Is there a clever way to implement cascaded moving average filters?
[EDIT:20181225, added a couple of references] This topic is coming back. P. Getreuer provides C code, and lists an handful of them in A survey of Gaussian convolution algorithms,  Image Processing On Line, 2013:


  Gaussian convolution is a common operation and building block for
  algorithms in signal and image processing. Consequently, its efficient
  computation is important, and many fast approximations have been
  proposed. In this survey, we discuss approximate Gaussian convolution
  based on finite impulse response filters, DFT and DCT based
  convolution, box filters, and several recursive filters. Since
  boundary handling is sometimes overlooked in the original works, we
  pay particular attention to develop it here. We perform numerical
  experiments to compare the speed and quality of the algorithms.


Here are a couple references:


Robinson, J. A., Efficient Gaussian filtering using Cascaded Prefix Sums, 2012



  This paper introduces a method for multidimensional Gaussian filtering
  using an efficient one-pass cascade of overlapping local average
  windows driven by prefix sums. Each local-average filter is
  implemented in n dimensions, with non-integer lengths, allowing
  accurate approximation of Gaussians of any variance. In axis oriented
  form the method has a scan-rate hardware realization and fast software
  implementation using minimal extra memory. In this latter case the new
  method consistently outperforms the fastest alternative Gaussian
  filtering method both in accuracy and speed.



Bernardo Rodrigues Pires, Karanhaar Singh, José M. F. Moura, Approximating image filters with box filters, 2011



  Box filters have been used to speed up many computation-intensive
  operations in Image Processing and Computer Vision. They have the
  advantage of being fast to compute, but their adoption has been
  hampered by the fact that they present serious restrictions to filter
  construction. This paper relaxes these restrictions by presenting a
  method for automatically approximating an arbitrary 2-D filter by a
  box filter. To develop our method, we first formulate the
  approximation as a minimization problem and show that it is possible
  to find a closed form solution to a subset of the parameters of the
  box filter. To solve for the remaining parameters of the
  approximation, we develop two algorithms: Exhaustive Search for small
  filters and Directed Search for large filters. Experimental results
  show the validity of the proposed method.



Elboher, Elhanan  and Werman, Michael, Efficient and Accurate Gaussian Image Filtering Using Running Sums, 2011



  This paper presents a simple and efficient method to convolve an image
  with a Gaussian kernel. The computation is performed in a constant
  number of operations per pixel using running sums along the image rows
  and columns. We investigate the error function used for kernel
  approximation and its relation to the properties of the input signal.
  Based on natural image statistics we propose a quadratic form kernel
  error function so that the output image l2 error is minimized. We
  apply the proposed approach to approximate the Gaussian kernel by
  linear combination of constant functions. This results in very
  efficient Gaussian filtering method. Our experiments show that the
  proposed technique is faster than state of the art methods while
  preserving a similar accuracy.



Kovesi, Peter, Fast Almost-Gaussian Filtering, 2010



  Image averaging can be performed very efficiently using either
  separable moving average filters or by using summed area tables, also
  known as integral images. Both these methods allow averaging to be
  performed at a small fixed cost per pixel, independent of the
  averaging filter size. Repeated filtering with averaging filters can
  be used to approximate Gaussian filtering. Thus a good approximation
  to Gaussian filtering can be achieved at a fixed cost per pixel
  independent of filter size. This paper describes how to determine the
  averaging filters that one needs to approximate a Gaussian with a
  specified standard deviation. The design of bandpass filters from the
  difference of Gaussians is also analysed. It is shown that difference
  of Gaussian bandpass filters share some of the attributes of log-Gabor
  filters in that they have a relatively symmetric transfer function
  when viewed on a logarithmic frequency scale and can be constructed
  with large bandwidths.



Wells, William M., Efficient Synthesis of Gaussian Filters by Cascaded Uniform Filters, 1986



  Gaussian filtering is an important tool in image processing and
  computer vision. In this paper we discuss the background of Gaussian
  filtering and look at some methods for implementing it. Consideration
  of the central limit theorem suggests using a cascade of simple''
  filters as a means of computing Gaussian filters. Amongsimple''
  filters, uniform-coefficient finite-impulse-response digital filters
  are especially economical to implement. The idea of cascaded uniform
  filters has been around for a while [13], [16]. We show that this
  method is economical to implement, has good filtering characteristics,
  and is appropriate for hardware implementation. We point out an
  equivalence to one of Burt's methods 1, 3 under certain
  circumstances. As an extension, we describe an approach to
  implementing a Gaussian Pyramid which requires approximately two
  addition operations per pixel, per level, per dimension. We examine
  tradeoffs in choosing an algorithm for Gaussian filtering, and finally
  discuss an implementation.