Posts Tagged ‘filtering’

## Separable convolutions

Thursday, August 19th, 2010

The convolution is an important tool in image processing. All linear filters are convolutions (like, for example, the Gaussian filter). One of the reasons linear filters are so prevalent is that they imitate physical systems. For example, an analog electric circuit (containing resistors, capacitors and inductors) can be described by a convolution. The projection of a cell on a slide, through the microscope’s optical systems, onto the CCD sensor, can be described by a convolution. Even the sampling and averaging that occur in the CCD can be described by a convolution.

Two properties of the convolution are quite interesting when looking for an efficient implementation:

• The convolution is a multiplication in the Fourier domain: f(x)⊗h(x) ⇒ F(ω)⋅H(ω) . This means that you can compute the convolution by applying the Fourier transform to both the image and the convolution kernel, multiplying the two results, then inverse transforming the result.
• The convolution is associative: (fh1)⊗h2 = f⊗(h1h2) . This post is about the repercussions of this property.

## Gaussian filtering with the Image Processing Toolbox

Tuesday, October 6th, 2009

Edit May 2018: Since publishing this post, the MATLAB Image Processing Toolbox has added the function `imgaussfilt` that correctly applies a Gaussian smoothing filter. For Gaussian derivatives, the recommendations here still apply.

If you don’t use DIPimage, you probably use MATLAB’s Image Processing Toolbox. This toolbox makes it really easy to do convolutions with a Gaussian in the wrong way. On three accounts. The function `fspecial` is used to create a convolution kernel for a Gaussian filter. This kernel is 2D. That’s the first problem. The other two problems are given by the default values of its parameters. The default value for the kernel size is `[3 3]`. The default value for the σ (sigma) is 0.5. (more…)