Image Features - Blob Detection

Goal

Independently scale interest points in each image, such that the detections are repeatable across different scales.

General Idea

Extract features as a variety of scales, e.g., by multiple resolutions in a image pyramid, and then matching features at the "corresponding" level

With the Harris corner detector we can find a maxima in a spatial search window, then find scale that gives local maxima of a function \(f\) in both position and scale.

"Blob" Detection - Laplacian of Gaussian

\[\nabla^2g(x,y,\sigma)=-(\pi\sigma^4)^{-1}(1-\frac{x^2+y^2}{2\sigma^2})\exp(-\frac{x^2+y^2}{2\sigma^2})\]

Define the characteristic scale as the scale that produced peak of the Laplacian response, so that such interest points are local maxima in both position and scale

Difference of Gaussian

Problem with LoG: LoG is not separable, and larger the \(\sigma\), larger the filter is.

Consider the approximation by finite differencing of Gaussian

\[DoG := G(x,y,k\sigma) - G(x,y,\sigma)\]

where \(G\) is the Gaussian function DoG is separable since Gaussian is separable, hence its difference.

Source code

import cv2
import numpy as np
from plotly.subplots import make_subplots
import plotly.graph_objects as go
import scipy.ndimage as scim

fig = make_subplots(3, 2, shared_xaxes=True)

N = 50
x = np.zeros(N)
x[N//3:N//3*2] = 1
sigma = 0
for i in range(3):
    for j in range(2):
        fig.add_trace(
            go.Scatter(
                x=np.arange(N), 
                y=scim.gaussian_laplace(x, sigma),
                name=f"sigma={sigma}"
            ),
            row=i+1, col=j+1
        )
        sigma += 1.5
fig.update_layout(margin={'t': 0, 'l': 0, 'b': 0, 'r': 0})
with open("../assets/blob_detection_log.json", "w") as f:
    f.write(fig.to_json())


x = np.linspace(-10, 10, N)
sigmas, ks = [1, 2, 3], [0.5, 0.7, 0.95]

fig = make_subplots(3, 3, shared_xaxes=True, shared_yaxes=True)

for i in range(9):
    sigma = sigmas[i % 3]
    k = ks[i // 3]
    g = 1 / sigma * np.exp(-(x ** 2) / (2 * (sigma ** 2)))
    g2 = 1 / (sigma * k) * np.exp(-(x ** 2) / (2 * ((sigma*k) ** 2)))
    fig.add_trace(
        go.Scatter(
            x=x,y=g-g2,name=f"k={k}, sigma={sigma}"
        ),
        row=i%3+1, col=i//3+1
    )
fig.update_layout(margin={'t': 0, 'l': 0, 'b': 0, 'r': 0})
with open("../assets/blob_detection_dog.json", "w") as f:
    f.write(fig.to_json())