = fspecial('laplacian', 0.2); % 0.2 is the alpha parameter
I_lap = imfilter(I, h_lap, 'replicate');

% 2. Laplacian of Gaussian (LoG) Filter (noise-robust)
h_log = fspecial('log', [5 5], 0.5); % 5x5 filter, sigma=0.5
I_log = imfilter(I, h_log, 'replicate');

% Display comparison
figure;
subplot(1, 3, 1); imshow(I); title('Original');
subplot(1, 3, 2); imshow(I_lap, []); title('Laplacian Only');
subplot(1, 3, 3); imshow(I_log, []); title('Laplacian of Gaussian');

Expected Output

A figure will show three images. The ‘Original’ is the input. The ‘Laplacian Only’ image will show edges but also a lot of speckled noise across the entire image. The ‘Laplacian of Gaussian’ image will have much cleaner, more distinct edges with the background noise suppressed, demonstrating the effectiveness of the initial Gaussian smoothing.