% -*- texinfo -*- % @deftypefn {Function File} @var{J} = imsmooth(@var{I}, @var{name}, @var{options}) % Smooth the given image using several different algorithms. % % The first input argument @var{I} is the image to be smoothed. If it is an RGB % image, each color plane is treated separately. % The variable @var{name} must be a string that determines which algorithm will % be used in the smoothing. It can be any of the following strings % % @table @asis % @item 'Gaussian' % Isotropic Gaussian smoothing. This is the default. % @item 'Average' % Smoothing using a rectangular averaging linear filter. % @item 'Disk' % Smoothing using a circular averaging linear filter. % @item 'Median' % Median filtering. % @item 'Bilateral' % Gaussian bilateral filtering. % @item 'Perona & Malik' % @itemx 'Perona and Malik' % @itemx 'P&M' % Smoothing using anisotropic diffusion as described by Perona and Malik. % @item 'Custom Gaussian' % Gaussian smoothing with a spatially varying covariance matrix. % @end table % % In all algorithms the computation is done in double precision floating point % numbers, but the result has the same type as the input. Also, the size of the % smoothed image is the same as the input image. % % @strong{Isotropic Gaussian smoothing} % % The image is convolved with a Gaussian filter with spread @var{sigma}. % By default @var{sigma} is @math{0.5}, but this can be changed. If the third % input argument is a scalar it is used as the filter spread. % % The image is extrapolated symmetrically before the convolution operation. % % @strong{Rectangular averaging linear filter} % % The image is convolved with @var{N} by @var{M} rectangular averaging filter. % By default a 3 by 3 filter is used, but this can e changed. If the third % input argument is a scalar @var{N} a @var{N} by @var{N} filter is used. If the third % input argument is a two-vector @code{[@var{N}, @var{M}]} a @var{N} by @var{M} % filter is used. % % The image is extrapolated symmetrically before the convolution operation. % % @strong{Circular averaging linear filter} % % The image is convolved with circular averaging filter. By default the filter % has a radius of 5, but this can e changed. If the third input argument is a % scalar @var{r} the radius will be @var{r}. % % The image is extrapolated symmetrically before the convolution operation. % % @strong{Median filtering} % % Each pixel is replaced with the median of the pixels in the local area. By % default, this area is 3 by 3, but this can be changed. If the third input % argument is a scalar @var{N} the area will be @var{N} by @var{N}, and if it's % a two-vector [@var{N}, @var{M}] the area will be @var{N} by @var{M}. % % The image is extrapolated symmetrically before the filtering is performed. % % @strong{Gaussian bilateral filtering} % % The image is smoothed using Gaussian bilateral filtering as described by % Tomasi and Manduchi [2]. The filtering result is computed as % @example % @var{J(x0, y0) = k * SUM SUM @var{I}(x,y) * w(x, y, x0, y0, @var{I}(x0,y0), @var{I}(x,y)) % x y % @end example % where @code{k} a normalisation variable, and % @example % w(x, y, x0, y0, @var{I}(x0,y0), @var{I}(x,y)) % = exp(-0.5*d([x0,y0],[x,y])^2/@var{sigma_d}^2) % * exp(-0.5*d(@var{I}(x0,y0),@var{I}(x,y))^2/@var{sigma_r}^2), % @end example % with @code{d} being the Euclidian distance function. The two paramteres % @var{sigma_d} and @var{sigma_r} control the amount of smoothing. @var{sigma_d} % is the size of the spatial smoothing filter, while @var{sigma_r} is the size % of the range filter. When @var{sigma_r} is large the filter behaves almost % like the isotropic Gaussian filter with spread @var{sigma_d}, and when it is % small edges are preserved better. By default @var{sigma_d} is 2, and @var{sigma_r} % is @math{10/255} for floating points images (with integer images this is % multiplied with the maximal possible value representable by the integer class). % % The image is extrapolated symmetrically before the filtering is performed. % % @strong{Perona and Malik} % % The image is smoothed using anisotropic diffusion as described by Perona and % Malik [1]. The algorithm iteratively updates the image using % % @example % I += lambda * (g(dN).*dN + g(dS).*dS + g(dE).*dE + g(dW).*dW) % @end example % % @noindent % where @code{dN} is the spatial derivative of the image in the North direction, % and so forth. The function @var{g} determines the behaviour of the diffusion. % If @math{g(x) = 1} this is standard isotropic diffusion. % % The above update equation is repeated @var{iter} times, which by default is 10 % times. If the third input argument is a positive scalar, that number of updates % will be performed. % % The update parameter @var{lambda} affects how much smoothing happens in each % iteration. The algorithm can only be proved stable is @var{lambda} is between % 0 and 0.25, and by default it is 0.25. If the fourth input argument is given % this parameter can be changed. % % The function @var{g} in the update equation determines the type of the result. % By default @code{@var{g}(@var{d}) = exp(-(@var{d}./@var{K}).^2)} where @var{K} = 25. % This choice gives privileges to high-contrast edges over low-contrast ones. % An alternative is to set @code{@var{g}(@var{d}) = 1./(1 + (@var{d}./@var{K}).^2)}, % which gives privileges to wide regions over smaller ones. The choice of @var{g} % can be controlled through the fifth input argument. If it is the string % @code{'method1'}, the first mentioned function is used, and if it is @var{'method2'} % the second one is used. The argument can also be a function handle, in which case % the given function is used. It should be noted that for stability reasons, % @var{g} should return values between 0 and 1. % % The following example shows how to set % @code{@var{g}(@var{d}) = exp(-(@var{d}./@var{K}).^2)} where @var{K} = 50. % The update will be repeated 25 times, with @var{lambda} = 0.25. % % @example % @var{g} = @@(@var{d}) exp(-(@var{d}./50).^2); % @var{J} = imsmooth(@var{I}, 'p&m', 25, 0.25, @var{g}); % @end example % % @strong{Custom Gaussian Smoothing} % % The image is smoothed using a Gaussian filter with a spatially varying covariance % matrix. The third and fourth input arguments contain the Eigenvalues of the % covariance matrix, while the fifth contains the rotation of the Gaussian. % These arguments can be matrices of the same size as the input image, or scalars. % In the last case the scalar is used in all pixels. If the rotation is not given % it defaults to zero. % % The following example shows how to increase the size of an Gaussian % filter, such that it is small near the upper right corner of the image, and % large near the lower left corner. % % @example % [@var{lambda1}, @var{lambda2}] = meshgrid (linspace (0, 25, columns (@var{I})), linspace (0, 25, rows (@var{I}))); % @var{J} = imsmooth (@var{I}, 'Custom Gaussian', @var{lambda1}, @var{lambda2}); % @end example % % The implementation uses an elliptic filter, where only neighbouring pixels % with a Mahalanobis' distance to the current pixel that is less than 3 are % used to compute the response. The response is computed using double precision % floating points, but the result is of the same class as the input image. % % @strong{References} % % [1] P. Perona and J. Malik, % 'Scale-space and edge detection using anisotropic diffusion', % IEEE Transactions on Pattern Analysis and Machine Intelligence, % 12(7):629-639, 1990. % % [2] C. Tomasi and R. Manduchi, % 'Bilateral Filtering for Gray and Color Images', % Proceedings of the 1998 IEEE International Conference on Computer Vision, Bombay, India. % % @seealso{imfilter, fspecial} % @end deftypefn

- fspecial % Create spatial filters for image processing.
- impad % Pad (augment) a matrix for application of image processing algorithms.
- medfilt2 % Two dimensional median filtering.

- edge % Detect edges in the given image using various methods. The first input @var{im}

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