BUUHWE_2D: Two-dimensional Bottom-Up Unbalanced Haar Wavelet Expansion...

In Bagidis: BAses GIving DIStances

Description

This function computes the two-dimensional Bottom-Up Unbalanced Haar Wavelet Expansion (2D-BUUHWE) of an image (encoded as a matrix). See Timmermans (2012) for details. The 2D-BUUHWE is termed SHaped-Adaptive Haar (SHAH) expansion in Timmermans and Fryzlewicz (2012).

Usage

BUUHWE_2D(im)
SHAH(im)
Signature_2D(out.BUUHWE_2D)

Arguments

im	im: a numeric matrix encoding an image. Missing values are not allowed.
out.BUUHWE_2D	out.BUUHWE_2D: the output of BUUHWE_2D or SHAH.

Details

More details can be found in Timmermans (2012), Chapter 4, or Timmermans and Fryzlewicz (2012). The function implements the 2D-BUUHWE (= SHAH) for images observed on a regular quadratic grid. The image is encoded as a matrix, of which the entries are labeled by column.

Value

The output of BUUHWE_2D or SHAH is a list with

d

a vector of two elements giving the dimension of the encoded image.

decomp.hist

The decomposition history of the image, which provides its BUUHWE expansion (=SHAH transform). For each i in d[1]*d[2]-1, matrix decomp.hist[,,i] is a matrix with the following entries: decomp.hist[1,1,i] is the label of the input node at step d[1]*d[2]-i, decomp.hist[1,2,i] is the label of the output node at step d[1]*d[2]-i, decomp.hist[2,1,i] is the weight of the input node at step d[1]*d[2]-i, decomp.hist[2,2,i] is the weight of the output node at step d[1]*d[2]-i , decomp.hist[3,1,i] is the detail d[1]*d[2]-i, decomp.hist[3,2,i] is always zero.

The output of Signature_2D is a matrix containing the signature of the image. This is a matrix of which row i is made of the components x1,y1,x2,y2, detail and abs(detail) related to rank i in the 2D-BUUHWE (SHAH) of an image.

Note

A graphical representation of the tranform is provided by BUUHWE_2D.plot ( or equivalently SHAH.plot). The 2D-BUUHWE (SHAH) is related to a basis expansion. The basis can be obtained and represented using BUUHWE_2D_Stepwise ( or equivalently SHAH_Stepwise). An efficient way to store the information defing the 2D-BUUHWE (SHAH) is through the signature of the image, obtained using Signature_2D.

BUUHWE provides with the Bottom-Up Unbalanced Haar Wavelet Expansion of a curve. Pairs of images can be compared through their 2D-BUUHWE, using the BAGIDIS semi-distance computed with semimetric.BAGIDIS_2D. See Timmermans (2012) for details.

Author(s)

Piotr Fryzlewicz, Department of Statistics, London School of Economics.

Catherine Timmermans, Institut de Statistique, Biostatistique et Sciences Actuarielles, Universite catholique de Louvain. <[e-mail]>

References

The main references are

• Timmermans C., 2012, Bases Giving Distances. A new paradigm for investigating functional data with applications for spectroscopy. PhD thesis, Universite catholique de Louvain. http://hdl.handle.net/2078.1/112451

• Timmermans C. and von Sachs R., 2015, A novel semi-distance for investigating dissimilarities of curves with sharp local patterns, Journal of Statistical Planning and Inference, 160, 35-50. http://hdl.handle.net/2078.1/154928

• Fryzlewicz P. and Timmermans C., 2015, SHAH: Shape Adaptive Haar wavelets for image processing. Journal of Computational and Graphical Statistics. (accepted - published online 27 May 2015) http://stats.lse.ac.uk/fryzlewicz/shah/shah.pdf

• Timmermans C., Delsol L. and von Sachs R., 2013, Using BAGIDIS in nonparametric functional data analysis: predicting from curves with sharp local features, Journal of Multivariate Analysis, 115, p. 421-444. http://hdl.handle.net/2078.1/118369

Other references include

• Girardi M. and Sweldens W., 1997, A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces, J. Fourier Anal. Appl. 3, 457-474

• Fryzlewicz P., 2007, Unbalanced Haar Technique for Non Parametric Function Estimation, Journal of the American Statistical Association, 102, 1318-1327.

• Timmermans C., von Sachs, R. , 2010, BAGIDIS, a new method for statistical analysis of differences between curves with sharp patterns (ISBA Discussion Paper 2010/30). Url : http://hdl.handle.net/2078.1/91090

• Timmermans, C. , Fryzlewicz, P., 2012, SHAH: Shape-Adaptive Haar Wavelet Transform For Images With Application To Classification (ISBA Discussion Paper 2012/15). Url: http://hdl.handle.net/2078.1/110529

The function BUUHWE_2D in this package is similar to the function uh.bu.2d (copyrighted Fryzlewicz 2014) in the package "shah_code", available on the webpage of Piotr Fryzlewicz: http://stats.lse.ac.uk/fryzlewicz/shah/shah_code.R , which accompanies the paper Fryzlewicz and Timmermans (2015).

See Also

BUUHWE_2D.plot,SHAH.plot ,BUUHWE_2D_Stepwise, SHAH_Stepwise, BUUHWE.

Examples

im = rbind(c(1,2,3),c(2,7,8),c(1,0,0))
BUUHWE_2D(im)
SHAH(im)
Signature_2D(SHAH(im))

SHAH_Stepwise(im)$details
Signature_2D(SHAH(im))

Bagidis documentation built on May 29, 2017, noon