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2D convolution with an M × N kernel requires M × N multiplications for each sample (pixel). If the kernel is separable, then the computation can be reduced to M + N multiplications. Using separable convolutions can significantly decrease the computation by doing 1D convolution twice instead of one 2D convolution. [2]
The Hough transform [3] can be used to detect lines and the output is a parametric description of the lines in an image, for example ρ = r cos(θ) + c sin(θ). [1] If there is a line in a row and column based image space, it can be defined ρ, the distance from the origin to the line along a perpendicular to the line, and θ, the angle of the perpendicular projection from the origin to the ...
where here denotes the 2-dimensional convolution operation. Since the Prewitt kernels can be decomposed as the products of an averaging and a differentiation kernel, they compute the gradient with smoothing. Therefore, it is a separable filter. For example, can be written as
Its impulse response is defined by a sinusoidal wave (a plane wave for 2D Gabor filters) multiplied by a Gaussian function. [6] Because of the multiplication-convolution property (Convolution theorem), the Fourier transform of a Gabor filter's impulse response is the convolution of the Fourier transform of the harmonic function (sinusoidal function) and the Fourier transform of the Gaussian ...
Image derivatives can be computed by using small convolution filters of size 2 × 2 or 3 × 3, such as the Laplacian, Sobel, Roberts and Prewitt operators. [1] However, a larger mask will generally give a better approximation of the derivative and examples of such filters are Gaussian derivatives [2] and Gabor filters. [3]
A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of f and g is differentiable as many times as f and g are in total. These identities hold for example under the condition that f and g are absolutely integrable and at least one of them has an absolutely integrable (L 1 ) weak ...
An example of an image blurred using a box blur. A box blur (also known as a box linear filter) is a spatial domain linear filter in which each pixel in the resulting image has a value equal to the average value of its neighboring pixels in the input image. It is a form of low-pass ("blurring") filter.
Values from this distribution are used to build a convolution matrix which is applied to the original image. This convolution process is illustrated visually in the figure on the right. Each pixel's new value is set to a weighted average of that pixel's neighborhood. The original pixel's value receives the heaviest weight (having the highest ...