Proceedings; XXI International Congress for Photogrammetry and Remote Sensing: Proceedings; XXI International Congress for Photogrammetry and Remote Sensing

Chen, Jun

Bibliographic data

Monograph

Persistent identifier:: 1759415847

Author:: Roscher, Wilhelm

Title:: Geistliche Gedanken eines National-Ökonomen

Sub title:: mit einem Bildnisse des Verfassers aus dem Jahre 1893, in Heliogravüre

Scope:: XXIX, 187 Seiten

Edition title:: Zweites Tausend

Year of publication:: 1895

Place of publication:: Dresden

Publisher of the original:: v. Zahn & Jaensch

Identifier (digital):: 1759415847

Illustration:: 1 Porträt

Signature of the source:: a 1979

Language:: German

Usage licence:: Public Domain Mark 1.0

Publisher of the digital copy:: Technische Informationsbibliothek Hannover

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2021

Document type:: Monograph

Collection:: General

Section

Title:: Geistliche Gedanken.

Document type:: Monograph

Structure type:: Section

Chapter

Title:: Christus mehr als ein großer Theolog und edler Held.

Document type:: Monograph

Structure type:: Chapter

The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 homogeneous regions. to have the same distribution contribute to this value: var(n) w(n, n ) d S pec(v2(n'),Vi(n))w(n, n')(13) n'Gdes(n) w(n, n') ■ a{n') Y2, w(n,n*) ■ a(n*) n* £des(n) (14) The weight w(n, n) between a node n and its descendent n' have to be normalized, so that Yln'edes(n) w ( n i n ') = 1 holds. The variability var(n) of a node n becomes therefore the weighted average of the spectral distances d spe c to its descendents des(n). The area a{n) of node n will be explained in more detail in the next section. 3.3 Node value recalculation V2 (n) ô(w(n, n )) z \ yz v 2 (n) • ö(w(n,ri)) (19) n'€<ies(n) f 1, if w(n, n') > 9 I 0, else S(w(n,n)) n' Edes(n) (20) (21) 3.4 Iterative processing All calculations, in particular adjustments of link strengths and recalculations of the values of each node are done iteratively. The following gives an overview of the algorithm: After the weights of each connection have been adjusted, the val ues of every parent have to be recalculated. Starting at the level 1=1 of the pyramid the values of all nodes in all levels have to be recomputed. The weights used have to depend on the link strength between the two nodes. However, they should also de pend on the size of the image area a(n') represented by the de scended n': Consider a node at a particular level of the pyramid, that has only one strong connection down the pyramid to only one image pixel. This node should have less influence than another node, that covers a large area within the image. a ( n ) = n' £des(n) w(n, n') ■ a(n') Y2 w(n'* ,n') n'* Epar(n') (15) As the sum in the denominator is computed over the parents of n', the area of a node is distributed among its parents in a nor malized way. This ensures that the total area of all nodes at each level is the same as the area of the original image. Every node in the pyramid plays two roles. On the one hand it is the sample covariance matrix of its descendents estimated by weighted averaging. On the other hand it is the descendent of a node on the next level. In order to apply the distance measure (11) it has to be Wishart distributed and the number of looks has to be known. One can show, that the sum of Wishart distributed random variables Xi is again Wishart distributed: Xi ~ W(m, X), * = 1,..., k => Xi ~ W m, X j (16) However, this holds only if all Xi have the same covariance ma trix X and are independent. This assumption should be strongly violated at higher levels of the pyramid, because their nodes cover large regions of the image. Furthermore, a multiplication with a scalar changes the distribution: X~W(n,Z) a-X ~W(n,a-Z) (17) The weighted average can therefore not be assumed to be Wishart distributed. That is why each node holds two values. The first one is simply the weighted average of the values of its descendents and therefore an estimation of the true covariance matrix: Ui(n) = ^2 V2 (n')-w(n,n) (18) n' Gdes(n) where w(n,n') is defined by (14). The second one is the (un weighted) average of the descendents with the strongest connec tions. This ensures, that only descendents which are very likely 0: INIT: Construct pyramid 1: While: levels not converged 1.1: FOR 1=1 TOL 1.1.1: Adjust weights w{n\n l ~ l ) 1.1.2: Recalculate area a(n l ) 1.1.3: Recalculate values v\(n l ) and V2(n l ) 1.1.4: Recalculate variability var{n l ) 2: Construct tree => Extract segments After a few iterations the link strengths will stabilise and not change anymore. At first the level Z = 1 of the pyramid con verges. At this time each node at this level will have strong links only to that subset of pixels in the set of descendents, which are very likely to have the same distribution governed by covariance matrix X of which the node value is an estimation. The sec ond value of a node at level l = 1 will now be an (unweighted) average of Wishart distributed random variables of the same dis tribution. That is why its own distribution can be assumed to be Wishart, too. However, the averaged variables cannot be assumed to be independent, because of the possible overlap of their areas in the image. Therefore the number of looks is estimated by the area the node covers in the original image and not by the number of looks of its descendents. All levels will converge in ascending order after a few iterations. 3.5 Tree construction If the whole pyramid has converged, meaning that link strengths and, therefore, values of all nodes do not change anymore, there are two general types of nodes in the pyramid. On the one hand, nodes that have strong connections to one or more parents and, on the other hand, nodes which have no strong connection to any par ent at all. Latter ones define roots of independent subtrees within the pyramid and represent homogeneous regions in the image. All nodes at the top level of the pyramid or nodes whose link strengths to all their parents are below a certain threshold are con sidered as such roots. However, because of the different statistics at each level, e.g. the mean number of looks decreases with de creasing height, one cannot use a global threshold. But there is in each level an abrupt rise in the number of roots for a certain value t r ■ This value is used as threshold to define the roots in each level. Figure 1 shows an example of the relationship be tween the number of roots at a certain level and the threshold to define them.

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