employed Canny edge detector, w hich was also applied to the cell detection problem.
3.1.2
Laplacian
The Laplacian of a 2-D function f(x,y) is a second-order derivative defined as:
V Y =
dy- [3.1]
This operator is invariant to rotation, and hence insensitive to the direction in which the discontinuity runs. It highlights the points, lines and edges in the image and suppresses uniform and smoothly varying regions.
Equation [3.1] may be im plem ented in digital form in various ways. For a 3x3 region (Fig. 3.2(b)), the form m ost frequently encountered in practice is
V ^ / - 4Z() - ( Z j +Z4 +Zj +Z7) [3.2] W here the z's are defined in Fig. 3.2(a).
(a) Zi = -1 Z j= -1 Zq = 360 0 -1 0 -1 4 -1 0 -1 0 (b) Z i 9 = -1 Z36O— "1 (c)
Figure 3.2 (a) 3 x 3 kernel, (b) Most frequently used mask to compute Laplacian. (c) 19 x 19 Laplacian operator.
chapter 3__________________________________________________ Tumour Cell N uclei Detection
Although the Laplacian responds to transitions in intensity, it is governed by a major disadvantage, which is that of noise enhancement. Thus, all discontinuities in the image will respond to an edge, even those that correspond to noise, rather than real edges.
For this reason, together w ith the common Laplacian operator, a large rectangular kernel (19 x 19, Zq = 360) w as also constructed to suppress any random rapid increase in grey-level, w hich will be mistakenly recognised as an edge (see Fig. 2(c)). Thus, only differences that are large enough will produce a central pixel ( Z q ) w ith high negative grey-values followed by high
positive ones (for all transitions from light to dark), and vice versa for all transitions from dark to lig h t\ Zq equals to 360 since the basic requirem ent in
defining the digital Laplacian is that the coefficient associated w ith the central pixel be positive and the coefficients associated w ith the outer pixels be negative [Gonzalez and Woods, 1992]. H ere it should be m entioned that other Laplacian filters of different sizes w ere also constructed as well as their effects on the images being investigated, b u t a 19 x 19 Laplacian proved to be the best one for the particular scale and magnification of the histological images tested.
3.2.2
D ifference of Boxes (DoB) and D ifference of Gaussians
(DoG)
The DoB operator utilises the difference betw een tw o different size box-filters (the simplest smoothing filters which average pixels w ithin a small neighbourhood, a process called neighborhood averaging). Smoothing an image using box-kernels of different size suppresses high-frequency information, which corresponds to small details and spacing that are present. The difference betw een the two filtered im ages keeps only those structures (lines, points, etc.) that are in the interm ediate size range between the tw o operators.
’ It is well known that increasing mask size decreases noise sensitivity because of the inherent noise averaging performed by the operator, at the cost of greater edge displacement. However, in the problem of cell detection the importance of the latter is minimal compared to the demand for greater noise suppression.
Chapter 3__________________________________________________ Tumour Cell N uclei Detection
However, a box filter has a very high bandw idth and tends to exhibit many maxima in its response to noisy step edges, which is a serious problem w hen the imaging system adds noise, or w hen the image itself contains textured regions^. In [Canny, 1986] it w as dem onstrated that the response of a DoB edge detector to a noisy step edge is a roughly triangular peak w ith num erous sharp maxima in the vicinity of the edge, im plying the generation of too m any responses, which is obviously undesirable.
A Gaussian filter is similar to the box filter, except that the values of the neighbouring pixels are given different weighting, that being defined by a spatial Gaussian distribution:
- (
G( x, y) = A e [3.3]
where A is simply a constant used to regulate the G aussian's m agnitude and o is the standard deviation that determ ines the spatial scale of the Gaussian. Smoothing an image using Gaussian kernels w ith different standard deviations also suppresses high-frequency information. Large structures are affected to a lesser extent. The Gaussian filter gives m uch better results in smoothing applications, since it appears to be m ore isotropic (i.e. the smoothing effect should be the same in all directions in order not to prefer any direction). Yet, working in the wave num ber domain, the attenuation of the transfer function increases monotonically, and does not oscillate as it does in the box-filter [fahne, 1997]. The latter also implies more accurate detection of edges, as opposed to the DoB m ethod w here the application of a box filter may shift the objecf s position significantly. Thus, it is very clear that a DoG operator will have analogous advantages for edge detection, especially for images w ith a high level of noise, such as those encountered in histology. The DoG filter calculates the difference betw een two local 2D Gaussians of different standard deviation, convoluted w ith square neighbourhoods of the same size (Fig. 3.3(b)). The difference betw een the two convolved images
^ A detailed discussion regarding the properties of several smoothing operators, including box and Gaussian filters, and their wave number response can be found in [jahne, 1997].
Chapter 3 Tumour Cell Nuclei Detection
keeps only those structures (lines, points, etc.) that are in the intermediate size range between the two operators.
The DoB operator was investigated by convoluting the grey-level image separately with a 19 x 19 and 4 x 4 size box-filters. The two resulting images were then subtracted and the final image was searched for zero-crossings in a 3 x 3 neighbourhood (Fig. 3.3(a)). The DoG operator was studied similarly with DoB. Two 2D Gaussian filters of various standard deviations were applied, but 2.5 and 0 2= 6 was proven to be the optim um combination.
Magnitude
Z, Z2 Z, Z, Z„ Z , Z , Z.2 Z .
/ \V -
Gaussians with two — different sigmas DoG (b) V 3 x 3 neighborhood and V l < i <4 search for: Z, x Z., <0 (a)
Figure 3.3 (a) Zero-crossing search, (b) The DoG operator in one dimension.
Two Gaussians are shown, with their difference.
3.2.3
Laplacian of Gaussian (LoG)
The LoG operator w as first proposed in 1980 by Marr and Hildreth [Marr and Hildreth, 1980], and is characterised by certain properties of the hum an visual system. It actually constitutes the basis of a physiological model of the m anner that eye view edges. This concept was translated in convoluting an image with the Laplacian of a 2D Gaussian function, that is, a rotationally symmetric convolution filter. The usefulness of applying a Gaussian filter before a Laplacian, is due to greater smoothing effects that match the observed requirements of biological vision. Yet, the filter's spectrum is smooth and