**2.1 Bandelets**

Bandelets (Le Pennec & Mallat, Apr 2005), (Le Pennec & Mallat, Dec 2005) belong to a second generation of wavelet transforms and are composed of anisotropic wavelets, which are in fact a combination of geometric flow of an image corresponding to local directions of its gray levels. This geometric flow represents a regularity of a vector field along edges contained in the image. Typical example of this geometric flow can be seen on Fig. 1, where it can be observed that all directions are aligned to object's edges at the boundary of two different areas.

Edges inside an image are often hard to determine. First generation of bandelet transform uses the vector field (Le Pennec & Mallat, 2001), which determines image regularities and irregularities. Therefore bandelet coefficients represent geometric flow defined by polynomial function. This geometric flow consists of directions of variations in image grey levels, where linear geometric flow is preferred. Bandelet transform image is divided into

Information Extraction and Despeckling of

2005).

**2.2 Contourlets** 

transform.

SAR Images with Second Generation of Wavelet Transform 377

<sup>2</sup> <sup>2</sup> , *d d dR G B* <sup>L</sup> *<sup>f</sup> <sup>R</sup> f f*

where *fdR* is the recovered signal from quantized coefficients acquired by inverse 1D wavelet transform, *RG* is the number of bits needed to code geometric parameter *d*, *RB* is the number of bits needed to code the quantized coefficients and λ = 3/28 (Le Pennec & Mallat, Apr

When there are gathered all approximations over each individual dyadic square, the quadtree can be constructed. The algorithm starts with the smallest squares that represent a leaf in quadtree and initialize the cumulative Lagrangian of the sub-tree. Within these dyadic squares, a best bandelet approximation is obtained by minimizing a Lagrangian cost function (Le Pennec & Mallat, Dec 2005). Fig. 2 shows an example of denoising obtained with the bandelet transform including dyadic squares that indicate a progress of dyadic levels. This image is represented by indexing a dyadic level used in bandelet transform,

a) b) c) Fig. 2. An example of image denoising using a bandelet transforms. a) Original image, b)

Contourlet transform (Do & Vetterli, 2005) is also classified as a second generation wavelet transform for which a Fourier transform is not needed anymore. Main advantage of second generation wavelet transform over the first generation is a true discrete 2D transformation, which is able to capture geometry of an image, but the first generation wavelet transform does not perform very well on edge regions. This transformation therefore results in

Best performance of wavelets is achieved in 1D case which is for example only one row of a 2D picture. Because pictures are not simply stacks of rows, discontinuities evolve along smooth regions. 2D wavelet transform thus captures edge points, but on the other hand the throughput on smooth regions is not quite as good anymore. Moreover wavelet transform can only capture a fraction of image directionality, which is clearly seen in Fig. 3 where wavelet transform needs a lot more subdivisions and information then a contourlet

adaptive multi-resolution and directional image expansion using contour segments.

where white indicates the first level and black the last level achieved.

Denoised image using a bandelet transforms, and c) Dyadic squares tree

*TR R* (1)

regions with corresponding vector fields, which describes directions of regularity inside a predefined neighborhood.

If the image intensity is uniformly regular in the neighborhood of a point, then this direction is not uniquely defined, and some form of global regularity is therefore imposed on the flow to specify it uniquely.

In literature it has been proven that the first generation of bandelets has minimum distortion for images whose edges correspond to geometric regularity. However, the first generation of bandelets is composed in continuous space, thus not being able to represent a multiresolution of the geometric regularity. Thus, the second generation of bandelets (Le Pennec & Mallat, Apr 2005) was introduced, which is an orthogonal multiscale transform constructed directly in discrete domain. The bandelet transform first creates a composition of smaller images representing subbands, and then uses fast subband-filtering algorithms. For applications including speckle-noise removal, the geometric flow is optimized in a way that bandelet transform produces minimum distortion in reconstructed images. The decomposition on a bandelet basis is computed using a wavelet filter bank followed by adaptive geometric orthogonal filters, which require *O*((log2*N*)3) operations.

The key parameters in bandelet transform are: the estimation of basis shapes, the partition of images, and the optimization of geometric flows (Yang et al., 2007). To represent image with as little as possible information, the complex edges must be divided into simpler smaller shapes so that linear geometric flows can represent them sufficiently. The image is commonly divided into smaller square regions that are being divided until there is only one contour inside a square region. It must be noted that the geometric regularity should be discrete, so dyadic decomposition by successive subdivisions of square regions into four smaller sub-squares of twice smaller width can be made. There is a defined maximum and minimum block size (Le Pennec & Mallat, Apr 2005), where the first division produces blocks of maximum size, while later iterations divide those blocks up until minimum size is reached. This partition result can be viewed as a quadtree, where each block is represented by its corresponding leaf in a tree. At each scale the resulting geometry is multiscale and calculated by a procedure that minimizes the Lagrangian cost function.

#### **Implementation**

The bandelet transform first computes the 2D wavelet transform of the input original image (Peyré & Mallat, Apr 2005). This transform is based on orthogonal or biorthogonal filter banks and results in four smaller images (children) containing low- and high-frequency components. By selecting a dyadic square and recursively splitting input wavelet image four new sub-squares are created. Further on geometric flow parameterization is performed in each of these sub-squares in every possible direction. Let us assume that each of these squares has a width of *k* pixels then the number of potential directions *d* is a little less than 2*k*2. The sampling location is then projected along potential direction and afterwards sorting the resulting 1D points is performed from left to right direction. These points define 1D discrete signal *fd* (Le Pennec & Mallat, Apr 2005) which is later on transformed with 1D discrete wavelet transform. For a given user defined threshold *T*, the bandelet transform has to find the best available direction, which in fact produces the less approximation error. Best geometry is obtained by choosing best direction *d* that minimizes the Lagrangian

regions with corresponding vector fields, which describes directions of regularity inside a

If the image intensity is uniformly regular in the neighborhood of a point, then this direction is not uniquely defined, and some form of global regularity is therefore imposed on the flow

In literature it has been proven that the first generation of bandelets has minimum distortion for images whose edges correspond to geometric regularity. However, the first generation of bandelets is composed in continuous space, thus not being able to represent a multiresolution of the geometric regularity. Thus, the second generation of bandelets (Le Pennec & Mallat, Apr 2005) was introduced, which is an orthogonal multiscale transform constructed directly in discrete domain. The bandelet transform first creates a composition of smaller images representing subbands, and then uses fast subband-filtering algorithms. For applications including speckle-noise removal, the geometric flow is optimized in a way that bandelet transform produces minimum distortion in reconstructed images. The decomposition on a bandelet basis is computed using a wavelet filter bank followed by

The key parameters in bandelet transform are: the estimation of basis shapes, the partition of images, and the optimization of geometric flows (Yang et al., 2007). To represent image with as little as possible information, the complex edges must be divided into simpler smaller shapes so that linear geometric flows can represent them sufficiently. The image is commonly divided into smaller square regions that are being divided until there is only one contour inside a square region. It must be noted that the geometric regularity should be discrete, so dyadic decomposition by successive subdivisions of square regions into four smaller sub-squares of twice smaller width can be made. There is a defined maximum and minimum block size (Le Pennec & Mallat, Apr 2005), where the first division produces blocks of maximum size, while later iterations divide those blocks up until minimum size is reached. This partition result can be viewed as a quadtree, where each block is represented by its corresponding leaf in a tree. At each scale the resulting geometry is multiscale and

The bandelet transform first computes the 2D wavelet transform of the input original image (Peyré & Mallat, Apr 2005). This transform is based on orthogonal or biorthogonal filter banks and results in four smaller images (children) containing low- and high-frequency components. By selecting a dyadic square and recursively splitting input wavelet image four new sub-squares are created. Further on geometric flow parameterization is performed in each of these sub-squares in every possible direction. Let us assume that each of these squares has a width of *k* pixels then the number of potential directions *d* is a little less than 2*k*2. The sampling location is then projected along potential direction and afterwards sorting the resulting 1D points is performed from left to right direction. These points define 1D discrete signal *fd* (Le Pennec & Mallat, Apr 2005) which is later on transformed with 1D discrete wavelet transform. For a given user defined threshold *T*, the bandelet transform has to find the best available direction, which in fact produces the less approximation error. Best

geometry is obtained by choosing best direction *d* that minimizes the Lagrangian

adaptive geometric orthogonal filters, which require *O*((log2*N*)3) operations.

calculated by a procedure that minimizes the Lagrangian cost function.

predefined neighborhood.

to specify it uniquely.

**Implementation** 

$$\mathcal{L}^{\prime\prime}\left(f\_d, \mathbf{R}\right) = \left\|f\_d - f\_{dR}\right\|^2 + \mathcal{L}T^2\left(\mathbf{R}\_G + \mathbf{R}\_B\right) \tag{1}$$

where *fdR* is the recovered signal from quantized coefficients acquired by inverse 1D wavelet transform, *RG* is the number of bits needed to code geometric parameter *d*, *RB* is the number of bits needed to code the quantized coefficients and λ = 3/28 (Le Pennec & Mallat, Apr 2005).

When there are gathered all approximations over each individual dyadic square, the quadtree can be constructed. The algorithm starts with the smallest squares that represent a leaf in quadtree and initialize the cumulative Lagrangian of the sub-tree. Within these dyadic squares, a best bandelet approximation is obtained by minimizing a Lagrangian cost function (Le Pennec & Mallat, Dec 2005). Fig. 2 shows an example of denoising obtained with the bandelet transform including dyadic squares that indicate a progress of dyadic levels. This image is represented by indexing a dyadic level used in bandelet transform, where white indicates the first level and black the last level achieved.

Fig. 2. An example of image denoising using a bandelet transforms. a) Original image, b) Denoised image using a bandelet transforms, and c) Dyadic squares tree
