Fast Point Feature Histograms (FPFH) descriptors
The theoretical computational complexity of the Point Feature Histogram (see
Point Feature Histograms (PFH) descriptors) for a given point cloud
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points
is
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tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
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is the number of neighbors for each point
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. For real-time or near real-time applications, the
computation of Point Feature Histograms in dense point neighborhoods can
represent one of the major bottlenecks.
This tutorial describes a simplification of the PFH formulation, called Fast
Point Feature Histograms (FPFH) (see [RusuDissertation] for more information),
that reduces the computational complexity of the algorithm to
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,
while still retaining most of the discriminative power of the PFH.
Theoretical primer
To simplify the histogram feature computation, we proceed as follows:
- in a first step, for each query point
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a set of tuples
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between itself and its neighbors are computed
as described in Point Feature Histograms (PFH) descriptors - this will be called the Simplified
Point Feature Histogram (SPFH);
- in a second step, for each point its k neighbors are re-determined, and the
neighboring SPFH values are used to weight the final histogram of pq
(called FPFH) as follows:
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where the weight
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represents a distance between the query point
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and a neighbor point
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in some given metric space, thus
scoring the (
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LaTeX2e <2005/12/01>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar
abic, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutc
h, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, mono
greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la
tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
sian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turkish, ukenglis
h, ukrainian, uppersorbian, welsh, loaded.
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) pair, but could just as well be selected as a
different measure if necessary. To understand the importance of this weighting
scheme, the figure below presents the influence region diagram for a
k-neighborhood set centered at
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This is pdfTeXk, Version 3.1415926-1.40.9 (Web2C 7.5.7)
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entering extended mode
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LaTeX2e <2005/12/01>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar
abic, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutc
h, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, mono
greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la
tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
sian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turkish, ukenglis
h, ukrainian, uppersorbian, welsh, loaded.
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Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-dist/tex/latex/base/size12.clo))
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.
Thus, for a given query point
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This is pdfTeXk, Version 3.1415926-1.40.9 (Web2C 7.5.7)
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LaTeX2e <2005/12/01>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar
abic, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutc
h, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, mono
greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la
tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
sian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turkish, ukenglis
h, ukrainian, uppersorbian, welsh, loaded.
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Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-dist/tex/latex/base/size12.clo))
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, the algorithm first estimates its
SPFH values by creating pairs between itself and its neighbors (illustrated
using red lines). This is repeated for all the points in the dataset, followed
by a re-weighting of the SPFH values of pq using the SPFH values of its
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This is pdfTeXk, Version 3.1415926-1.40.9 (Web2C 7.5.7)
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LaTeX2e <2005/12/01>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar
abic, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutc
h, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, mono
greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la
tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
sian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turkish, ukenglis
h, ukrainian, uppersorbian, welsh, loaded.
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Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
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neighbors, thus creating the FPFH for
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This is pdfTeXk, Version 3.1415926-1.40.9 (Web2C 7.5.7)
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LaTeX2e <2005/12/01>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, german-x-2008-06-18, ngerman-x-2008-06-18, ancientgreek, ibycus, ar
abic, basque, bulgarian, catalan, pinyin, coptic, croatian, czech, danish, dutc
h, esperanto, estonian, farsi, finnish, french, galician, german, ngerman, mono
greek, greek, hungarian, icelandic, indonesian, interlingua, irish, italian, la
tin, mongolian, mongolian2a, bokmal, nynorsk, polish, portuguese, romanian, rus
sian, sanskrit, serbian, slovak, slovenian, spanish, swedish, turkish, ukenglis
h, ukrainian, uppersorbian, welsh, loaded.
(/usr/share/texmf-dist/tex/latex/base/article.cls
Document Class: article 2005/09/16 v1.4f Standard LaTeX document class
(/usr/share/texmf-dist/tex/latex/base/size12.clo))
(/usr/share/texmf-dist/tex/latex/base/inputenc.sty
(/usr/share/texmf-dist/tex/latex/ucs/utf8x.def))
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. The extra FPFH
connections, resultant due to the additional weighting scheme, are shown with
black lines. As the diagram shows, some of the value pairs will be counted
twice (marked with thicker lines in the figure).