Jens Lund
February 23, 1999
This is a note explaining the status of my PhD project and the plans for the thesis. One article is submitted, two is well underway, and one or two more are planned. The requirement for PhD courses is fulfilled and there has been two longer visits at foreign departments.
This paper is the written note required in connection with the midterm seminars for PhD students at KVL. This paper is organized as follows: this introduction explains the background for the project and the overall plan for the thesis. Sections 2 and 3 explains the work already done or well underway, whereas Section 4 discusses the plan for the future work. Appendices A, B, and C list my planned publications, longer visits, and PhD courses, respectively.
The overall idea in the project is to identify tree tops in aerial photos of Norway spruce. The project can be seen as a continuation of [Dra97] (especially [DR96] and [DR97]), and is very closely connected to [Lar97] and [LR98a].
In forestry there is a wish to get information about the number and volume of trees in the forest. This will help management and planning of the forest, as well as the positions of the trees. So far, the expenses in terms of working hours and money to obtain this information has prevented a detailed inventory. Analysis of aerial photos seem to promise a less expensive way to get detailed information on the individual tree. Further background information can be found in [Dra97].
The plan is to make the thesis from the papers listed in Appendix A together with one or more introductory sections and a discussion. One of the five publications listed in Appendix A is submitted and two is well underway. Work on the publications 4 and 5 has not started yet, and they are further discussed in Section 4.
This section describes the work in connection with estimation of tree top positions in aerial photos.
The upper part of Figure 1 shows an aerial photo from a flight 560m above a thinning experiment in Norway spruce. The goal is to identify the positions of tree tops in the image.
Figure: Image with sidelighted trees, and, in the lower part,
171 X-points (centres of circles) corresponding to `true' tree
tops and 206 Y-points (dots) corresponding to template
matching. The area of the delineated subplot is 4454 m 
,
and the unit of the axes in the lower part is linear pixel size,
0.15 m.
Morten Larsen has in [Lar97] and [LR98a] developed a template model for one tree taking into account the positions of the camera and light sources. The resulting template, shown in the right part of Figure 2 bounded by an ellipse, was moved pixelwise over the image. Local maxima of the correlation between template and image pixel grey levels were considered as candidate positions of tree tops. The left part of Figure 2 sketches the model for light reflection within a tree.
Figure 2: Model tree and, in the right part, template with optimal
bounding ellipse.
The lower part of Figure 1 shows a map of the
image in the upper part. Both the true positions of tree tops
(denoted by circles) and the estimated positions as found by the
template matching method (denoted by dots) are shown. Let
, 
, denote the positions of
the true points and let 
, 
denote the positions found by the template matching method. We want
to estimate the true positions X but do instead observe Y. The
template method in [LR98a] gives 570 candidate positions
for tree tops, but only the 206 best candidates are used here.
See [LR98b] for full details.
As noted above, the X and Y point sets are not identical. The idea in the inclusion of more Y points than the present number of X points in the data is two fold:
we
hope to improve the estimation of X (and derived statistics)
compared to [LR98a]. This is the goal of
Section 2.4.
.
Suppose that Y is generated from the X-process by the following disturbance mechanisms:
, is thinned with
probability 
and retained with probability 
. If
an X-point is thinned, then there will not be any corresponding
Y-point. Thinnings are assumed to be independent for different
points.
a
corresponding Y-point is generated by displacement to a position
with probability density 
with respect to the
Lebesgue measure on . Given X, the displacements of
different points are independent, mutually and of the thinnings.
occurs with probability
. (Here 
denotes the complement
of the set A.)
, where X, as
above, denotes the entire X-process including thinned points.
Given X, the ghost points are assumed to be independent of
thinning, displacement and censoring.
So far I have used a homogeneous thinning probability, p(x)=p, a
normal distribution as the displacement distribution, and
homogeneous poisson noise 
.
We let 
denote the set of one-to-one mappings 
for two finite sets M and M' with the same
number of elements. The likelihood function for the observation of
Y given X is given by Theorem 1.
Estimation of the parameters in the model (1) is
difficult due to the very large sum. The sum is over all possible
ways to pair X and Y points. We will call such a pairing
a state. The state tells which Y points that
comes from the Poisson noise, and which Y points that are paired
to which X points. We will consider estimation of the parameters
in the model in case both X and Y is observed.
The problem can be considered as a missing data problem with the
information on the state s as the missing data. I have suggested a
method based on the concept of ``neighbours'' to a state s that
does approximate likelihood analysis. The crucial issue in the
approximate likelihood computation is to find states
such that the corresponding terms give large
contributions to (1), and then focus on only a small
number of terms. This is achieved by use of a
deterministic, iterative algorithm which consists of a starting
procedure for finding an initial set of states and local
maximizations over suitably chosen neighbourhoods of states until no
further improvement is obtained. A neighbour of a state s is a
state s' which is only slightly different from s.
The paper [LR98b] covers the problems presented in Section 2.2 and 2.3. This paper is finished and submitted.
The paper [LPR99] covers the problems described in this section. It is work in progress and it is estimated to be finished in April.
The idea is to use the model (1) to reconstruct X when
the parameters p, 
, ..., are known. The knowledge of
the parameters can e.g. be obtained through training data sets
observed under similar conditions as the current data.
Problems similar to this one can be found in [BvL93], [CL98], [DR95]. The first is about a point process description of an image, and the latter two is about detection of mines in minefields.
We will use a Bayesian approach and have a prior distribution on X. The prior distribution should be a regular point pattern, which expresses that trees in planted and managed forests (as our images) tend to be placed regularly in the area.
Next, our problem is to say something about the posterier
distribution
. We
will explore this posterior distribution by MCMC samples from the
distribution. We make a sampler of Metropolis-Hasting type along the
lines in [Gre95] and [GM94]. Some inital
experiments with this sampler is shown in Figures 3,
4, and 5.
Figure 3: The number of simulated X point and the number of matched X points in a simulation of length 10000 starting from the empty point configuration.
Figure: Same as Figure 3, but without the first 1000 observations.
Figure 5: Simulation number 9999. Both the true X points (circles),
the observed Y points (dots) and the simulated X points
(x's) are marked.
The above simulations are made with a Poisson process as the prior distribution. The plan for the paper is to use a prior with regular point patterns, but some minor problems must be solved. Also some work on how to best present the results remain.
The work described in this section is documented in [Lun98] and is not connected to the work described in Section 2. A model for survival times of Sitka spruce based on a discretised version of a Cox model is suggested. The positions af all the trees are known, and a competition index is used as a covariate to take the spatial competition between trees into account. The plan is to write a joint paper together with Jens Peter Skovsgaard based on the course report.
In this section some ideas for the future work is outlined.
In continuation of the work in Section 2 it is quite natural to improve the estimation of the positions by use of several images at the same time. The idea is to estimate the tree top positions in 3-dimensional real world coordinates. Each image only contains information on 2 dimensions and it might not be the same trees that are found in every image. My impression is that a pretty good reconstruction can be made by, say, 3-4 images. A lot of the ground work for this is already done, but of course there is still a lot of outstanding issues.
The thesis could need a more theoretical paper than the ones mentioned above. One idea might be to look into the area of perfect simulation. During my stay in Jyväskylä I participated in a study group about some articles on perfect simulation, and the summer school on coupling methods in Göteborg this summer mentioned in Appendix C is also relevant for this subject.
This summer I have the following plans for conferences, etc.:
Under consideration is the following events:
This section describes the planned articles for the thesis.
Two longer visits have taken place:
A visit at Chalmers during the visitors programme in April-May 1999 is considered.
The requirement for PhD courses is fulfilled by these courses.
I have considered participating in a summer school ``Coupling methods in probability'', Göteborg, June 14-19, 1999.
Spatial models related to trees
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