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Hough Line Transform

Goal

In this tutorial you will learn how to:

Theory

Note

The explanation below belongs to the book Learning OpenCV by Bradski and Kaehler.

Hough Line Transform

  1. The Hough Line Transform is a transform used to detect straight lines.
  2. To apply the Transform, first an edge detection pre-processing is desirable.

How does it work?

  1. As you know, a line in the image space can be expressed with two variables. For example:

    1. In the Cartesian coordinate system: Parameters:

      System Message: WARNING/2 ((m,b))

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      .
    2. In the Polar coordinate system: Parameters:

      System Message: WARNING/2 ((r,\theta))

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    Line variables

    For Hough Transforms, we will express lines in the Polar system. Hence, a line equation can be written as:

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Arranging the terms:

System Message: WARNING/2 (r = x \cos \theta + y \sin \theta)

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  1. In general for each point

    System Message: WARNING/2 ((x_{0}, y_{0}))

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    , we can define the family of lines that goes through that point as:

    System Message: WARNING/2 (r_{\theta} = x_{0} \cdot \cos \theta + y_{0} \cdot \sin \theta)

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    Meaning that each pair

    System Message: WARNING/2 ((r_{\theta},\theta))

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    represents each line that passes by

    System Message: WARNING/2 ((x_{0}, y_{0}))

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    .

  2. If for a given

    System Message: WARNING/2 ((x_{0}, y_{0}))

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    we plot the family of lines that goes through it, we get a sinusoid. For instance, for

    System Message: WARNING/2 (x_{0} = 8)

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    and

    System Message: WARNING/2 (y_{0} = 6)

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    we get the following plot (in a plane

    System Message: WARNING/2 (\theta)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    -

    System Message: WARNING/2 (r)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    ):

    Polar plot of a the family of lines of a point

    We consider only points such that

    System Message: WARNING/2 (r > 0)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    and

    System Message: WARNING/2 (0< \theta < 2 \pi)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    .

  3. We can do the same operation above for all the points in an image. If the curves of two different points intersect in the plane

    System Message: WARNING/2 (\theta)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    -

    System Message: WARNING/2 (r)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    , that means that both points belong to a same line. For instance, following with the example above and drawing the plot for two more points:

    System Message: WARNING/2 (x_{1} = 9)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    ,

    System Message: WARNING/2 (y_{1} = 4)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    and

    System Message: WARNING/2 (x_{2} = 12)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    ,

    System Message: WARNING/2 (y_{2} = 3)

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    , we get:

    Polar plot of the family of lines for three points

    The three plots intersect in one single point

    System Message: WARNING/2 ((0.925, 9.6))

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.

    , these coordinates are the parameters (

    System Message: WARNING/2 (\theta, r)

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    ) or the line in which

    System Message: WARNING/2 ((x_{0}, y_{0}))

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    ,

    System Message: WARNING/2 ((x_{1}, y_{1}))

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    and

    System Message: WARNING/2 ((x_{2}, y_{2}))

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    lay.

  4. What does all the stuff above mean? It means that in general, a line can be detected by finding the number of intersections between curves.The more curves intersecting means that the line represented by that intersection have more points. In general, we can define a threshold of the minimum number of intersections needed to detect a line.

  5. This is what the Hough Line Transform does. It keeps track of the intersection between curves of every point in the image. If the number of intersections is above some threshold, then it declares it as a line with the parameters

    System Message: WARNING/2 ((\theta, r_{\theta}))

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    of the intersection point.

Standard and Probabilistic Hough Line Transform

OpenCV implements two kind of Hough Line Transforms:

  1. The Standard Hough Transform
  • It consists in pretty much what we just explained in the previous section. It gives you as result a vector of couples

    System Message: WARNING/2 ((\theta, r_{\theta}))

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  • In OpenCV it is implemented with the function HoughLines
  1. The Probabilistic Hough Line Transform
  • A more efficient implementation of the Hough Line Transform. It gives as output the extremes of the detected lines

    System Message: WARNING/2 ((x_{0}, y_{0}, x_{1}, y_{1}))

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  • In OpenCV it is implemented with the function HoughLinesP

Code

  1. What does this program do?
    • Loads an image
    • Applies either a Standard Hough Line Transform or a Probabilistic Line Transform.
    • Display the original image and the detected line in two windows.
  2. The sample code that we will explain can be downloaded from here. A slightly fancier version (which shows both Hough standard and probabilistic with trackbars for changing the threshold values) can be found here.
#include "opencv2/highgui/highgui.hpp"
#include "opencv2/imgproc/imgproc.hpp"

#include <iostream>

using namespace cv;
using namespace std;

void help()
{
 cout << "\nThis program demonstrates line finding with the Hough transform.\n"
         "Usage:\n"
         "./houghlines <image_name>, Default is pic1.jpg\n" << endl;
}

int main(int argc, char** argv)
{
 const char* filename = argc >= 2 ? argv[1] : "pic1.jpg";

 Mat src = imread(filename, 0);
 if(src.empty())
 {
     help();
     cout << "can not open " << filename << endl;
     return -1;
 }

 Mat dst, cdst;
 Canny(src, dst, 50, 200, 3);
 cvtColor(dst, cdst, CV_GRAY2BGR);

 #if 0
  vector<Vec2f> lines;
  HoughLines(dst, lines, 1, CV_PI/180, 100, 0, 0 );

  for( size_t i = 0; i < lines.size(); i++ )
  {
     float rho = lines[i][0], theta = lines[i][1];
     Point pt1, pt2;
     double a = cos(theta), b = sin(theta);
     double x0 = a*rho, y0 = b*rho;
     pt1.x = cvRound(x0 + 1000*(-b));
     pt1.y = cvRound(y0 + 1000*(a));
     pt2.x = cvRound(x0 - 1000*(-b));
     pt2.y = cvRound(y0 - 1000*(a));
     line( cdst, pt1, pt2, Scalar(0,0,255), 3, CV_AA);
  }
 #else
  vector<Vec4i> lines;
  HoughLinesP(dst, lines, 1, CV_PI/180, 50, 50, 10 );
  for( size_t i = 0; i < lines.size(); i++ )
  {
    Vec4i l = lines[i];
    line( cdst, Point(l[0], l[1]), Point(l[2], l[3]), Scalar(0,0,255), 3, CV_AA);
  }
 #endif
 imshow("source", src);
 imshow("detected lines", cdst);

 waitKey();

 return 0;
}

Explanation

  1. Load an image

    Mat src = imread(filename, 0);
if(src.empty())
{
  help();
  cout << "can not open " << filename << endl;
  return -1;
}

  2. Detect the edges of the image by using a Canny detector

    Canny(src, dst, 50, 200, 3);

    Now we will apply the Hough Line Transform. We will explain how to use both OpenCV functions available for this purpose:

  3. Standard Hough Line Transform

    1. First, you apply the Transform:

      vector<Vec2f> lines;
HoughLines(dst, lines, 1, CV_PI/180, 100, 0, 0 );

      with the following arguments:

      • dst: Output of the edge detector. It should be a grayscale image (although in fact it is a binary one)
      • lines: A vector that will store the parameters

        System Message: WARNING/2 ((r,\theta))

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        of the detected lines
      • rho : The resolution of the parameter

        System Message: WARNING/2 (r)

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        in pixels. We use 1 pixel.
      • theta: The resolution of the parameter

        System Message: WARNING/2 (\theta)

        latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.
        in radians. We use 1 degree (CV_PI/180)
      • threshold: The minimum number of intersections to “detect” a line
      • srn and stn: Default parameters to zero. Check OpenCV reference for more info.

    2. And then you display the result by drawing the lines.

      for( size_t i = 0; i < lines.size(); i++ )
{
  float rho = lines[i][0], theta = lines[i][1];
  Point pt1, pt2;
  double a = cos(theta), b = sin(theta);
  double x0 = a*rho, y0 = b*rho;
  pt1.x = cvRound(x0 + 1000*(-b));
  pt1.y = cvRound(y0 + 1000*(a));
  pt2.x = cvRound(x0 - 1000*(-b));
  pt2.y = cvRound(y0 - 1000*(a));
  line( cdst, pt1, pt2, Scalar(0,0,255), 3, CV_AA);
}

  4. Probabilistic Hough Line Transform

    1. First you apply the transform:

      vector<Vec4i> lines;
HoughLinesP(dst, lines, 1, CV_PI/180, 50, 50, 10 );

      with the arguments:

      • dst: Output of the edge detector. It should be a grayscale image (although in fact it is a binary one)
      • lines: A vector that will store the parameters

        System Message: WARNING/2 ((x_{start}, y_{start}, x_{end}, y_{end}))

        latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.
        of the detected lines
      • rho : The resolution of the parameter

        System Message: WARNING/2 (r)

        latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.
        in pixels. We use 1 pixel.
      • theta: The resolution of the parameter

        System Message: WARNING/2 (\theta)

        latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) entering extended mode (./math.tex LaTeX2e <2009/09/24> Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh yphenation, loaded. (/usr/share/texmf-texlive/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf-texlive/tex/latex/base/size12.clo)) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty (/usr/share/texmf-texlive/tex/latex/ucs/utf8x.def)) (/usr/share/texmf-texlive/tex/latex/ucs/ucs.sty (/usr/share/texmf-texlive/tex/latex/ucs/data/uni-global.def)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty (/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty)) (/usr/share/texmf-texlive/tex/latex/amscls/amsthm.sty) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty (/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texmf-texlive/tex/latex/tools/bm.sty) (/usr/share/texmf-texlive/tex/latex/euler/euler.sty Package: `euler’ v2.5 <1995/03/05> (FJ and FMi) ) (/usr/share/texmf-texlive/tex/latex/graphics/color.sty (/etc/texmf/tex/latex/config/color.cfg) (/usr/share/texmf-texlive/tex/latex/graphics/dvips.def) (/usr/share/texmf-texlive/tex/latex/graphics/dvipsnam.def)) ! LaTeX Error: File `bbm.sty’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: sty) Enter file name: ! Emergency stop. <read *> l.11 ...ackage{amsmath}\usepackage{bbm}\usepackage {colortbl}^^M No pages of output. Transcript written on math.log.
        in radians. We use 1 degree (CV_PI/180)
      • threshold: The minimum number of intersections to “detect” a line
      • minLinLength: The minimum number of points that can form a line. Lines with less than this number of points are disregarded.
      • maxLineGap: The maximum gap between two points to be considered in the same line.

    2. And then you display the result by drawing the lines.

      for( size_t i = 0; i < lines.size(); i++ )
{
  Vec4i l = lines[i];
  line( cdst, Point(l[0], l[1]), Point(l[2], l[3]), Scalar(0,0,255), 3, CV_AA);
}

  5. Display the original image and the detected lines:

    imshow("source", src);
imshow("detected lines", cdst);

  6. Wait until the user exits the program

    waitKey();

Result

Note

The results below are obtained using the slightly fancier version we mentioned in the Code section. It still implements the same stuff as above, only adding the Trackbar for the Threshold.

Using an input image such as:

Result of detecting lines with Hough Transform

We get the following result by using the Probabilistic Hough Line Transform:

Result of detecting lines with Hough Transform

You may observe that the number of lines detected vary while you change the threshold. The explanation is sort of evident: If you establish a higher threshold, fewer lines will be detected (since you will need more points to declare a line detected).

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