Method and apparatus for representing image data using polynomial approximation method and iterative transformation-reparametrization technique

In a computer system having a processor means and a memory coupled to said processor means, a method for representing a curve, said method comprising the steps of:

• i) storing in a first memory means a sampled data point curve representation;
• thereafter
• ii) approximating with said processor means said curve utilizing said curve representation by:
• a) demarcating with said processor means a current segment;
• b) lengthening with said processor means said current segment by appending an adjacent fragment of said curve representation to said current segment, thereby lengthening said current segment;
• c) parameterizing with said processor means said lengthened current segment employing a first parametrization table;
• d) deriving with said processor means a set of cosine coefficients via a forward cosine-type transform on said lengthened current segment;
• e) truncating with said processor means said set of cosine coefficients, thereby obtaining a set of truncated cosine coefficients;
• f) modifying with said processor means said set of truncated cosine coefficients to satisfy at least two specific edge conditions, thereby obtaining a set of truncated and modified cosine coefficients;
• g) obtaining with said processor means a parametric representation of an approximation of said lengthened current segment via an inverse cosine-type transform on said set of truncated and modified cosine coefficients;
• h) computing with said processor means a first approximation error and storing said first approximation error in a memory array;
• i) if a trend of approximation errors tends toward a first predetermined goodness-of-fit threshold:
• i1) recomputing with said processor means said first parametrization table utilizing said approximation of said lengthened current segment;
• i2) if said first approximation error is worse than a second predetermined goodness-of-fit threshold, repeating with said processor means the method starting at step c;
• i3) if said first approximation error is not worse than said second predetermined goodness-of-fit threshold, forming a first set of segment descriptors representing said current segment and storing in a second memory means said first set of segment descriptors and repeating with said processor means the method starting at step b, taking said lengthened current segment as a new said current segment;
• and
• j) if said trend of approximation errors does not tend toward said first predetermined goodness-of-fit threshold, designating with said processor means the ending point of said current segment the starting point of a new said current segment, and repeating with said processor means the method starting at step b.

In a computer system having a memory means, processor means, and output means, a method for compressing a source curve comprising the steps of:

• a) digitizing said source curve to obtain a binary representation of said source curve;
• b) designating a first segment of said source curve;
• c) parameterizing said first segment to obtain an initial parametric representation of said first segment;
• d) transforming said initial parametric representation to obtain a cosine coefficient representation;
• e) truncating said cosine coefficient representation to obtain a truncated cosine coefficient representation;
• f) inverse transforming said truncated cosine coefficient representation to obtain a parametric representation of an approximated segment;
• g) computing a parametrization of said approximated segment from said parametric representation of said approximated segment;
• h) normalizing said parametrization of said approximated segment to obtain a successor parametrization of said first segment;
• i) reparameterizing said first segment in accordance with said successor parametrization to obtain a successor parametric representation of said first segment;
• j) transforming said successor parametric representation to obtain a new cosine coefficient representation;
• k) comparing said approximated segment to said first segment to determine if a predetermined goodness-of-fit criterion or convergence criterion has been met;
• l) repeating said e), f), g), h), i), j) and k) steps until one of said predetermined criteria has been met;
• m) storing a final truncated cosine coefficient representation generated in said j) step as a compressed representation of said first segment;
• and
• n) entropy encoding said compressed representation to obtain a further compressed representation of said first segment.

A computer system for representing image data using polynomial approximation and iterative transformation-reparametrization, said computer system having a memory means and output means, said computer system comprising:

• a digitizer for inputting and converting a curve into a binary format;
• first memory location in said memory means for storing a first set of data points representing said curve including a first curve end in said binary format;
• processor means, coupled to said memory means, comprising means for representing successive abutting segments of said curve as first sets of cosine coefficients and means for optimizing said first sets of cosine coefficients to obtain second sets of cosine coefficients;
• and
• second memory location in said memory means for storing said second sets of cosine coefficients.

In a computer system having a processor and a memory means coupled to said processor means, a method for representing a curve comprising the steps of:

• digitizing a curve into a binary format with a digitizer means and storing at a first portion of said memory means a first set of data points representative of a first curve end;
• demarcating a piecewise segment of said curve with said processor means and storing said segment at a second portion of said memory means;
• deriving a first set of cosine coefficients approximately representing said piecewise segment with said processor means and storing said first cosine coefficients set at a third portion of said memory means;
• storing at a fourth portion of said memory means a second set of data points designating a piecewise segment ending point;
• and
• optimizing with said processor means said first cosine set and storing said optimized second cosine set at a sixth portion of said memory means.

In a computer system having a processor and a memory means coupled to said processor means, a method for representing a curve comprising the steps of:

• digitizing a curve into a binary format with a digitizer means and storing at a first portion of said memory means a first set of data points representative of a first curve end;
• forming a current segment having as a first segment end said first curve end with said processor means and storing said current segment at a second portion of said memory means;
• iteratively lengthening said current segment with said processor means to obtain a lengthened current segment and storing said lengthened current segment at a third portion of said memory means, parameterizing said lengthened current segment according to a parametrization table with said processor means and storing said parameterized current segment at a fourth portion of said memory means, performing a forward cosine-type transform on said parameterized current segment to derive a first set of cosine coefficients with said processor means and storing said first cosine set at a fifth portion of said memory means, truncating said first cosine set to form a truncated set according to a predetermined degree of approximation with said processor means and storing said truncated first cosine set at a sixth portion of said memory means, modifying said truncated set to satisfy at least two edge conditions defining said lengthened current segment to obtain a modified set with said processor means and storing said modified set at a seventh portion of said memory means, performing an inverse cosine-type transform on said modified set to form an approximation of said lengthened current segment with said processor means and storing said approximation at an eighth portion of said memory means, deriving a parametrization of said approximation and storing said parametrization of said approximation at a ninth portion of said memory means, normalizing said parametrization of said approximation to obtain a normalized parametrization and storing said normalized parametrization at a tenth portion of said memory means, applying said normalized parametrization to said lengthened current segment to obtain a successor parametrized current segment and storing said successor parametrized current segment at said fourth location of said memory means, computing an approximation error between said approximation and said lengthened current segment with said processor means and storing said approximation error at a eleventh portion of said memory means;
• designating a next-to-last-lengthened current segment of said piecewise segment if a trend of approximation errors does not tend toward a predetermined goodness-of-fit threshold with said processor means and storing said designated next-to-last-lengthened current segment at a twelfth portion of said memory means as a current piecewise segment;
• deriving a second set of cosine coefficients approximately representing said piecewise segment with said processor means and storing said second cosine coefficients set at an thirteenth portion of said memory means;
• and
• storing at a fourteenth portion of said memory means a second set of data points designating a piecewise segment ending point.