Multiscale representation

Multiscale techniques have been developed to provide a way to isolate, analyse and interpret structures of different scales within an image [13]. The main idea of a multiscale representation is to generate a one-parameter family of derived images $I(x,y;s)$ by convolving the original image $I(x,y)$ with a Gaussian kernel $G(x,y;s)$ of variance $s^{2}$:

$$I(x,y;s)= I(x,y) \otimes G(x,y;s)$$ (1)

where $G$ is:

$$G(x,y;s)=\frac{1}{2\pi s^{2}} e^{-\frac{x^2 + y^2}{2 s^{2}}}$$ (2)

and s is a length scale factor. The effect of convolving a signal by a Gaussian kernel is to suppress most of the structures in the signal with a characteristic length less than s. Figure 2 shows different scales representations of a portion of a negative red-free retinal image, for s = 2, 8 and 14 pixels, showing the progressive blurring of the image as the scale factor increases.

Figure 2: Multiscale analysis for $s=$ 0, 2, 8 and 14 pixels of a portion ($720\times 580$ pixels) of a red-free retinal image ( $2800\times 2400$ pixels).

The use of Gaussian kernels to generate multiscale information ensures that the image analysis is invariant with respect to translation, rotation and size [14,15]. The derivatives of an image are defined as the linear convolution of the image with scale-normalised derivatives of the Gaussian kernel.

$$\partial^{n} I(x,y;s)=I(x,y) \otimes s^{n} \partial^{n} G(x,y;s)$$ (3)

where $n$ indicates the order of the derivative.

The normalisation by scale makes the derivatives dimensionless which means that the derivatives will have the proper behaviour under spatial rescaling of the original image and that structures at different scales will be treated in a similar manner. An approach advocated by Koenderink [14] is to describe image properties in terms of differential geometric descriptors which constitutes a natural framework for expressing both physical processes and geometrical properties. For retinal blood vessels we suggest that gradient magnitudes and principal curvatures of the Hessian tensor are the most useful geometrical descriptors of the vessels which can be extracted.