In the context of fine structure extraction, lots of methods have been introduced, and, particularly in pavement
crack detection. We can distinguish approaches based on a threshold, employing mathematical morphology
tools or neuron networks and, more recently, techniques with transformations, like wavelet decomposition. The
goal of this paper is to introduce a 2D matched filter in order to define an adapted mother wavelet and, then,
to use the result of this multi-scale detection into a Markov Random Field (MRF) process to segment fine
structures of the image. Four major contributions are introduced. First, the crack signal is replaced by a more
real one based on a Gaussian function which best represents the crack. Second, in order to be more realistic,
i.e. to have a good representation of the crack signal, we use a 2D definition of the matched filter based on
a 2D texture auto-correlation and a 2D crack signal. The third and fourth improvements concern the Markov
network designed in order to allow cracks to be a set of connected segments with different size and position.
For this part, the number of configurations of sites and potential functions of the MRF model are completed.
The Filtered BackProjection is still questionable since
many discrete versions have been derived from the continuous Radon formalism.
From a continuous point of view, a previous work has made a link between continuous and discrete FBP versions
denoted as Spline 0-FBP model leading to a regularization of the infinite Ramp filter by the Fourier transform of a
trapezoidal shape. However, projections have to be oversampled (compared to the pixel
size) to retrieve the theoretical properties of Sobolev and Spline spaces. Here we obtain a novel version of the Spline 0 FBP
algorithm with a complete continuous/discrete correspondence using a specific discrete Radon transform, the Mojette transform.
From a discrete point of view, the links toward the FBP algorithm are shaped with the morphological
description and the extended use of discrete projection angles. The resulting equivalent FBP scheme uses a selected set of angles
which covers all the possible discrete Katz's directions issued from the pixels of the (square) shape under reconstruction: this is
implemented using the corresponding Farey's series. We present a new version of a discrete FBP method using a finite number
of projections derived from discrete geometry considerations.
This paper makes links between these two approaches.
In this paper, we propose to perform a novel discrete implementation of the filtered back projection algorithm. For this, we use a version of the discrete exact Radon transform called the Mojette transform that has been developed in our team for few years. The initial questioning was centered about the angular distribution needed for the continuous Radon reconstruction. Because of the discrete set of angles used in the FBP algorithm, discrete angles issued from Farey's series were used. Our version of the FBP algorithm is compared with the classical FBP algorithm. The choice of the set of projection angles is discussed in order to produce a good and efficient angular sampling. Finally, the very different behaviors between the classical FBP and our algorithm justify our study.