The Tasseled Cap Transformation in remote sensing is the conversion of the readings in a set of channels into composite values; i.e., the weighted sums of separate channel readings. One of these weighted sums measures roughly the brightness of each pixel in the scene. The other composite values are linear combinations of the values of the separate channels, but some of the weights are negative and others positive. One of these other composite values represents the degree of greenness of the pixels and another might represent the degree of yellowness of vegetation or perhaps the wetness of the soil. Usually there are just three composite variables.

Something very much like the tasseled cap transformation could have arisen from strictly empirical observations. Principal component analysis could have provided the inspiration and guide for specifying the tasseled cap transformations. Principal component analysis creates new variables as weighted sums of the different channel readings. Typically the first few components contain most of the information in the data so that four channels of LANDSAT MSS data or the six channels of the Thematic Mapper data may be reduced to just three principal components. The componenets higher than three are usually treated as being informationless noise.

The weights used in principal component analysis are determined statistically from the data but it was soon observed that typically the first principal component typically corresponded to roughly equal weights. In other words, the data generally fall along the diagonal when channel values are plotted together. If the weights used in a weighted-sum transformation are equal then the values obtained are proportional to the sum of the channel values and hence correspond to "brightness."

Principal component analysis is equivalent to transforming the data to a new coordinate system with a new set of orthogonal axes. The tasseled cap transformation also corresponds to a transformation of the data to a new set of orthogonal axes. While the tasseled cap transformation was inspired by the method of principal component analysis combined with generalization from empirical observations the actual details had a more analytical basis.

The tasseled cap transformation was presented in 1976 by R.J. Kauth and G.S. Thomas of Environmental Research Institute of Michigan in an article entitled "The tasseled Cap -- A Graphic Description of the Spectral-Temporal Development of Agricultural Crops as Seen by LANDSAT." This paper was published in the Proceedings of the Symposium on Machine Processing of Remotely Sensed Data, which was printed by Purdue University of West Lafayette, Indiana. In this article Kauth and Thomas provide the rationale for the patterns found for LANDSAT data from crop lands as a function of the life cyle of the crop.

The data for bare soil can vary because of the character of the soil or as a function of sun angle in relation to the slope of the fields. In the diagram shown below the bare soil values A1 and B1 fall along a line through the origin. As a crop such as wheat emerges from dark soil there is an increase in reflectance in the near IR band because of the reflectance of chlorophyll but a decrease in the red band because of chlorophyll's absorption of red light. Also the plants create shadows that result in lower readings from the soil. The shadowing of the soil by the plants will depend upon the orientation of the crop rows compared to the angle of the sun. If the sun is shining down the rows the shadows on the soil will be less that if the crop rows are prependicular to the direction of the sun.

The net result is that as the wheat plants grow the near IR readings have a net gain and the red readings have a net loss as shown by the points A2 and A3 in the diagram below. For light soils the pattern may be different. The light soil has a high enough reflectance that as the wheat plants grows the reflectance of the plant even in the near IR band is not enough replace the loss of reflectance from the light soil so readings decrease in both bands but more so in the red band than in the near IR band. This is shown by points B2 and B3 in the diagram.

When the wheat reaches maturity and in effect creates a closed canopy the reflectances for the dark and light soil fields converge to points A4 and B4. From this point the readings are the same for the dark and light soil fields. As the wheat crop ripens it starts to turn yellow, as in points A5 and B5. When the wheat is fully ripens the reflectance in the near IR drops substantially as is the case for points A6 and B6.

In a real wheat field there will be a distribution of the readings for bare soil along a ellipsoid as a result of variations in soil types and land angle relative to the sun angle. With growing wheat there will be variation among the wheat plants as to the stage in the life cycle. Thus the plot of the data for a wheat field may resemble a cap as shown in the diagram below, which is based up on the depiction in Kauth and Thomas' article.

Kauth and Thomas define their tasseled cap transformation relying upon the above diagram. One component of their transfromation is the weighted sum where the weights are statistically derived. In the original formulation the weights are not all equal. Later analyses simlified the transformation to be the sum of the channel readings and the result is characterized as "brightness." The second component is perpendicular to the first and its axis passes through the point of maturity of the plants, corresponding to points A4 and B4 in the previous diagram. The third component corresponds to an axis perpendicular to the first and second and passing through the point which represents ripened wheat, the "yellow stuff." The fourth component represents projection onto an axis perpendicular to the other three. Kauth and Thomas call it "non-such" and treat it as random noise.

The weights used by Kauth and Thomas the tasseled cap transformation of LANDSAT MSS data are shown below:

The weights for the first component were found from data for Fayette County, Illinois. The green point was based upon a cluster for corn field data in Fayette County as well. Kauth and Thomas had no yellow point for the Fayette County data so they used the spectrum of yellow corn to approximate what the yellow point would be.

Crist and Cicone adapted the tasseled cap transformation to the six channels of Thematic Mapper data. The weights are different and the third component is taken to represent soil wetness rather than yellowness as in Kauth and Thomas' original formulation. The weights found by Crist and Cicone for the Thematic Mapper bands were:

Bibliography:

• R.J. Kauth and G.S. Thomas, "The tasseled Cap -- A Graphic Description of the Spectral-Temporal Development of Agricultural Crops as Seen by LANDSAT." Proceedings of the Symposium on Machine Processing of Remotely Sensed Data, Purdue University of West Lafayette, Indiana, 1976, pp. 4B-41 to 4B-51.
• E.P. Crist and R.J. Kauth, "The tasseled Cap De-Mystified," in Photogrammetric Engineering and Remote Sensing, Vol. 52, No. 1, January 1986, pp. 81-86.
• Eric P. Crist and Richard C. Cicone, "Application of the tasseled Cap Concept to Simulated Thematic Mapper Data," in Photogrammetric Engineering and Remote Sensing, 1984, pp. 343-352.
• Eric P. Crist and Richard C. Cicone, "A Physically-Based Transformation of Thematic Mapper Data--The TM Tasseled Cap," in IEEE Transactions on Geoscience and Remote Sensing, Vol. 22, No. 3, May 1984, pp. 256-263