Keywords: Real 3D-City Surface, Image Inversion, Wavelets, Fractal

Abstract
This paper proposes a wavelet-based algorithm to describe efficiently any 'real' signal with some distinctive properties which generally cannot be represented accurately by conventional methods that describe only a smooth approximation. This paper takes a real 3D-city surface as an example of such a signal and presents a way to automatic and high precision reconstruction of a real 3D-city surface using an image inversion method. For that purpose, some preliminary tests using simulated data and real aerial images are done to examine the applicability of the proposed algorithm. The test results show that high precision representation of a real 3D-city surface is realizable.

Introduction
The state-of-the-art computer science and technology enable an operational realization of the so-called 'virtual reality'. It causes a revolutionary progress on the representation form of surveyed object(s), namely from a conventional 2D-map to an easy-to-learn 3D virtual reality about surveyed object(s). A conventional 2D-map is not very user-friendly and loses much object information, whereas a 3D virtual reality model is really easy-to-learn and is a minified 3D model of surveyed object(s). To realize dynamical simulation analysis and representation of such a virtual reality of interest objects, many researches in the field of surveying aim also at automation technologies utilizing multi-sensor, multi-source and/or multi-temporal data (and information as well). For example, an automatic reconstruction and thematic information extraction of a geometric and radiometric 3D-city model is a hot topic.

This paper presents some preliminary study results. They aim at a research project, namely a (semi-) automatic reconstruction and analysis of a real 3D-city surface. The proposed models and algorithms should also be available for another kinds of real signals, e.g. a 3D sea surface, which have some distinctive properties that traditional methods can not describe exactly.

Properties of a Real 3D-City Surface
A 3D-city surface can be regarded as a continuous 'signal' of surface height or surface radiometric intensity, where the so-called 'signal' is generally interpreted as 'a sequence of digital data' (Meyer and Ryan, 1994). Those real signals such as a geometrical 3D-city surface have almost always the following four distinctive properties:

1. having local surfaces with (pseudo) break points or –lines,
2. multi-resolution,
3. various degree of continuity on different surface points, and
4. various fractal dimension, where detailed definition of 'fractal dimension' please see (Mandelbrot, 1982; Farge, et. al., 1993).

For example, (pseudo) break points or –lines often can be found on walls right round a building. Blocks of great and high buildings, blocks of small and low houses, visible surfaces of separate local trees/forests and some local naked natural earth surface as well compose a multi-resolution city surface with the properties (2) ~ (4).

An Idea
An idea aims at a research project, namely automatic and high precision reconstruction and analysis of a 'real' 3D-city surface having the above-mentioned properties (1) ~ (4), where one uses orthonormal fractal wavelets, e.g. the compactly supported Daubechies wavelets derived from minimum phase filters (Daubechies, 1992), to define a feasible functional model in the object space to describe a real geometric and/or radiometric city surface, and utilizes the principles of 'image inversion', 'sampling theorem' and 'least squares estimation technique’ under the application of the 'from-coarse-to-fine' strategy. The main idea is similar to the method of facets stereo vision (Tsay, 1998). It is a method for performing digital terrain reconstruction and ortho image computation, where digital images are the basic observations. It can also integrate in principle multi-sensor and multi-source data to provide reliable and accurate results for photogrammetry and remote sensing. The first draft of facets stereo vision was proposed publicly in 1987 (Wrobel, 1987).

The idea would further extend and improve the method of (Tsay, 1998) with the new capability for representing a real signal e.g. with the fore-mentioned properties (1)~(4). More explicitly to say, the idea aims for an automatic image inversion method to get object information. The method must be able to represent fractal geometry. For the present, only fractal geometry can correctly describe and analyze real natural objects. Therefore, fractal geometry is apparently the correct mathematics (Jaehne, 1991).

For that purpose, a thorough research of models and algorithms for representing a signal having the properties (1)~(4) is done, where some known simulated and representative 2D signals, chiefly 2D-city profile functions, and a 3D-city surface function are used to do the van study, especially about the characteristics and accuracy of the proposed algorithm for representing the above-mentioned signals.

Mathematical Models for Representing a Real Signal
The algorithm starts from the mathematical models based on some special orthonormal wavelets with the ability for displaying fractal signals. In fact, the compactly supported orthonormal wavelets of Daubechies display a fractal geometry (Kaiser, 1994).

Mathematical Models
An approximation of a real signal f with finite energy can be represented by:

where

f and y. are an orthonormal father and mother wavelet function, respectively (Antonini, et. al., 1992; Daubechies, 1994; Meyer and Ryan, 1994).

The 2D-model can be expressed similarly to the equation (1), e.g. see (Mallat, 1998).

Models' Characteristics
In general, such models are based on those orthonormal fractal wavelets that cannot, in general, be written in a closed analytic form. Their graph and the other mathematical operations in practice, such as integration, differentiation, addition, multiplication, division, etc., can be computed with arbitrarily high precision by some specified methods, please see (Tsay, 1996).