On gray scale image measurements. I. Arc length and area

We investigate methods for approximating the arc length of a planar curve and the area of a region whose boundary is a closed curve where the data are taken from a rectangular lattice of gray scale values. The commonly used sample-count method is a simple idea. Arc length is estimated by counting pixels along the curve. Typically the relative error is larger than desired. Area is estimated by counting interior pixels. The relative error is usually smaller than that for arc length, but identifying interior pixels may be a difficult geometric problem and adds computational overhead. The sample-distance methods for measuring arc length is also standard. This method requires the curve pixels to be ordered geometrically. The distances between consecutive pixels are summed to give an estimate of arc length. Although the relative errors are small, the required geometric ordering adds complexity to the problem. We introduce the sample-normal method for estimating arc length and area from gray level data. This method requires construction of unit normal vectors for the sampled curve points. Local estimates of arc length at the pixels are made from the normal vectors. The area can be approximated by using the normal vectors in the divergence theorem from calculus. Two major advantages for this method are that no geometric ordering of data points is required and that the algorithms are easily implemented. We compare the sample-normal method to the sample-count and sample-distance methods using both artificially created data and actual 8-bit digital images.