{"id":1954,"date":"2014-02-05T17:32:47","date_gmt":"2014-02-05T17:32:47","guid":{"rendered":"https:\/\/www.anagram.at\/en\/diplomarbeit\/the-pinhole-camera\/"},"modified":"2014-02-05T17:32:47","modified_gmt":"2014-02-05T17:32:47","slug":"the-pinhole-camera","status":"publish","type":"page","link":"https:\/\/www.anagram.at\/en\/diplomarbeit\/the-pinhole-camera\/","title":{"rendered":"The Pinhole Camera"},"content":{"rendered":"


\n
\n Next:<\/b> Camera Model with Lens<\/a>
\n Up:<\/b> Perspective Projection<\/a>
\n Previous:<\/b> Perspective Projection<\/a>
\n<\/p>\n


\nThe Pinhole Camera<\/a>
\n<\/h2>\n

\nAn ideal model of a camera is the pinhole camera, as seen in Figure 2.2<\/a>. This kind of
\ncamera can be imagined as a box with a pinhole, through which
\nlight enters and forms a two-dimensional image on the opposite site. A point
\n\"$ in the three-dimensional \"$-space is projected to an image-point
\n\"$ in the two-dimensional \"$-space (wall). If the coordinate system of the
\n\"$-space is aligned at the pinhole so that the Z-axis coincides with the optical axis and the image
\nplane has its origin at \"$, then the projection equations are
\ngiven by<\/p>\n<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.4)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\nand <\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.5)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\n

<\/p>\n\n\n
Figure 2.2:<\/strong>
\nPinhole camera model<\/caption>\n
\n
\n \"Image<\/div>\n<\/td>\n<\/tr>\n<\/table>\n<\/div>\n

\nTo represent Equation 2.4<\/a> and Equation 2.5<\/a> in a linear way, we transform the point \"$ in the Euclidean plane to a point \"$ in the projective plane. This represents the same point, we simply added a new coordinate \"$. Overall scaling is unimportant. The point
\n\"$ can be re-transformed by dividing through \"$. Thus \"$ is similar to
\n\"$. Because scaling is unimportant, the coordinate \"$ is called homogeneous coordinate<\/i>. Homogenous coordinates can also be used within a higher dimensional domain. Now we can combine Equation 2.4<\/a> and Equation 2.5<\/a> to <\/p>\n<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.6)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\n

\nIf we want to know the image coordinates we have to take four more values into account. Namely\n<\/p>\n

\n
\"$<\/strong><\/dt>\n
image principal point, which is the intersection between the camera’s optical axis and the image plane.\n<\/dd>\n
\"$<\/strong><\/dt>\n
distance between two sensor elements in \"$ and \"$ direction\n<\/dd>\n<\/dl>\n

\"$ and \"$ can normally be found in the datasheet2.2<\/sup><\/a> of the sensor chip. The image principal point has to be found by calibration (see Section 2.2<\/a> for more information). If the parameters and the point in camera coordinates are known, we can compute the image coordinates with the following formulation<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.7)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\nand <\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.8)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\nwhere \"$ and \"$ are the coordinates of the point in the camera coordinate system. If we combine Equation 2.6<\/a> with Equation 2.7<\/a> and Equation 2.8<\/a> we can formulate the translation from Point \"$ to the image coordinates
\n\"$<\/p>\n\n

<\/p>\n\n\n
\"$displaystyle<\/td>\n\n(2.9)<\/td>\n<\/tr>\n<\/table>\n<\/div>\n


<\/p>\n\n\n

Next:<\/b> Camera Model with Lens<\/a>
\n Up:<\/b> Perspective Projection<\/a>
\n<\/p>\n

<\/body><\/p>\n","protected":false},"excerpt":{"rendered":"

The Pinhole Camera<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1946,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":""},"categories":[],"featured_image_src":null,"featured_image_src_square":null,"_links":{"self":[{"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/pages\/1954"}],"collection":[{"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/comments?post=1954"}],"version-history":[{"count":0,"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/pages\/1954\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/pages\/1946"}],"wp:attachment":[{"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/media?parent=1954"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.anagram.at\/en\/wp-json\/wp\/v2\/categories?post=1954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}