page_title: Mojette Transform
section_title: Mathematics
hierarchical_section_title: Mojette Transform / Mathematics
caption_attribution_description: Français&#160;: les projections de direction (p,q) de la grille 4x4
context_page_description: The Mojette Transform is an application of discrete geometry. More specifically, it is a discrete and exact version of the Radon transform, thus a projection operator.
  The IRCCyN laboratory - UMR CNRS 6597 in Nantes, France has been developing it since 1994.
  The first characteristic of the Mojette Transform is using only additions and subtractions. The second characteristic is that the Mojette Transform is redundant, spreading the initial geometrical information into several projections.
  This transform uses discrete geometry in order to dispatch information onto a discrete geometrical support. This support is then projected by the Mojette operator along discrete directions. When enough projections are available, the initial information can be reconstructed.
  The Mojette transform has been already used in numerous applications domains:
  Medical tomography
  Network packet transfer
  Distributed storage on disks or networks
  Image fingerprinting and image cryptography schemes
context_section_description: The "raw" transform Mojette definition is this:
  The following figure 1 helps to explain the “raw” transform Mojette.
  We start with the function f represented by 16 pixels from p1 to p16. The possible values of the function at the point (k, l) are different according to the applications. This can be a binary value of 0 or 1 that it often used to differentiate the object and the background. This can be a ternary value as in the Mojette game. This can be also a finite set of integers value from 0 to (n-1), or more often we take a set of cardinality equal to a power of 2 or a prime number. But it can be integers and real numbers with an infinite number of possibilities, even though this idea has never been used.
  With the index "k" as "kolumn" and “l” as a “line”, we define a Cartesian coordinate system. But here we will only need the integer coordinates. On Figure 2, we have arbitrarily chosen the left bottom point as the origin (0,0) and the direction of the two axes. The coordinates of each pixel are denoted in red on Figure 2.
  For the projections, the coordinate system is derived from the one of the grid. Indeed, it meets two requirements:
  1) The pixel (0,0) is always projected on the point 0 of the projection (this is a consequence of linearity of the Mojette operator)
  2) The direction of the projection is fixed "counterclockwise" as in trigonometry when going from 0 ° to 180 °.
  Altogether, it necessarily gives the positions of the bins like the ones in blue color on the Figure 2.
  Let’s head back to the formula (1): the red dots correspond to the index (k, l) and the blue dots to the index b. The only elements remaining to clarify are the (p, q) values.
  These two values (p, q) are precisely those characterizing the Mojette Transform. They define the projection angle. Figure 3 shows colored arrows corresponding with the color code to the projection indexed by (p, q). For the 90° angle, the projection is shown below the grid for convenience but the direction is upward. Table 1 shows the correspondence between the angles in degrees and the values of p and q.
  The only valid Mojette angles are given by the following rules:
  An angle is given by the direction of projection in line and column
  A direction is composed of two integers (p, q) with gcd (p, q) = 1
  An angle is always between 0 and 180 °, which means that q is never negative
  These rules ensure the uniqueness in the correspondence of an angle and (p, q) values. For example, the 45 ° angle, the Rule 2 forbid to define the angle pairs (2,2) or (3,3) and Rule 3 prohibits to use (-2, -2) and (-1, -1). Only the angle (p = 1, q = 1) satisfies the three rules.
caption: Figure 3: Projections of direction (p, q) of the 4x4 grid
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