1 Introduction

  Systems that preserve scale-space causality are usually associated with Gaussian filters [ 1 , 2 ] and diffusion [ 3 , 4 ] in which the image forms the initial conditions for a discretisation of the continuous diffusion equation,

If the conduction coefficient, c , is a constant this becomes the linear diffusion equation, which may be implemented by convolving the image with the Green's function of the diffusion equation: a Gaussian filter. Of course, care is needed when discretising ( 1 ) but, if it is done correctly [ 5 ], a scale-space with discrete space and continuous scale may be formed with separable filters ( in [ 5 ]) as,

f ( x , y ) is the pixel value at position ( x , y ) and is the pixel value after smoothing to scale s . T ( n ; s ) is the discrete approximation to the Gaussian kernel and is the modified Bessel function of the first kind.

Such scale-space systems have several problems:

1. The edges of sharp-edged objects become blurred at large scales. This leads to poor localisation and a necessity to track edges back through scale-space   .
2. They introduce extrema, albeit in rather pathological cases   .
3. They require scale calibration since an image filtered to scale s contains features at many scales   .
4. Multiple convolutions may require significant amounts of computation using real arithmetic   .

Problem  1 may be reduced by reverting to ( 1 ) and allowing the conduction coefficient, c , to be a function of thus . If this function is carefully chosen (several have been suggested) then the effect is to allow diffusion in low contrast regions but not at sharp-edges. Unfortunately anisotropic diffusion requires even more computation than linear diffusion and problem  3 is exacerbated - sharp-edged small scale objects persist to large scales.

Problem  2 is illustrated in Figure  1 . On the left is a three-dimensional intensity plot of an image. A disc and a square connected by thin isthmus form its main feature. After diffusion filtering (centre) the single maximum associated with the two discs and isthmus has now become two maxima. This example, due to Pizer [ 6 ], is well known and led to a new definition of scale-space causality: the ``non-enhancement of existing regional extrema'' principle which is satisfied by diffusion systems. The right-hand plot in Figure  1 shows the result of applying an M -sieve. The smaller area disc is removed and the large area feature is unaffected.

   
Figure 1: Original image (left), Gaussian filtered image (centre), sieved image (right)

Figure  1 also illustrates problem  3 . In the Gaussian processor the intensity of the output at scale s is proportional to the scale and intensity of the original feature. So recovering the parameters of the original signal may be tricky. As far as image interpretation goes, having peaks representing extrema from objects of a variety of scales is inconvenient. It is possible to design morphological filters that produce, at scale s , only objects of size s .