### Abstract

We consider the inverse problem associated with IFSM: Given a target function f, find an IFSM, such that its fixed point f is sufficiently close to f in the L^{p} distance. Forte and Vrscay [1] showed how to reduce this problem to a quadratic optimization model. In this paper, we extend the collage-based method developed by Kunze, La Torre and Vrscay ([2][3][4]), by proposing the minimization of the 1-norm instead of the 0-norm. In fact, optimization problems involving the 0-norm are combinatorial in nature, and hence in general NP-hard. To overcome these difficulties, we introduce the 1-norm and propose a Sequential Quadratic Programming algorithm to solve the corresponding inverse problem. As in Kunze, La Torre and Vrscay [3] in our formulation, the minimization of collage error is treated as a multi-criteria problem that includes three different and conflicting criteria i.e., collage error, entropy and sparsity. This multi-criteria program is solved by means of a scalarization technique which reduces the model to a single-criterion program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented.

Original language | English |
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Title of host publication | ICNPAA 2016 World Congress |

Subtitle of host publication | 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences |

Publisher | American Institute of Physics Inc. |

Volume | 1798 |

ISBN (Electronic) | 9780735414648 |

DOIs | |

Publication status | Published - Jan 27 2017 |

Event | 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016 - La Rochelle, France Duration: Jul 4 2016 → Jul 8 2016 |

### Conference

Conference | 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016 |
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Country | France |

City | La Rochelle |

Period | 7/4/16 → 7/8/16 |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*ICNPAA 2016 World Congress: 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences*(Vol. 1798). [020090] American Institute of Physics Inc.. https://doi.org/10.1063/1.4972682

**IFSM fractal image compression with entropy and sparsity constraints : A sequential quadratic programming approach.** / Kunze, Herb; La Torre, Davide; Lin, Jianyi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*ICNPAA 2016 World Congress: 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences.*vol. 1798, 020090, American Institute of Physics Inc., 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, La Rochelle, France, 7/4/16. https://doi.org/10.1063/1.4972682

}

TY - GEN

T1 - IFSM fractal image compression with entropy and sparsity constraints

T2 - A sequential quadratic programming approach

AU - Kunze, Herb

AU - La Torre, Davide

AU - Lin, Jianyi

PY - 2017/1/27

Y1 - 2017/1/27

N2 - We consider the inverse problem associated with IFSM: Given a target function f, find an IFSM, such that its fixed point f is sufficiently close to f in the Lp distance. Forte and Vrscay [1] showed how to reduce this problem to a quadratic optimization model. In this paper, we extend the collage-based method developed by Kunze, La Torre and Vrscay ([2][3][4]), by proposing the minimization of the 1-norm instead of the 0-norm. In fact, optimization problems involving the 0-norm are combinatorial in nature, and hence in general NP-hard. To overcome these difficulties, we introduce the 1-norm and propose a Sequential Quadratic Programming algorithm to solve the corresponding inverse problem. As in Kunze, La Torre and Vrscay [3] in our formulation, the minimization of collage error is treated as a multi-criteria problem that includes three different and conflicting criteria i.e., collage error, entropy and sparsity. This multi-criteria program is solved by means of a scalarization technique which reduces the model to a single-criterion program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented.

AB - We consider the inverse problem associated with IFSM: Given a target function f, find an IFSM, such that its fixed point f is sufficiently close to f in the Lp distance. Forte and Vrscay [1] showed how to reduce this problem to a quadratic optimization model. In this paper, we extend the collage-based method developed by Kunze, La Torre and Vrscay ([2][3][4]), by proposing the minimization of the 1-norm instead of the 0-norm. In fact, optimization problems involving the 0-norm are combinatorial in nature, and hence in general NP-hard. To overcome these difficulties, we introduce the 1-norm and propose a Sequential Quadratic Programming algorithm to solve the corresponding inverse problem. As in Kunze, La Torre and Vrscay [3] in our formulation, the minimization of collage error is treated as a multi-criteria problem that includes three different and conflicting criteria i.e., collage error, entropy and sparsity. This multi-criteria program is solved by means of a scalarization technique which reduces the model to a single-criterion program by combining all objective functions with different trade-off weights. The results of some numerical computations are presented.

UR - http://www.scopus.com/inward/record.url?scp=85013627084&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013627084&partnerID=8YFLogxK

U2 - 10.1063/1.4972682

DO - 10.1063/1.4972682

M3 - Conference contribution

AN - SCOPUS:85013627084

VL - 1798

BT - ICNPAA 2016 World Congress

PB - American Institute of Physics Inc.

ER -