Pooling All this "convolving" across an entire image generates a lot of information, and this can quickly become a computational nightmare.
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Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur) would more accurately reproduce the bokeh effect.
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First, polynomial multiplication exactly corresponds to discrete convolution, so that repeatedly convolving the sequence {..., 0, 0, 1, 1, 0, 0, ...} with itself corresponds to taking powers of 1 + x, and hence to generating the rows of the triangle. Second, repeatedly convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent copies of that variable; this is exactly the situation to which the central limit theorem applies, and hence leads to the normal distribution in the limit.
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Digital filtering generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. The surrounding samples may be identified with respect to time or space. The output of a linear digital filter to any given input may be calculated by convolving the input signal with an impulse response.
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Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few , and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase- space regions smaller than , and thus renders such "negative probabilities" less paradoxical.
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The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form. For more information, see control theory. The Laplace transform is invertible on a large class of functions.
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The Airy pattern, caused by Fraunhofer diffraction. Many special functions exhibit oscillatory decay, and thus convolving with such a function yields ringing in the output; one may consider these ringing, or restrict the term to unintended artifacts in frequency domain signal processing. Fraunhofer diffraction yields the Airy disk as point spread function, which has a ringing pattern. A few Bessel functions of the first kind, showing oscillatory decay.
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Input Shaping is an open-loop control technique for reducing vibrations in computer-controlled machines. The method works by creating a command signal that cancels its own vibration. That is, vibration excited by previous parts of the command signal is cancelled by vibration excited by latter parts of the command. Input shaping is implemented by convolving a sequence of impulses, known as an input shaper, with any arbitrary command.
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Image gradients can be used to extract information from images. Gradient images are created from the original image (generally by convolving with a filter, one of the simplest being the Sobel filter) for this purpose. Each pixel of a gradient image measures the change in intensity of that same point in the original image, in a given direction. To get the full range of direction, gradient images in the x and y directions are computed.
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B-spline windows can be obtained as k-fold convolutions of the rectangular window. They include the rectangular window itself (k = 1), the (k = 2) and the (k = 4). Alternative definitions sample the appropriate normalized B-spline basis functions instead of convolving discrete-time windows. A kth order B-spline basis function is a piece-wise polynomial function of degree k−1 that is obtained by k-fold self-convolution of the rectangular function.
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In order to place a sound at a certain position in 3D space, the set of FIR filters that correspond to the position is applied to the incoming sound, yielding spatial sound. The computations involved in convolving the sound signal from a particular point in space is fairly large. So a lot of work is done to reduce the complexity. One such work is based on combining grouped Principal Component Analysis (PCA) and Balanced Model Truncation (BMT).
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A Gaussian blur effect is typically generated by convolving an image with an FIR kernel of Gaussian values. In practice, it is best to take advantage of the Gaussian blur’s separable property by dividing the process into two passes. In the first pass, a one-dimensional kernel is used to blur the image in only the horizontal or vertical direction. In the second pass, the same one-dimensional kernel is used to blur in the remaining direction.
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At each point in the image, the result of the Sobel–Feldman operator is either the corresponding gradient vector or the norm of this vector. The Sobel–Feldman operator is based on convolving the image with a small, separable, and integer-valued filter in the horizontal and vertical directions and is therefore relatively inexpensive in terms of computations. On the other hand, the gradient approximation that it produces is relatively crude, in particular for high-frequency variations in the image.
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One method to remove noise is by convolving the original image with a mask that represents a low-pass filter or smoothing operation. For example, the Gaussian mask comprises elements determined by a Gaussian function. This convolution brings the value of each pixel into closer harmony with the values of its neighbors. In general, a smoothing filter sets each pixel to the average value, or a weighted average, of itself and its nearby neighbors; the Gaussian filter is just one possible set of weights.
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The Prewitt operator is used in image processing, particularly within edge detection algorithms. Technically, it is a discrete differentiation operator, computing an approximation of the gradient of the image intensity function. At each point in the image, the result of the Prewitt operator is either the corresponding gradient vector or the norm of this vector. The Prewitt operator is based on convolving the image with a small, separable, and integer valued filter in horizontal and vertical directions and is therefore relatively inexpensive in terms of computations like Sobel and Kayyali operators.
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For example, where image smoothing might convolve an intensity signal with a blur kernel formed using the Laplace operator, geometric smoothing might be achieved by convolving a surface geometry with a blur kernel formed using the Laplace-Beltrami operator. Applications of geometry processing algorithms already cover a wide range of areas from multimedia, entertainment and classical computer-aided design, to biomedical computing, reverse engineering, and scientific computing. Geometry processing is a common research topic at SIGGRAPH, the premier computer graphics academic conference, and the main topic of the annual Symposium on Geometry Processing.
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For instance, the Wigner distribution can and normally does take on negative values for states which have no classical model—and is a convenient indicator of quantum mechanical interference. (See below for a characterization of pure states whose Wigner functions are non-negative.) Smoothing the Wigner distribution through a filter of size larger than (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.
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The resulting effect is the same as convolving with a two- dimensional kernel in a single pass, but requires fewer calculations. Discretization is typically achieved by sampling the Gaussian filter kernel at discrete points, normally at positions corresponding to the midpoints of each pixel. This reduces the computational cost but, for very small filter kernels, point sampling the Gaussian function with very few samples leads to a large error. In these cases, accuracy is maintained (at a slight computational cost) by integration of the Gaussian function over each pixel's area.
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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 operates by convolving the image with the Laplacian of the Gaussian function, or, as a fast approximation by difference of Gaussians. Then, zero crossings are detected in the filtered result to obtain the edges. The Laplacian-of-Gaussian image operator is sometimes also referred to as the Mexican hat wavelet due to its visual shape when turned upside-down.
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In signal processing, a matched filter is obtained by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise. Matched filters are commonly used in radar, in which a known signal is sent out, and the reflected signal is examined for common elements of the out-going signal.
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Other fast convolution algorithms, such as the Schönhage–Strassen algorithm or the Mersenne transform, use fast Fourier transforms in other rings. If one sequence is much longer than the other, zero-extension of the shorter sequence and fast circular convolution is not the most computationally efficient method available. Instead, decomposing the longer sequence into blocks and convolving each block allows for faster algorithms such as the Overlap–save method and Overlap–add method. A hybrid convolution method that combines block and FIR algorithms allows for a zero input-output latency that is useful for real-time convolution computations.
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Example of a curve (red line) fit to a small data set (black points) with nonparametric regression using a Gaussian kernel smoother. The pink shaded area illustrates the kernel function applied to obtain an estimate of y for a given value of x. The kernel function defines the weight given to each data point in producing the estimate for a target point. Kernel regression estimates the continuous dependent variable from a limited set of data points by convolving the data points' locations with a kernel function—approximately speaking, the kernel function specifies how to "blur" the influence of the data points so that their values can be used to predict the value for nearby locations.
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An estimation of the original image is attained by convolving the coded image with the original coded aperture. In general, the recovered image will be the convolution of the object with the autocorrelation of the coded aperture and will contain artifacts unless its autocorrelation is a delta function. Some examples of coded apertures include the Fresnel zone plate (FZP), random arrays (RA), non-redundant arrays (NRA), uniformly redundant arrays (URA), modified uniformly redundant arrays (MURA), among others. Fresnel zone plates, called after Augustin-Jean Fresnel, may not be considered coded apertures at all since they consist of a set of radially symmetric rings, known as Fresnel zones, which alternate between opaque and transparent.
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For example, consider an image that is sent over some wireless network subject to electro-optical noise. Possible noise sources include errors in channel transmission, the analog to digital converter, and the image sensor. Usually noise caused by the channel or sensor creates spatially-independent, high-frequency signal components that translates to arbitrary light and dark spots on the actual image. In order to rid the image data of the high-frequency spectral content, it can be multiplied by the frequency response of a low-pass filter, which based on the convolution theorem, is equivalent to convolving the signal in the time/spatial domain by the impulse response of the low-pass filter.
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The Airy disc function falls off very quickly but has very wide tails (actually, infinitely wide tails). As long as the brightness of adjacent parts of the image are roughly in the same range, the effect of the blurring caused by the Airy disc is not particularly noticeable; but in parts of the image where very bright parts are adjacent to relatively darker parts, the tails of the Airy disc become visible, and can extend far beyond the extent of the bright part of the image. In HDRR images, the effect can be re-produced by convolving the image with a windowed kernel of an Airy disc (for very good lenses), or by applying Gaussian blur (to simulate the effect of a less perfect lens), before converting the image to fixed-range pixels. The effect cannot be fully reproduced in non-HDRR imaging systems, because the amount of bleed depends on how bright the bright part of the image is.
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