Image representation via a finite Radon transform

Matúš, František; Flusser, Jan

doi:10.1109/34.254058

Cited by 172 publications

(88 citation statements)

References 12 publications

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“…The majority of methods proposed in literature, for the 2-D case, have been devised for computerized tomography or to approximate to the continuous formula [10]- [18]. But, none of them were specifically designed to be invertible transforms for discrete images and cannot be used for the discrete ridgelet transform.…”

Section: B the Discrete Ridgelet Transformmentioning

confidence: 99%

Discrete analytical Ridgelet transform

Carré

Andrès

2004

Signal Processing

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In this paper, we propose an implementation of the 3-D ridgelet transform: The 3-D Discrete Analytical Ridgelet Transform (3-D DART). This transform uses the Fourier strategy for the computation of the associated 3-D discrete Radon transform. The innovative step is the definition of a discrete 3-D transform with the discrete analytical geometry theory by the construction of 3-D discrete analytical lines in the Fourier domain. We propose two types of 3-D discrete lines: 3-D discrete radial lines going through the origin defined from their orthogonal projections and 3-D planes covered with 2-D discrete line segments. These discrete analytical lines have a parameter called arithmetical thickness, allowing us to define a 3-D DART adapted to a specific application. Indeed, the 3-D DART representation is not orthogonal, It is associated with a flexible redundancy factor.The 3-D DART has a very simple forward/inverse algorithm that provides an exact reconstruction without any iterative method. In order to illustrate the potentiality of this new discrete transform, we apply the 3-D DART and its extension to the Local-DART (with smooth windowing) to the denoising of 3-D image and colour video. These experimental results show that the simple thresholding of the 3-D DART coefficients is efficient.

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Section: B the Discrete Ridgelet Transformmentioning

confidence: 99%

Discrete analytical Ridgelet transform

Carré

Andrès

2004

Signal Processing

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“…This contrasts with the projections of the general DRT in [4] which require an algebraic solution for the inversion process. The FRT applies to square arrays of prime size, p × p. A discrete form of both the Fourier slice theorem and convolution property of the continuous transform hold [1]. Thus the transformation and inversion can also be achieved very efficiently via the 2-D discrete Fourier transform of the function.…”

Section: Introductionmentioning

confidence: 99%

“…(c) Pixels on y ≡ 6x + t (mod 32) sum to give R ⊥ 3 (7), i.e., s = 3. N is now composite and the gradient 2s has a common factor with N , each row and column is no longer uniquely sampled 1993 [1]. This transform is an exact and invertible projective mapping, requiring only additive operations.…”

Section: Introductionmentioning

confidence: 99%

“…This paper presents a Discrete Radon Transform (DRT) based on congruent mathematics that applies to N × N arrays where N ∈ N. This definition incorporates and is a natural extension of the more restricted cases of the finite Radon transform [1] where N must be prime, the discrete periodic Radon transform [2] where N must be a power of 2, and the DRT over p n [3], where N must be a power of a single prime. The DRT exactly and invertibly maps a 2-D image to a set of 1-D projections of length N .…”

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confidence: 99%

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A Discrete Modulo N Projective Radon Transform for N × N Images

Kingston

Svalbe

2005

Lecture Notes in Computer Science

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Abstract. This paper presents a Discrete Radon Transform (DRT) based on congruent mathematics that applies to N × N arrays where N ∈ N. This definition incorporates and is a natural extension of the more restricted cases of the finite Radon transform [1] where N must be prime, the discrete periodic Radon transform [2] where N must be a power of 2, and the DRT over p n [3], where N must be a power of a single prime. The DRT exactly and invertibly maps a 2-D image to a set of 1-D projections of length N . Projections are found as the sum of the pixels centred on a parallel set of discrete lines. The image is assumed to be periodic and these lines wrap around the array under modulo N arithmetic. Properties of the continuous Radon transform are preserved in the DRT; a discrete form of the Fourier slice theorem applies, as does the convolution property. A formula is given to find the projection set required to be exactly invertible for arrays with N any composite number, as well as a means to determine the level of redundancy in sampling that is introduced on such composite arrays.

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“…The samples are oriented at integer array displacements of x m horizontally and y m vertically on the lattice at each translate position (t). Arrays of prime size [3] generate unique pixel sampling patterns for each DRT projection. This means that digital images can be projected and reconstructed exactly with the DRT using only simple (and hence fast) addition operations.…”

Section: Introductionmentioning

confidence: 99%

Intertwined Digital Rays in Discrete Radon Projections Pooled over Adjacent Prime Sized Arrays

Svalbe

Kingston

2003

Discrete Geometry for Computer Imagery

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Abstract. Digital projections are image intensity sums taken along directed rays that sample whole pixel values at periodic locations along the ray. For 2D square arrays with sides of prime length, the Discrete Radon Transform (DRT) is very efficient at reconstructing digital images from their digital projections. The periodic gaps in digital rays complicate the use of the DRT for efficient reconstruction of tomographic images from real projection data, where there are no gaps along the projection direction. A new approach to bridge this gap problem is to pool DRT digital projections obtained over a variety of prime sized arrays. The digital gaps are then partially filled by a staggered overlap of discrete sample positions to better approximate a continuous projection ray. This paper identifies primes that have similar and distinct DRT pixel sampling patterns for the rays in digital projections. The projections are effectively pooled by combining several images, each reconstructed at a fixed scale, but using projections that are interpolated over different prime sized arrays. The basis for the pooled image reconstruction approach is outlined and we demonstrate the principle of this mechanism works.

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Image representation via a finite Radon transform

Cited by 172 publications

References 12 publications

Discrete analytical Ridgelet transform

Discrete analytical Ridgelet transform

A Discrete Modulo N Projective Radon Transform for N × N Images

Intertwined Digital Rays in Discrete Radon Projections Pooled over Adjacent Prime Sized Arrays

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