Modelling surface topography from reflected light.

This thesis is concerned with the use of the modulus of the Fourier spectrum to characterise object features and also to reconstruct object surfaces in the complete absence of phase information. In general, a phaseless synthesis is completely meaningless and many characteristic features of the object are obliterated when the modulus of the spectral components is inverse Fourier transformed with zero phase. However, the outcome is different when the object possesses some form of regularity and repetition in its characteristics. In such circumstances, the utilisation of both the modulus and the intensity of the spatial spectrum can reveal information regarding the characteristic features of the object surface.
The first part of this research has utilised the intensity of the spectral components as a means of surface feature characterisation in the study of a machined surface. Two separate approaches were adopted for assessing the zero-phase images. Both the optically recorded Fourier spectrum and the computer simulated Fourier spectrum were used to extract surface related parameters in the zero-phase synthesis. Although merely a characterisation, the zero-phase synthesis of the spectral components revealed periodic behaviour very similar to that present in the original surface. The presence of such cyclic components was confirmed by their presence in travelling microscope images and in scanning electron microscope images of the surface.
Additionally, a novel approach has been adopted to recover finer periodicities on the surface. The scale sensitivity of the frequency domain fosters an exceptional means through which digital magnification can be performed with the added advantage that it is accompanied by enhanced resolution. Magnification realised through spatial frequency data is by far superior to any spatial domain magnification. However, there are limitations to this approach.
The second part of this research has been centred around the possible use of a non-iteratively based approach for extracting the unknown phases from the modulus of the Fourier spectrum and thus retrieving the 3-D geometrical structure of the unknown object surface as opposed to characterising its profile. The logarithmic Hilbert transform is one such approach which allows a non-iterative means of extracting unknown phases from the modulus of the Fourier spectrum. However, the technique is only successful for object surfaces which are well-behaved and display well-behaved spectral characteristics governed by continuity. For real object surfaces where structure, definition and repetition governs the characteristics, the spectrum is not well behaved. The spectrum is populated by maxi ma, minima and many isolated regions which are occupied by colonies of zeros disrupting the continuity.
A new and unique approach has been devised by the author to reform the spectral behaviour of real object surfaces without affecting the fidelity that it conveys. The resultant information enables phase extraction to be achieved through the logarithmic Hilbert transform. It is possible to reform the spread of spectral behaviour to cultivate better continuity amongst its spectral components through an object scale change. The combination of the logarithmic Hilbert transform and the Fourier scaling principle has led to a new approach for extracting the unknown phases for real object structures which would otherwise have been impossible to perform through the use of Hilbert transformation alone. The validity of the technique has been demonstrated in a series of simulations conducted on one-dimensional objects as well as the two-dimensional object specimens. The limitations of the approach, improvements and the feasibility for practical implementation are ail issues which have been addressed.

A thesis submitted to Middlesex University in partial fulfilment of the requirements for the degree of Doctor of Philosophy.