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:
After one year of research, the first communication introducing the Mojette Transform was held in May 1995 in the first edition of CORESA National Congress CCITT Rennes. Many others will follow it for 18 years of existence. In 2011, the book The Mojette Transform: Theory and Applications at ISTE-Wiley was well received by the scientific community. All this support has encouraged the IRCCyN research team to continue the research on this topic.
Jeanpierre Guédon, professor and inventor of the transform called it: "Mojette Transform". The word "Mojette" comes from the name of white beans in Vendee, originally written "Moghette" or "Mojhette". In many countries, bean is a basic educational tool representing an exact unit that teaches visually additions and subtractions. Therefore, the choice of the name "Mojette" serves to emphasize the fact that the transform uses only exact unit in additions and subtractions.
The original purpose of the Mojette Transform was to create a discrete tool to divide the Fourier plane into angular and radial sectors. The first attempt of application was the psychovisual encoding of image, reproducing the human vision channel. However, it was never realized.
Mathematics
The "raw" transform Mojette definition is this:
The following figure 1 helps to explain the “raw” transform Mojette.
Figure 1: 4x4 grid with 16 pixels, 4 projections and 22 bins
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.
Figure 2: Mark of 4x4 grid and projections
For the projections, the coordinate system is derived from that 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.
Figure 3: Projections of direction (p, q) of the 4x4 grid
0°
p=1
q=0
b-l=0
45°
p=1
q=1
b+k-l=0
90°
p=0
q=1
b+k=0
135°
p=-1
q=1
b+k+l=0
Table 1: The correspondence of the angles projections with direction equation b + qk - pl = 0
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.
Applications and achievements
The distributed storage disk or network
The most important area of application using the "Mojette Transform" is distributed storage. Particularly, this method is used in RozoFS, an open-source distributed file system. In this application, the "Mojette Transform" is used as an erasure code in order to provide reliability, while significantly reducing the total amount of stored data when compared to classical techniques like replication (typically by a factor of 2). Thus, it significantly reduces the cost of the storage cluster in terms of hardware, maintenance or energy consumption for example.
In 2010, Pierre Evenou, research engineer of the IVC team IRCCyN laboratory, decided to create the start-up Fizians (currently known as Rozo Systems) using this application. The start-up offers storage solutions in cloud computing, virtualization, storage servers, file servers, backup and archiving.
Networks packets transfer
Thanks to the redundancy of the transform, sent packets can be fragmented without loss. Additionally, the fact of using only additions and subtractions increases the speed of information transmission. Finally, the information cannot be reconstructed without having the initial angle of the projections, so it also provides data security.
This application has been selected by Thales Cholet for its ad hoc network (using wireless network and terminals to transmit messages between them) in order to secure the information and has multiple paths between the source and destination. In 2002, the start-up PIBI has used this technology to provide secure Internet payment services.
The Medical tomography
In the field of medical imaging, the properties of the “Transform Mojette” create a direct mapping and solve the missing wedge problem. However, the image acquisition using the Mojette transform has not been yet developed. The problem of obtaining exact “Mojette” values while using approximated data acquisition has been studied but has to be continued. Besides, the post-processing of medical images is doing well since data acquisition is already done.
These results are used by the company Keosys in 2001 with Jerome Fortineau and the company Qualiformed created in 2006 by Stephen Beaumont. Prof. Guédon and the IRCCyN laboratory were heavily involved in the creation of these two companies. The companies have already financed several PhD students and participated in research projects in order to continue the development of the application in medical tomography. The results have led to apply patents and implementation on their equipment of image processing.
The watermarking and image encryption
Cryptography and watermarking were also part of the research conducted in the IRCCyN laboratory. It provides solutions for security and authentication.
In cryptography, the instability of the transformed Mojette secures data. The fact that the transform is exact encrypts information and allows no deviation even minimal. For watermarking, the transform is very effective in fingerprinting. By inserting "Mojette Transform" marks in images, one can authenticate documents using the same properties as in cryptography.
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H. Der Sarkissian, Jp. Guédon, P. Tervé, N. Normand and I. Svalbe. (2012)." Evaluation of Discrete Angles Rotation Degradation for Myocardial Perfusion Imaging", EANM Annual Congress 2012.
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H. Der Sarkissian, B. Recur, N. Normand, Jp. Guédon. (2013), "Rotations in the Mojette Space" in 2013 IEEE International Conference on Image Processing.
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A light field, or lightfield, is a vector function that describes the amount of light flowing in every direction through every point in a space. The space of all possible light rays is given by the five-dimensional plenoptic function, and the magnitude of each ray is given by its radiance. Michael Faraday was the first to propose that light should be interpreted as a field, much like the magnetic fields on which he had been working. The term light field was coined by Andrey Gershun in a classic 1936 paper on the radiometric properties of light in three-dimensional space.
Tomographic reconstruction is a type of multidimensional inverse problem where the challenge is to yield an estimate of a specific system from a finite number of projections. The mathematical basis for tomographic imaging was laid down by Johann Radon. A notable example of applications is the reconstruction of computed tomography (CT) where cross-sectional images of patients are obtained in non-invasive manner. Recent developments have seen the Radon transform and its inverse used for tasks related to realistic object insertion required for testing and evaluating computed tomography use in airport security.
Stransform as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the S transform is a generalization of the short-time Fourier transform (STFT), extending the continuous wavelet transform and overcoming some of its disadvantages. For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in S transform. Moreover, the S transform doesn't have a cross-term problem and yields a better signal clarity than Gabor transform. However, the S transform has its own disadvantages: the clarity is worse than Wigner distribution function and Cohen's class distribution function.
Phase correlation is an approach to estimate the relative translative offset between two similar images or other data sets. It is commonly used in image registration and relies on a frequency-domain representation of the data, usually calculated by fast Fourier transforms. The term is applied particularly to a subset of cross-correlation techniques that isolate the phase information from the Fourier-space representation of the cross-correlogram.
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:
The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain. This algorithm was introduced in 1989 by Stéphane Mallat.
The stationary wavelet transform (SWT) is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet transform (DWT). Translation-invariance is achieved by removing the downsamplers and upsamplers in the DWT and upsampling the filter coefficients by a factor of in the th level of the algorithm. The SWT is an inherently redundant scheme as the output of each level of SWT contains the same number of samples as the input – so for a decomposition of N levels there is a redundancy of N in the wavelet coefficients. This algorithm is more famously known as "algorithme à trous" in French which refers to inserting zeros in the filters. It was introduced by Holschneider et al.
In mathematics, a wavelet series is a representation of a square-integrable function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.
Discrete tomography focuses on the problem of reconstruction of binary images from a small number of their projections.
Tomosynthesis, also digital tomosynthesis (DTS), is a method for performing high-resolution limited-angle tomography at radiation dose levels comparable with projectional radiography. It has been studied for a variety of clinical applications, including vascular imaging, dental imaging, orthopedic imaging, mammographic imaging, musculoskeletal imaging, and chest imaging.
In mathematics, log-polar coordinates is a coordinate system in two dimensions, where a point is identified by two numbers, one for the logarithm of the distance to a certain point, and one for an angle. Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry. In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates.
RozoFS is a free software distributed file system. It comes as a free software, licensed under the GNU GPL v2. RozoFS uses erasure coding for redundancy.
The computed tomography imaging spectrometer (CTIS) is a snapshot imaging spectrometer which can produce in fine the three-dimensional hyperspectral datacube of a scene.
The Fly Algorithm is a computational method within the field of evolutionary algorithms, designed for direct exploration of 3D spaces in applications such as computer stereo vision, robotics, and medical imaging. Unlike traditional image-based stereovision, which relies on matching features to construct 3D information, the Fly Algorithm operates by generating a 3D representation directly from random points, termed "flies." Each fly is a coordinate in 3D space, evaluated for its accuracy by comparing its projections in a scene. By iteratively refining the positions of flies based on fitness criteria, the algorithm can construct an optimized spatial representation. The Fly Algorithm has expanded into various fields, including applications in digital art, where it is used to generate complex visual patterns.
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