ALIMENTADOR PRIMARIO 19_B

localized in the spatial dom ain and sm ooth and band-lim ited in the frequency domain^.

Thus, from Eq. [3.3] if w e set r^= x^+y^ the Laplacian of G (that is, the second derivative of G w ith respect to r) is:

LoG=V^G = A'{2 ) e

(7

[3.4]

(a)

3W

Figure 3.4 (a) Parameters associated with LoG masks, (b) 33 x 33

Laplacian of Gaussian with a=4. The function crosses x-axis at the 5^’’ (with regard to the central pixel) position.

The overall algorithm enables the definition of an excitatory region W, w hich

is defined as fV = l 4 2 a . The LoG w as im plem ented as a square m atrix of size

^ These two localization requirements, the one in the spatial and the other in the frequency domain, are conflicting though. Choosing a small filter mask results in a poor transfer function. Conversely, starting with an ideal transfer function results in large filter masks and overshooting filter responses. The Gaussian distribution is the only one that optimises such a relation [M arr and H ildreth, 1980].

Chapter 3__________________________________________________ Tumour Cell N uclei Detection

about 8.5a, as suggested in [Huertas and Medioni, 1986], and used successfully

in [Suza et a l, 2000]; 99.7 percent of the area under a ID Gaussian lies between

plus and minus 3 standard deviations from the mean, giving an area very close to zero for the V^G. These param eters are illustrated in Fig. 3.4.

One unusual aspect of the LoG filter is its use of very large operators [Chen et

ah, 1987]. In contrast to more conventional edge detection algorithm s in which

the operators rarely exceed a size of 5 x 5 pixels, the LoG filters used in this theory typically span several hundreds of pixels.

Another consideration is the selection of appropriate param eter values. Specifically, after the image is convolved w ith the filter, intensity changes are detected by finding the zero crossings generated by the convolution. The filter sensitivity to an intensity change having a certain w idth is prim arily associated w ith W, the w idth of the central excitatory region of the operator

[Huertas and Medioni, 1986]. The smaller the value of o, the space constant of

the Gaussian, the more sensitive the filter and the m ore detail is seen at the cost of poor noise tolerance. As an opposite effect, large a values can merge nearby edges belonging to distinct elements. Reference [Huertas and Medioni,

1986] provides a very good discussion on the effect of convolving a LoG filter

w ith step edges of various sizes. For our application a reasonable compromise between more edge features and fewer non-meaningful details w as found using a = 4, which provides a 33 x 33 matrix (see Fig. 3.4(b)).

3.2.4

M odifying the LoG

As mentioned previously, the great advantage of applying a Gaussian filter before a Laplacian is to produce greater sm oothing effects m atching the characteristics of hum an vision. The LoG filter provides a prom ising tool that seems to satisfy the dem and for applying an autom ated m ethod readily applicable in a series of images from a whole tissue section and w ith m inim um user interaction.

However, like all second order edge detectors, it should be followed by a zero crossings procedure that detects local edgels (i.e. edge elements), w hich are

Chapter 3___________________________________________________Tumour Cell N uclei Detection

generally unconnected. Thus, some further processing is required to link them into boundaries. As will be show n in the experimental section, due to the large num ber of cells and histological noise present in the images, LoG, as well as the other aforementioned filters, was found to produce excessive edge responses which were impossible to link and thus form meaningful objects. The previous observations led to the idea of m odifying the LoG in order to tackle the problem of nuclear edgel linking. The first attem pt w as based on applying a negative threshold, instead of a zero-crossing routine, following edge detection. The intention w as to detect all negative variations w hich are caused whenever an edge is present (due to the transition of grey level intensity from the histological background to nuclear boundaries), enabling localisation of "thick' internal boundaries around the areas of interest (dark in this case). The binary image can then be easily hole-filled to complete the identification of actual cell nuclei.

However, since the LoG is m ainly designed to generate a zero-crossing and not a negative value w here an edgel is present, it w as observed to produce some false responses that corresponded to negative variations other than those generated from the edges of the cell nuclei (mainly due to histological noise). Thus, it was initially attem pted to replace the central pixel value of LoG by the difference betw een its negative' A, and positive' A^ p art (see Fig. 3.5), so as to make the filter have a zero response w hen assimilated w ith a field of constant grey levels:

Z„ = L o G { x „ , y ^ ) - Y , ' L ^ o G { x , y ) [3.5]

(-k, k)

where the kernel size is 2k+l and Xq and yoare the central pixel co-ordinates. However, for the particular size and sigma value of the LoG filters that were used, the new central pixel value Zq w as m uch smaller than the original one,

LoG(xQ,yo), resulting in distortion of the smooth 'Mexican hat' shape of the

filter, and also generating m any erroneous results.

As an alternative approach, the entire LoG filter w as shifted dow nw ards in such a way that its shape w as retained intact. The new m odified version of

chapter 3__________________________________________________ Tumour Cell Nuclei Detection

LoG was named OLGA, which is an acronym for Optimum Laplacian of Gaussian Assimilator.

-CLC 2

= = ^ ( 1 ---[3.6]

(J-

Thus, this procedure achieved the condition: < 0 , since I A \ I > A \; where

A \ and A'2 are the negative and positive parts of the OLGA respectively, and

Z ' are the new coefficients of the filter (see Fig. 3.5). The new central pixel

value is now replaced by + | ^ Z j = , so that

= 0 , and the filter's response is zero whenever convolved w ith a

constant grey-level neighbourhood (note that K corresponds to the original central pixel value of OLGA).

'Positive part Aj

Negative part Positive part A' 2

Negative part A'^

-1

Figure 3.5 A n LoG function with its corresponding OLGA. A new value is

assigned to the central pixel Zq as described in the text.

Finally, Fig. 3.6 illustrates graphically the effect of OLGA in a 2D step edge of size d = 19 pixels. Figure 3.6(d) shows the profile of OLGA's response, whereas the grey area under the peak shows the region detected after applying a negative threshold on the OLGA-convolved image, which results eventually in the detection of the entire black line feature. The application of this concept in a tissue section image will be shown in a later section. Finally, it is worth noticing that the absolute values of a negative, and its closest