results in a blur that preserves boundaries and edges better than other, more uniform blurring filters; see also scale space implementation. In theory, the Gaussian function at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels at a distance of more than 3σ have a small enough influence to be considered effectively zero. Thus contributions from pixels outside that range can be ignored. Typically, an image processing program need only calculate a matrix with dimensions × (where is the ceiling function) to ensure a result sufficiently close to that obtained by the entire Gaussian distribution. In addition to being circularly symmetric, the Gaussian blur can be applied to a two-dimensional image as two independent one-dimensional calculations, and so is termed a separable filter. That is, the effect of applying the two-dimensional matrix can also be achieved by applying a series of single-dimensional Gaussian matrices in the horizontal direction, then repeating the process in the vertical direction. In computational terms, this is a useful property, since the calculation can be performed in time (where h is height and w is width; see Big O notation), as opposed to for a non-separable kernel. Applying successive Gaussian blurs to an image has the same effect as applying a single, larger Gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied. For example, applying successive Gaussian blurs with radii of 6 and 8 gives the same results as applying a single Gaussian blur of radius 10, since . Because of this relationship, processing time cannot be saved by simulating a Gaussian blur with successive, smaller blurs — the time required will be at least as great as performing the single large blur. Gaussian blurring is commonly used when reducing the size of an image. When downsampling an image, it is common to apply a low-pass filter to the image prior to resampling. This is to ensure that spurious high-frequency information does not appear in the downsampled image (aliasing). Gaussian blurs have nice properties, such as having no sharp edges, and thus do not introduce ringing into the filtered image. Low-pass filter Gaussian blur is a low-pass filter, attenuating high frequency signals. Its amplitude Bode plot