Elder and Zucker begin by making the assumption that focally or penumbrally blurred edges can be characterized by a sigmoidal intensity gradient. Given this approximation, call the point of inflection in the sigmoidal edge (where the derivative changes concavity) the single pixel localization of the blurred edge. The basic Elder and Zucker algorithm therefore computes the second derivative of the intensity at every pixel in the direction of the intensity gradient. When this second derivative changes sign (along the direction of the gradient) consider this to be the inflection of a sigmoidal edge and label it an edge pixel.

What we find is that the extent of the blurring can cause these sigmoidal edges to vary in size across shallow field photographs or obliquely shadowed images. While in principle, one could easily compute "edges" at various different resolutions by simply running a standard edge detector on each image in a multiscale pyramid, little is known about how to combine this multiscale information. Most scale-space methods that attempt to characterize edges at different scales either require artificial scale thresholds or make an a priori assumption as to the relative importance of scales (e.g., coarse-to-fine tracking). Additionally, most methods do not compute two different scales for each locale in the image (Elder and Zucker do so twice: once for the gradient estimation operator and once for the second derivative estimate).

In order to perform the scale assignment at each pixel, the algorithm computes threshold values for the derivative and second derivative filters at various scales by analyzing the effects of these filters on normally distributed sensor noise. They argue that the most general method of dealing with multiscale edges it to use, at each point in the image, the filter with the smallest standard deviation which still yields results surpassing the threshold for that filter and for that image. Intuitively, this means that very tight filters should be used to detect sharp, high signal-to-noise-ratio edges, and broader filters may be required for wide edges that have large separation from neighboring events. Running a small filter on such a wide edge should not yield gradient magnitudes that are significantly above the noise threshold -- which, in the algorithm, would directly translate to the use of a higher scale.

Therefore, the choice of thresholding values is critical to the success of this algorithm. Since we are interested in minimizing the scale as much as possible, and since smaller scales are more sensitive to noise than larger ones, the choice of critical values is especially important for the lower scales.

The algorithm is implemented as follows:

• Choose a series of scales which are considered for local scales
• Compute the gradient of the image using a steerable filter.
  • Compute two directional derivatives of the gaussian, in the x and y direction.
  • Convolve the image with the two directional derivatives above to obtain the basis directional derivatives of the image in the x and y direction
  • Compute the angle of the gradient as the arctangent of the response in the x direction divided by the response in the y direction.
  • Steer the basis above to get the magnitude of the first derivative (gradient) of the image.
  • Optimize this step by taking the fourier transform of the image and the gaussian filter and multiplying them in the frequency domain as opposed to convolving them in the space domain.
• As the standard deviation of the gaussian in the filter is decreased, attribute a 'scale' to each pixel.
  • Try to find the minimum reliable scale for which the likelihood of error due to sensor noise is statisically determined to be below a specified confidence (determined from an estimate of the sensor noise and assuming that this noise is normally distributed). The smallest standard deviation derivative filter which yields a gradient above the threshold necessary to satisfy this confidence is selected based on the magnitude of the gradient.
  • Update the minimum reliable gradient image, substituting in newly computed values of gradient magnitude and direction whenever a smaller standard deviation directional derivative yields a gradient that passes threshold.
  • What this step actually does: Note that sharp edges exhibit a larger signal to noise ratio.

    In large gradient operators, there is better signal to noise detection.

    If the edge is blurry, a large scale gradient operator is necessary to detect the edge since the signal to noise ratio is poor.

    Start with the large scale (lowest resolution), and continue down to smaller and smaller scales which still give a reliable output for that locale in the image.

    As proposed by Elder and Zucker, the minimum reliable scale for estimating the gradient of the edge is defined by the smallest scale at which the edge response exceeds the significance threshold. (more about the significance threshold below)

• Choose a series of scales which will be considered local scales of the second derivative.
• Compute the laplacian filters for the various scales.
• The laplacian of the image in the direction of the gradient is computed on each scale.
  • Steer the image response to the laplacian filter in the direction of the gradient (as determined above), to compute this.
  • Again, optimize this step by performing matrix multiplication in the frequency domain as opposed to convolution in the spatial domain.
• As the size of the filter is decreased(scale down), again attribute a 'scale' to each pixel.
• The choice of scale is dependent on the minimum reliable scale for estimating the laplacian.
• Define 'edgels' at all the zero crossings of the approximation to the second-derivative of the image.