Quaternion-Based Texture Analysis of Multiband Satellite Images: Application to the Estimation of Aboveground Biomass in the East Region of Cameroon

Abstract

Within the last decade, several approaches using quaternion numbers to handle and model multiband images in a holistic manner were introduced. The quaternion Fourier transform can be efficiently used to model texture in multidimensional data such as color images. For practical application, multispectral satellite data appear as a primary source for measuring past trends and monitoring changes in forest carbon stocks. In this work, we propose a texture-color descriptor based on the quaternion Fourier transform to extract relevant information from multiband satellite images. We propose a new multiband image texture model extraction, called FOTO++, in order to address biomass estimation issues. The first stage consists in removing noise from the multispectral data while preserving the edges of canopies. Afterward, color texture descriptors are extracted thanks to a discrete form of the quaternion Fourier transform, and finally the support vector regression method is used to deduce biomass estimation from texture indices. Our texture features are modeled using a vector composed with the radial spectrum coming from the amplitude of the quaternion Fourier transform. We conduct several experiments in order to study the sensitivity of our model to acquisition parameters. We also assess its performance both on synthetic images and on real multispectral images of Cameroonian forest. The results show that our model is more robust to acquisition parameters than the classical Fourier Texture Ordination model (FOTO). Our scheme is also more accurate for aboveground biomass estimation. We stress that a similar methodology could be implemented using quaternion wavelets. These results highlight the potential of the quaternion-based approach to study multispectral satellite images.

Appendices

Appendix 1

We present here a complete scheme to compute the polar form of any quaternion numbers.

In Sect. 3.3, when dealing with quaternion algebra, we have shown how to compute the polar representation of a quaternion only in the case where it is non-equal to zero. We present here how to compute the polar form when either the scalar or the modulus of its vector part is equal to zero.

Let

$$q = q_{0} + q_{1} i + q_{2} j + q_{3} k,$$

(A.1)

with $\left| q \right| \ne 0$. The polar representation of the quaternion $q$ can be expressed as:

$$q = \left| q \right|e^{\alpha } ,$$

(A.2)

where α is a pure quaternion with $\left| {V \left( \alpha \right)} \right| \ne 0$. It follows that:

• 1. if $S\left( q \right) = q_{0} = 0$ then

  • $q = \left| q \right|e^{{\left( {\frac{\pi }{2} + 2k\pi } \right).\frac{V\left( q \right)}{\left| q \right|}}} ,\,{\text{for}},\quad k \ge 0, \;k \in {\mathbb{N}}.$

  • $q = \left| q \right|e^{{ - \left( {\frac{3\pi }{2} + 2k\pi } \right) \frac{V\left( q \right)}{\left| q \right|} }} ,\;{\text{for}}, \quad k \ge 0,\;k \in {\mathbb{N}}.$

• 2.
  $$if\,S\left( q \right) = q_{0} \ne 0$$

  then

  • a. if $V \left( q \right) = 0$, then

    • i. if $S\left( q \right) > 0$ then

      $$q = S\left( q \right) = \left| q \right|e^{\alpha } ,$$

      (A.3)

      with $S\left( \alpha \right) = 0$ and $\left| {V \left( \alpha \right)} \right| = 2\pi k,\;{\text{for}}\quad k \ge 1,\;k \in {\mathbb{N}}.$

    • ii. if $S\left( q \right) < 0$ then

      $$q = S\left( q \right) = \left| q \right|e^{\alpha } ,$$

      (A.4)

      with $S\left( \alpha \right) = 0$ and $\left| {V \left( \alpha \right)} \right| = \left( {2k + 1} \right)\pi , \,{\text{for}} \,k \ge 0,k \in {\mathbb{N}}.$

  • b. if $V \left( q \right) \ne 0$ then

    • i. if ${ \cos }\left( {\left| {V \left( q \right)} \right|} \right) > 0$ and ${ \sin }\left( {\left| {V \left( q \right)} \right|} \right) > 0$ then

      $$q = \left| q \right|e^{{ \arctan \left( { \frac{V\left( q \right)}{S\left( q \right)}} \right) \cdot \frac{V\left( q \right)}{\left| q \right|}}} ,$$

      (A.5)

      with $\arctan \left( {\frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \in \left( {2k\pi ,\frac{\pi }{2} + 2k\pi } \right), \quad k \ge 0,\;k \in {\mathbb{N}}.$

    • ii. If ${\text{cos}}\left( {\left| {V\left( q \right)} \right|} \right) < 0\;{\text{and}}\;\sin \left( {\left| {V \left( q \right)} \right|} \right) > 0$

      $$q = \left| q \right|e^{{\arctan \left( {\frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \cdot \frac{V\left( q \right)}{\left| q \right|}}} ,$$

      (A.6)

      with $\arctan \left( { \frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \in \left( {\frac{\pi }{2} + 2k\pi ,\left( {2k + 1} \right)\pi } \right),\quad k \ge 0,\;k \in {\mathbb{N}}.$

    • iii. if $\cos \left( {\left| {V \left( q \right)} \right|} \right) < 0\;{\text{and}}\;{ \sin }\left( {\left| {V \left( q \right)} \right|} \right) < 0$

      $$q = \left| q \right|e^{{\arctan \left( { - \frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \cdot - \frac{V\left( q \right)}{\left| q \right|}}} ,$$

      (A.7)

      with $\arctan \left( { - \frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \in \left( {\pi + 2k\pi ,\frac{3\pi }{2} + 2k\pi } \right),\quad k \ge 0,\;k \in {\mathbb{N}}.$

    • iv. if $\cos \left( {\left| {V \left( q \right)} \right|} \right) > 0\;{\text{and}}\;\sin \left( {\left| {V \left( q \right)} \right|} \right) < 0$

      $$q = \left| q \right|e^{{ \arctan \left( { - \frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \cdot \frac{V\left( q \right)}{\left| q \right|} }} ,$$

      (A.8)

      with $\arctan \left( { - \frac{{\left| {V\left( q \right)} \right|}}{S\left( q \right)}} \right) \in \left( {\frac{3\pi }{2} + 2k\pi , 2\left( {k + 1} \right)\pi } \right), \quad k \ge 0,\;k \in {\mathbb{N}}.$

Appendix 2a

We describe here below an algorithm to convert a RGB-color image to Lab color space.

Appendix 2b

We present here an algorithm to convert a RGB-color image to HSV color space.