Constraints and proofs are provided by Shih and Pu (see ibid., vol.43, no.2, p.539, 1995) to justify their previous analysis of the idempotent soft morphological filters. It is shown that for a special case, the idempotency in Theorems 3 and 4 of a previous paper by the authors will not hold in the first stage, but the root signal will be produced in the second stage. The exact constraint is added to ensure the idempotency to be valid for the soft morphological closing and opening

We present the properties of soft morphological operations and the new definitions of binary soft morphological operations. It is shown that soft morphological filtering an arbitrary signal is equivalent to decomposing the signal into binary signals, filtering each binary signal with a binary soft morphological filter, and then reversing the decomposition. This equivalence allows problems in the analysis and the implementation of soft morphological operations in real time by using only logic gates for binary signals instead of sorting the numbers. The architectures of logic-gate implementation of soft morphological operations are also presented. Furthermore, unlike standard morphological filters, the soft morphological closing and opening are in general not idempotent. We develop the conditions and properties for a new class of idempotent soft morphological filters

Gray-scale soft mathematical morphology is the natural extension of binary soft mathematical morphology which has been shown to be less sensitive to additive noise and to small variations. But gray-scale soft morphological operations are difficult to implement in real time. In this Note, a superposition property called threshold decomposition and another property called stacking are applied successfully on gray-scale soft morphological operations. These properties allow gray-scale signals and structuring elements to be decomposed into their binary sets respectively and operated by only logic gates in new VLSI architectures, and then these binary results are combined to produce the desired output as of the time-consuming gray-scale processing.

In this paper, we introduce efficient pipeline archi- tectures for the recursive morphological operations. The standard morphological operation is applied directly on the original in- put image and produces an output image. The order of image scanning in which the operator is applied to the input pixels is irrelevant. However, the intent of the recursive morphological operations is to feed back the output at the current scanning pixel to overwrite its corresponding input pixel to be considered into computation at the following scanning pixels. The resultant output image by recursive morphology inherently depends on the image scanning sequence. Two pipelined implementations of the recursive morphological operations are presented. The design of an application-specific systolic array is first introduced. The systolic array uses 3 n cells to process an n x n image in 6 n-2 cycles. The cell utilization rate is 100%. Second, a parallel program implementing the recursive morphological operations and running on distributed-memory multicomputers is described. Performance of the program can be finely tuned by choosing appropriate partition parameters.

A simple and efficient algorithm using the maxima tracking approach on Euclidean distance transform to detect skeleton points is presented. The advantages of the skeleton obtained are: (1) connectivity preservation; (2) single-pixel in width; and (3) its locations as close as to the most symmetrical axes. Besides, the condition of the least slope change of skeleton is used to ensure the fairness of the digital medial axes. With the least effort, the algorithm can be modified to eliminate non-significant short skeletal branches originating from the object contour while the critical shape-informative medial axes are preserved.

A useful morphological shape description tool called geometric spectrum (G-spectrum) for quantifying the geometric features of multidimensional binary images is described. The basis of this tool relies upon the cardinality of a set of non-overlapping segments in an image using morphological operations. G-spectrum is not only used as a shape descriptor for describing the geometric features but also as a tool for shape recognition. In this paper, a shape recognition algorithm using G-spectrum is presented. The test results show that shape recognition using G-spectrum is satisfactory

The properties of soft morphological operations and the new definitions of binary soft morphological operations are presented. It is shown that soft morphological filtering on an arbitrary signal is equivalent to decomposing the signal into binary signals, filtering each binary signal with a binary soft morphological filter, and then reversing the decomposition. This equivalence allows problems in the analysis and the implementation of soft morphological operations in real time by using only logic gates for binary signals instead of sorting numbers.<>

A useful morphological shape description tool is presented called geometric spectrum or G-spectrum , for quantifying the geometric features on multidimensional binary images. The basis of this tool relies upon the cardinality of a set of non-overlapping segments in an image using morphological operations. The G-spectrum preserves the translation invariance property. With a chosen set of isotropic structuring elements the G-spectrum also preserves the rotation invariance. After the procedure of normalization, the G-spectrum can also preserve the scaling invariance. The properties and proofs of the G-spectrum are discussed.

A skeletonization algorithm based on the Euclidean distance function using the sequential maxima-tracking method is described which, when applied to a connected image, generates a connected skeleton composed of simple digital arcs. With a slight modification, the algorithm can preserve the more important features in the skeletal branches which touch the object boundary at corners. Therefore its application to shape recognition can be easily achieved

A novel concept for determining the root called backpropagation morphology is defined. The backpropagation morphology feeds back the immediate result to replace the input sample and continues until it scans to the end. The theorems of a two-scan algorithm using backpropagation morphology to derive the root generation without recursively applying forward morphology are developed. The algorithm's operation is independent of the object's size and saves significant computation time compared with the number of iterations in forward morphology. A systolic array for efficiently processing the two-scan operation is also presented. The array uses 3n cells to process an n×n image in 6n-2 cycles. The cell utilization is 100%. Also studied is the implementation of the two-scan algorithm on a distributed-memory multicomputer. A programming paradigm called pipelined data parallelism is used to develop the parallel program, which is asynchronous, data driven, and very efficient. Performance of the program can be tuned by choosing appropriate partition parameters

A novel medial axis transformation (MAT) algorithm extracted from the Euclidean distance transform of a binary image is presented. The extracted MAT satisfies the following properties: reconstructivity, rotation-invariance, connectivity, and single-pixel width. The preservation of properties is proved, and some experimental results are shown. The skeleton is trimmed by removing short branches to make it simpler and useful for object recognition.<>