The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination. It is a widely used effect in graphics software, typically to reduce image noise and reduce detail. Under composition, an Identity function is "neutral": if f is any function from X to Y, then f o I = f and I o f = f.Ī Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function. The Identity function is a bijective (both injective and surjective).įor example, the Identity function on a set A is the function that does nothing to each element of A. The domain and range of the Identity function are both equal to R. So, the function f:R->R defined by f(x) = x for all x in R is called the identity function. The function that assigns every real number to itself is called an Identity function and is usually denoted by I. In equations, the function is given by f(x) = x The identity function is a function that always returns the same value that was used as its argument. Please, have a look at this link: Read More If you are in love with Matrices :) I've created this tool with a Matrix. When you will find this Textfield, you can copy-paste the values directly in the Matrix 3x3 node in Nuke. If Normalize is deselected, the value of the pixel will be 375 in the example above, otherwise, if selected the result will be Normalized, then divided by the divisor, so 46,875 In Nuke you can introduce a Matrix Node from here: Filter → Matrix. So the final value of the pixel is 375 / Divisor = 375 / 8 = 46,875 = 47 This is useful if result may be negative. The Offset is added to the division result. You will hardly use 1, which lets result unchanged, and 9 or 25 according to matrix size, which gives the average of pixel values.ĭivisor is equal to: 0 + 1 + 0 + 1 + 4 + 1 + 0 + 1 + 0 = 8 The result of previous calculation will be divided by this divisor. (the filter doesn't work on the image but on a copy). It multiplied the value of each of them by the kernel corresponding value and added results. Here is what happened: the filter read successively, from left to right and from top to bottom, all the pixels of the kernel action area. In the middle is the kernel (Edge Detection matrix) and, on the right is the convolution result. The kernel action area has a blue border. The element at coordinates is the central element with red color. On the left is the image matrix: each pixel is marked with its value. You can see how to apply a Edge Detection matrix 3x3 to an image. If we want to calculate the Output value, we have to apply this formula: OUTPUT = (i * 1) + (h * 2) + (g * 3) + (f * 4) + (e * 5) + (d * 6) + (c * 7) + (b * 8) + (a * 9)Īt this link you can find one of the best example about the 2D Convolution. If the kernel is centered (aligned) exactly at the sample that we are interested in, multiply the kernel data by the overlapped input data. (L=3 if kernel is 3x3)įirst, flip the kernel, which is the shaded box, in both horizontal and vertical direction. This is the sum of the coefficient of a convolution kernel, or 1 if the sum is equal to 0 The coefficient of a convolution kernel at position i,jĭ = Divisor (or 1/F). The data value of a pixel that corresponds to the position i,jį = Filter Matrix. So, in general, we can say that the final formula for computing a pixel at (y,x) is: Finally, sometimes we also add a constant value (Offset) to the result O. In those cases we need to avoid division by D = 0. Some filters, like edge-detections ones, have a coefficients’ sum of 0. D is the the sum of all coefficients in the Filter Matrix. The filter F should be normalized (sum of all weights equals 1), otherwise we need to divide O with a divisor (or factor) D. So, if we apply the convolution operation on a pixel (y,x) of the input image I, the resulting pixel in the output image O will be: In the next Formula you will find points as instead of just because that’s the way the matrices are declared/used in the library. We will consider only 3x3 matrices, they are the most used and they are enough for all effects you want. You can use different kind of matrix: 3x3, 5x5, 7x7, and so on. Convolution is the treatment of a matrix by another one which is called “kernel”. This is accomplished by doing a convolution between a kernel and an image. It is used for blurring, sharpening, embossing, edge detection, and more. In image processing, a kernel, convolution matrix, or mask is a small matrix.
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