3D image analysis by separable discrete orthogonal moments based on Krawtchouk and Tchebichef polynomials

Batioua, Imad; Benouini, Rachid; Zenkouar, Khalid; Zahi, Azeddine; Fadili, Hakim El

doi:10.1016/j.patcog.2017.06.013

Cited by 58 publications

(23 citation statements)

References 37 publications

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“…In this work, up to the third‐order moments are used in the most of the experiments. It is observed in some other works also that up to third‐order moments are sufficient for effective modelling of features [8, 36]. In one experiment, the impact of the order of moments is analysed by varying the order.…”

Section: Experiments and Discussionmentioning

confidence: 99%

“…The number of feature images depends on the order of moments. A maximum of third‐order moments are recommended as it is good enough for the most of the applications [36], i.e.

p + q + r = 3

.…”

Section: Proposed Methodsmentioning

confidence: 99%

“…The higher order moments represent the comparatively global features. In discrete case, the geometric moments for a digital image I of size

N \times M

are defined using discrete sum approximation as

m_{p, q} = \sum_{i = 0}^{N - 1} \sum_{j = 0}^{M - 1} i^{p} j^{q} I false(i, j false)

For volumetric image of size

N \times M \times K

, the 2D moments can be extended to 3D form [36] as follows:

m_{p, q, r} = \sum_{i = 0}^{N - 1} \sum_{j = 0}^{M - 1} \sum_{k = 0}^{K - 1} i^{p} j^{q} k^{r} I false(i, j, k false)

where I is the volumetric image,

m_{p, q, r}

is the 3D moment of

P th

order, and

p, q, r = 0, 1, \dots normal∞

such that

p + q + r = P

. The translation invariant central geometric moments can be determined using the centroid of the object.…”

Section: Geometric Momentsmentioning

confidence: 99%

“…The translation invariant central geometric moments can be determined using the centroid of the object. The coordinates of the centroid of the 3D objects denoted by

false(i_{0}, j_{0}, k_{0} false)

are defined using zeroth and first‐order moments as

i_{0} = \frac{m_{1, 0, 0}}{m_{0, 0, 0}}, j_{0} = \frac{m_{0, 1, 0}}{m_{0, 0, 0}}, k_{0} = \frac{m_{0, 0, 1}}{m_{0, 0, 0}}

The translation invariant central geometric moments are defined using the coordinates

false(i_{0}, j_{0}, k_{0} false)

μ_{p, q, r} = \sum_{i = 0}^{N - 1} \sum_{j = 0}^{M - 1} \sum_{k = 0}^{K - 1} false(i - i_{0} false)^{p} false(j - j_{0} false)^{q} false(k - k_{0} false)^{r} I false(i, j, k false)

If a 3D object is rotated by angle

ϕ

along x axis, by an angle

θ

along y axis, and by an angle

ψ

along axis z , its rotation matrix is given by a series of three 2D rotations [36] as follows:

R_{x y z} false(ϕ, θ, ψ false) = R_{x} false(ϕ false) R_{y} false(θ false) R_{z} false(ψ false)

where

R_{x}

R_{y}

, ...…”

Section: Geometric Momentsmentioning

confidence: 99%

See 3 more Smart Citations

Hyperspectral image classification using three‐dimensional geometric moments

Kumar

2020

IET image process

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Section: Experiments and Discussionmentioning

confidence: 99%

“…The number of feature images depends on the order of moments. A maximum of third‐order moments are recommended as it is good enough for the most of the applications [36], i.e.

p + q + r = 3

.…”

Section: Proposed Methodsmentioning

confidence: 99%

“…The higher order moments represent the comparatively global features. In discrete case, the geometric moments for a digital image I of size

N \times M

are defined using discrete sum approximation as

m_{p, q} = \sum_{i = 0}^{N - 1} \sum_{j = 0}^{M - 1} i^{p} j^{q} I false(i, j false)

For volumetric image of size

N \times M \times K

, the 2D moments can be extended to 3D form [36] as follows:

m_{p, q, r} = \sum_{i = 0}^{N - 1} \sum_{j = 0}^{M - 1} \sum_{k = 0}^{K - 1} i^{p} j^{q} k^{r} I false(i, j, k false)

where I is the volumetric image,

m_{p, q, r}

is the 3D moment of

P th

order, and

p, q, r = 0, 1, \dots normal∞

such that

p + q + r = P

. The translation invariant central geometric moments can be determined using the centroid of the object.…”

Section: Geometric Momentsmentioning

confidence: 99%

“…The translation invariant central geometric moments can be determined using the centroid of the object. The coordinates of the centroid of the 3D objects denoted by

false(i_{0}, j_{0}, k_{0} false)

are defined using zeroth and first‐order moments as

i_{0} = \frac{m_{1, 0, 0}}{m_{0, 0, 0}}, j_{0} = \frac{m_{0, 1, 0}}{m_{0, 0, 0}}, k_{0} = \frac{m_{0, 0, 1}}{m_{0, 0, 0}}

The translation invariant central geometric moments are defined using the coordinates

false(i_{0}, j_{0}, k_{0} false)

μ_{p, q, r} = \sum_{i = 0}^{N - 1} \sum_{j = 0}^{M - 1} \sum_{k = 0}^{K - 1} false(i - i_{0} false)^{p} false(j - j_{0} false)^{q} false(k - k_{0} false)^{r} I false(i, j, k false)

If a 3D object is rotated by angle

ϕ

along x axis, by an angle

θ

along y axis, and by an angle

ψ

along axis z , its rotation matrix is given by a series of three 2D rotations [36] as follows:

R_{x y z} false(ϕ, θ, ψ false) = R_{x} false(ϕ false) R_{y} false(θ false) R_{z} false(ψ false)

where

R_{x}

R_{y}

, ...…”

Section: Geometric Momentsmentioning

confidence: 99%

See 2 more Smart Citations

Hyperspectral image classification using three‐dimensional geometric moments

Kumar

2020

IET image process

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“…DTT is one of the derivatives of the orthonormal Tchebichef polynomial [9,28,29]. DTT transforms the image with a polynomial recursive technique ( ).…”

Section: Research Methods 21 Discrete Tchebichef Transform (Dtt)mentioning

confidence: 99%

Image watermarking using discrete wavelet-tchebichef transform

Setyono

Setiadi

2019

IJEECS

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<span>Image watermarking is one of the most popular techniques for authenticating copyright on the digital image. Many research on image watermarking has proved that the joint of Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT) combinations can improve both imperceptibility and robustness when compared to DCT or DWT only. Discrete Tchebichef Transform (DTT) denotes an alternative transformation that has a similarity property with DCT. DTT has an advantage in reducing memory requirements during computing, so the calculation speed is much faster than DCT. This study tested the performance of DTT and DCT on non-blind image watermarking method, where DTT and DCT are performed after DWT. Based on the experimental results, this research proved that the DTT was combined successfully with DWT and very potential for further investigation because it has a computing performance much better than DCT. While the image watermarking quality, both the imperceptibility and robustness aspects were completely identical with the combination of DCT and DWT transformation.</span>

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Quaternionic Moments

Hitzer

2023

Trends in Mathematics

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3D image analysis by separable discrete orthogonal moments based on Krawtchouk and Tchebichef polynomials

Cited by 58 publications

References 37 publications

Hyperspectral image classification using three‐dimensional geometric moments

Hyperspectral image classification using three‐dimensional geometric moments

Image watermarking using discrete wavelet-tchebichef transform

Quaternionic Moments

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