Difference of Gaussians
Encyclopedia
In computer vision
, Difference of Gaussians is a grayscale
image enhancement algorithm that involves the subtraction of one blurred version of an original grayscale image from another, less blurred version of the original. The blurred images are obtained by convolving
the original grayscale image with Gaussian kernels having differing standard deviations. Blurring an image using a Gaussian
kernel 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 difference of Gaussians is a band-pass filter
that discards all but a handful of spatial frequencies that are present in the original grayscale image.
and for the centered two-dimensional case (see Gaussian blur
):
operate by enhancing high frequency detail, but because random noise 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 employed to smooth
the sample image were 10 pixels and 5 pixels. The algorithm can also be used to obtain an approximation of the Laplacian of Gaussian 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 the scale-invariant feature transform
. In fact, the DOG as the difference of two Multivariate normal distribution 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) with K~1.6 and the receptive fields of ganglion cells in the retina
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.
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
, Difference of Gaussians is a grayscale
Grayscale
In photography and computing, a grayscale or greyscale digital image is an image in which the value of each pixel is a single sample, that is, it carries only intensity information...
image enhancement algorithm that involves the subtraction of one blurred version of an original grayscale image from another, less blurred version of the original. The blurred images are obtained by convolving
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
the original grayscale image with Gaussian kernels having differing standard deviations. Blurring an image using a Gaussian
Gaussian blur
A Gaussian blur is the result of blurring an image by a Gaussian function. It is a widely used effect in graphics software, typically to reduce image noise and reduce detail...
kernel 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 difference of Gaussians is a band-pass filter
Band-pass filter
A band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....
that discards all but a handful of spatial frequencies that are present in the original grayscale image.
Mathematics of Difference of Gaussians
The Difference of Gaussians (DOG) is a wavelet mother function of null total sum which approximates the Mexican Hat wavelet by subtracting a wide Gaussian from a narrow Gaussian, as defined by this formula in one dimension:and for the centered two-dimensional case (see Gaussian blur
Gaussian blur
A Gaussian blur is the result of blurring an image by a Gaussian function. It is a widely used effect in graphics software, typically to reduce image noise and reduce detail...
):
Details and applications
As an image 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 filtersEdge detection
Edge detection is a fundamental tool in image processing and computer vision, particularly in the areas of feature detection and feature extraction, which aim at identifying points in a digital image at which the image brightness changes sharply or, more formally, has discontinuities...
operate by enhancing high frequency detail, but because random noise 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 employed to smooth
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
the sample image were 10 pixels and 5 pixels. The algorithm can also be used to obtain an approximation of the Laplacian of Gaussian 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
Blob detection
In the area of computer vision, blob detection refers to visual modules that are aimed at detecting points and/or regions in the image that differ in properties like brightness or color compared to the surrounding...
in the scale-invariant feature transform
Scale-invariant feature transform
Scale-invariant feature transform is an algorithm in computer vision to detect and describe local features in images. The algorithm was published by David Lowe in 1999....
. In fact, the DOG as the difference of two Multivariate normal distribution 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) with K~1.6 and the receptive fields of ganglion cells in the retina
Retina
The vertebrate retina is a light-sensitive tissue lining the inner surface of the eye. The optics of the eye create an image of the visual world on the retina, which serves much the same function as the film in a camera. Light striking the retina initiates a cascade of chemical and electrical...
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 algorithmMarr-Hildreth algorithmIn 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 operates by convolving the image with the Laplacian of...
- Treatment of the difference of Gaussians approach in blob detection.
- Blob detectionBlob detectionIn the area of computer vision, blob detection refers to visual modules that are aimed at detecting points and/or regions in the image that differ in properties like brightness or color compared to the surrounding...
- Scale-space
- Scale-invariant feature transformScale-invariant feature transformScale-invariant feature transform is an algorithm in computer vision to detect and describe local features in images. The algorithm was published by David Lowe in 1999....
- Notes by Bryan S. Morse on Edge Detection and Gaussian related mathematics from the University of Edinburgh.