In
imaging science
Imaging is the representation or reproduction of an object's form; especially a visual representation (i.e., the formation of an image).
Imaging technology is the application of materials and methods to create, preserve, or duplicate images.
...
, difference of Gaussians (DoG) is a
feature
Feature may refer to:
Computing
* Feature (CAD), could be a hole, pocket, or notch
* Feature (computer vision), could be an edge, corner or blob
* Feature (software design) is an intentional distinguishing characteristic of a software item ...
enhancement algorithm that involves the subtraction of one
Gaussian blurred version of an original image from another, less blurred version of the original. In the simple case of
grayscale images, the blurred images are obtained by
convolving the original
grayscale images with
Gaussian kernel
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real constants , and non-zero . It is ...
s having differing width (standard deviations). Blurring an image using a Gaussian
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
suppresses only
high-frequency spatial information. Subtracting one image from the other preserves spatial information that lies between the range of frequencies that are preserved in the two blurred images. Thus, the DoG is a spatial
band-pass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-por ...
that attenuates frequencies in the original grayscale image that are far from the band center.
["Molecular Expressions Microscopy Primer: Digital Image Processing – Difference of Gaussians Edge Enhancement Algorithm", ''Olympus America Inc., and Florida State University''](_blank)
Michael W. Davidson, Mortimer Abramowitz
Mathematics of difference of Gaussians
Given an ''m''-channel, ''n''-dimensional image
:
The difference of Gaussians (DoG) of the image
is the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
:
obtained by subtracting the image
convolved
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
with the Gaussian of variance
from the image
convolved with a Gaussian of narrower variance
, with
. In one dimension,
is defined as:
:
and for the centered two-dimensional case:
:
which is formally equivalent to:
:
which represents an image convoluted to the difference of two Gaussians, which approximates a
Mexican hat function.
The relation between the difference of Gaussians operator and the
Laplacian of the Gaussian operator (the
Mexican hat wavelet
In mathematics and numerical analysis, the Ricker wavelet
:\psi(t) = \frac \left(1 - \left(\frac\right)^2 \right) e^
is the negative normalized second derivative of a Gaussian function, i.e., up to scale and normalization, the second Hermite fu ...
) is explained in appendix A in Lindeberg (2015).
Details and applications
As a
feature
Feature may refer to:
Computing
* Feature (CAD), could be a hole, pocket, or notch
* Feature (computer vision), could be an edge, corner or blob
* Feature (software design) is an intentional distinguishing characteristic of a software item ...
enhancement algorithm, the difference of Gaussians can be utilized to increase the visibility of edges and other detail present in a digital image. A wide variety of alternative
edge sharpening filters operate by enhancing high frequency detail, but because
random noise
In electronics, noise is an unwanted disturbance in an electrical signal.
Noise generated by electronic devices varies greatly as it is produced by several different effects.
In particular, noise is inherent in physics, and central to the ...
also has a high spatial frequency, many of these sharpening filters tend to enhance noise, which can be an undesirable artifact. The difference of Gaussians algorithm removes high frequency detail that often includes random noise, rendering this approach one of the most suitable for processing images with a high degree of noise. A major drawback to application of the algorithm is an inherent reduction in overall image contrast produced by the operation.
When utilized for image enhancement, the difference of Gaussians algorithm is typically applied when the size ratio of kernel (2) to kernel (1) is 4:1 or 5:1. In the example images to the right, the sizes of the Gaussian
kernels
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...
employed to
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
the sample image were 10 pixels and 5 pixels.
The algorithm can also be used to obtain an approximation of the
Laplacian of Gaussian
In computer vision, blob detection methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some proper ...
when the ratio of size 2 to size 1 is roughly equal to 1.6. The Laplacian of Gaussian is useful for detecting edges that appear at various image scales or degrees of image focus. The exact values of sizes of the two kernels that are used to approximate the Laplacian of Gaussian will determine the scale of the difference image, which may appear blurry as a result.
Differences of Gaussians have also been used for
blob detection
In computer vision, blob detection methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some propert ...
in the
scale-invariant feature transform
The scale-invariant feature transform (SIFT) is a computer vision algorithm to detect, describe, and match local ''features'' in images, invented by David Lowe in 1999.
Applications include object recognition, robotic mapping and navigation, ima ...
. In fact, the DoG as the difference of two
Multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
has always a total null sum and convolving it with a uniform signal generates no response. It approximates well a second derivate of Gaussian (
Laplacian of Gaussian
In computer vision, blob detection methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some proper ...
) with K~1.6 and the receptive fields of ganglion cells in the
retina
The retina (from la, rete "net") is the innermost, light-sensitive layer of tissue of the eye of most vertebrates and some molluscs. The optics of the eye create a focused two-dimensional image of the visual world on the retina, which then ...
with K~5. It may easily be used in recursive schemes and is used as an operator in real-time algorithms for blob detection and automatic scale selection.
More information
In its operation, the difference of Gaussians algorithm is believed to mimic how neural processing in the retina of the eye extracts details from images destined for transmission to the brain.
See also
*
Marr–Hildreth algorithm In computer vision, the Marr–Hildreth algorithm is a method of detecting edges in digital images, that is, continuous curves where there are strong and rapid variations in image brightness. The Marr–Hildreth edge detection method is simple and ...
*Treatment of the
difference of Gaussians approach in blob detection.
*
Blob detection
In computer vision, blob detection methods are aimed at detecting regions in a digital image that differ in properties, such as brightness or color, compared to surrounding regions. Informally, a blob is a region of an image in which some propert ...
*
Gaussian pyramid
Pyramid, or pyramid representation, is a type of multi-scale signal representation developed by the computer vision, image processing and signal processing communities, in which a signal or an image is subject to repeated smoothing and subsam ...
*
Scale space
Scale-space theory is a framework for multi-scale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theor ...
*
Scale-invariant feature transform
The scale-invariant feature transform (SIFT) is a computer vision algorithm to detect, describe, and match local ''features'' in images, invented by David Lowe in 1999.
Applications include object recognition, robotic mapping and navigation, ima ...
References
Further reading
*Notes by Melisa Durmuş o
Edge Detection and Gaussian related mathematicsfrom the University of Edinburgh.
{{DEFAULTSORT:Difference Of Gaussians
Continuous wavelets
Feature detection (computer vision)