# Wavelet Analysis

Contributed by Eva Strand

## Other Names:

Spatial Wavelet Analysis (SWA)
Mexican-hat Wavelet Analysis

## Description

Wavelets have been used across a wide range of scientific disciplines from medical imaging to astronomy, to identify the size, shape, and location of individual objects of interest. Spatial wavelet analysis is an image processing techniques that has considerable potential to objectively and automatically quantify ecologically relevant objects at multiple scales across large landscapes. The image analysis method can be applied to any type of digital imagery, for example aerial photography, LiDAR (Light Detection and Ranging) or satellite images.

The technique can be explained in four simple steps: 1) Selection of a wavelet function that is similar in shape to the objects of interest in the image. We selected the Mexican Hat Wavelet because it has a shape that is similar to the western juniper trees in our aerial photos. 2) The wavelet is passed across the image and every time the wavelet crosses an object of similar size and shape a high score is recorded in a tabular database. Mathematically this is based on convolution of the wavelet function and the intensity (pixel values) of the object in the image. We used the Matlab software to conduct the analysis. 3) Wavelets of different sizes are passed across the image and the wavelet with the size most similar to the object receives the highest score. 4) Locations with wavelet scores and the size the wavelet that recorded the highest score are reported in the final output table, providing the user with a good approximation of the size and location of image objects.

## Output

The output is an ASCII file listing the diameter and spatial location (X and Y coordinates at the object center) for objects detected by the wavelet analysis algorithm. GIS software is needed to display the output data spatially. All objects are represented by a circle when the Mexican Hat wavelet is used, however other wavelet shapes could potentially identify objects of different shapes.

## Successful Rangeland Uses

Spatial wavelet analysis has been used successfully to quantify the crown diameter and location of individual juniper trees in a matrix of grassland or shrub steppe. Analysis of temporal sequences of photographs can provide information about rates of juniper expansion into steppe vegetation. Expansion rates in different biophysical settings and under a variety of human caused or natural disturbance regimes can be estimated.

Wavelet analysis can, with the use of allometric relationships relating the crown diameter of individual juniper trees to biomass, help quantify the above ground woody biomass contained within an area or estimate changes in above ground woody biomass over time if data from different time periods are compared.

## Application References

• Falkowski, M. J., A. M. S. Smith, A. T. Hudak, P. E. Gessler, L. A. Vierling, and N. L. Crookston (2006), Automated estimation of individual conifer tree height and crown diameter via two-dimensional spatial wavelet analysis of LIDAR data, Can. J. Remote Sens., 32(1), 153– 161.
• Strand E.K., A.P. Robinson, and S.C. Bunting 2007. Spatial patterns on the sagebrush steppe / Western juniper ecotone, Plant Ecology 190: 159-173.
• Strand E. K., L. A. Vierling, A. M. S. Smith, S. C. Bunting , 2008. Net changes in aboveground woody carbon stock in western juniper woodlands, 1946–1998, Journal of Geophysical Research Biogeosciences, 113, G01013, doi:10.1029/2007JG000544.
• Smith A.M.S., E.K. Strand, C.M. Steele, D.B. Hann, S.R. Garrity, M.J. Falkowski, and J.S. Evans, 2008. Production of vegetation spatial-structure maps by per-object analysis of juniper encroachment in multitemporal aerial photographs. Canadian Journal of Remote Sensing 34 (Suppl. 2):S268-S285.

## Technical References

• Addison, P.S., 2002. The Illustrated Wavelet Transform Handbook (London: Institute of Physics).
• Strand, E., Smith A.M.S., Bunting, S.C., Vierling, L.A., Hann, D.B., and Gessler, P.E., 2006. Wavelet estimation of plant spatial patterns in multi-temporal aerial photography, International Journal of Remote Sensing 27, 9-10, 2049-2054.

## Limitations

• Objects that are smaller than 2-3 times the image pixel size are difficult to detect using spatial 2-dimensional wavelet analysis.
• Objects that are clumped may be detected as one large object rather than several small objects. For example, individual trees in a high resolution aerial photo may be difficult to differentiate between if the tree canopy cover is over 50-60% canopy cover. The method works well for woodlands and open forests but has limitations in closed tree canopies.

## Data Inputs

Spatial wavelet analysis can be applied to any type of digital imagery, for example aerial photography, LiDAR (Light Detection and Ranging) or satellite images. Images must be converted to ASCII files for processing in the MATLAB software.

## Software/Hardware Requirements

Spatial wavelet analysis has been conducted on a Windows XP workstation using the MATLAB software from The MathWorks (http://www.mathworks.com/products/matlab/). ArcGIS from ESRI is helpful in displaying the results and calculating caopy cover and a spreadsheet software such as Microsoft Excel is needed to summarize data and use allometric equations to compute biomass for individual plants.

## Sample Graphic

Aerial photographs of a western juniper/sagebrush steppe landscape (15 ha). The dark dots are juniper plants in the matrix of sagebrush steppe. Upper left: Original black & white aerial photograph at 1-m resolution from 1939. Upper right: Original aerial photograph of the same area in 1998. Lower left: Projectedjuniper plant radii derived from wavelet analysis for the 1939 photograph, Lower right: Projected juniper plant radii derived from wavelet analysis for the 1998 photograph. The juniper canopy cover estimated from the 1939 photograph is 2.7% using wavelet analysis, and 7.3% in 1998 using the wavelet technique.

## Who Is Using This Method?

• Eva Strand, Department of Rangeland Ecology and Management, University of Idaho, Moscow, Idaho, 83844-1135, email: evas@uidaho.edu, voice: 208-885-5779
• Alistair Smith, Department of Forest Resources, University of Idaho, Moscow, Idaho, 83844-1133, email: alistair@uidaho.edu, voice: 208-885-1009
• http://www.cnr.uidaho.edu/measurements/Research/swa.htm

## Existing datasets

• Spatial wavelet analysis has been conducted on 1-m black and white aerial photography for 48 square kilometers spaced across the Owyhee Mountains of southwestern Idaho

## Implementation help

• Eva Strand, Department of Rangeland Ecology and Management, University of Idaho, Moscow, Idaho, 83844-1135, email: evas@uidaho.edu, voice: 208-885-5779
• Alistair Smith, Department of Forest Resources, University of Idaho, Moscow, Idaho, 83844-1133, email: alistair@uidaho.edu, voice: 208-885-1009