Determining Parameters for Images Amplification by ...

Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM (artículo arbitrado)

Determining Parameters for Images Amplification by Pulses Interpolation Determinación de parámetros para amplificación de imágenes mediante interpolación de pulsos Morera-Delfín Leandro Telecommunications’ Department Electrical Engineering Faculty Higher Polytechnic Institute Jose Antonio Echeverria E-mail: [email protected] Information on the article: received: June 2013, reevaluated: June, August, and September 2013, accepted: January 2014

Abstract This paper presents the implementation of a method for image samples interpolation based on a physical scanning model. It uses the theory to take digital image samples and to perform an implementation of such mechanism through software. This allows us to get the appropriate parameters for ‘Žȱ’–ŠŽœȱŠ–™•’ęŒŠ’˜—ȱžœ’—ȱŠȱ›ž—ŒŠŽȱœŠ–™•Ž›ȱŠ››Š—Ž–Ž—ǯȱ‘Žȱœ‘˜ —ȱ process copies the physical model of image acquisition in order to incorpo›ŠŽȱ‘Žȱ›Žšž’›ŽȱœŠ–™•Žœȱ˜›ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ǯȱ‘’œȱ™›˜ŒŽœœȱ’œȱžœŽž•ȱ’—ȱ‘Žȱ reconstruction of details in low resolution images and for images compression. The proposed method studies the conservation of high frequency in the ‘’‘ȱ›Žœ˜•ž’˜—ȱ™•Š—Žȱ˜›ȱ‘ŽȱŽ—Ž›Š’˜—ȱ˜ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ȱ”Ž›—Ž•ǯȱȱ—Ž ȱ way of direct application of the physical model for scanning images in analytic mode is presented.

Keywords: ‡ ‡ ‡ ‡ ‡

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Resumen Este trabajo muestra la aplicación de un método para la interpolación de muestras sobre una imagen basado en un modelo físico de digitalización de las mismas. Se utiliza la fundamentación teórica de muestreo de la imagen digital y se realiza una implementación de dicho mecanismo a través de un software. Esto permite obtener •˜œȱ ™Š›¤–Ž›˜œȱ ŠŽŒžŠ˜œȱ ™Š›Šȱ Š–™•’ęŒŠ›ȱ •Šœȱ ’–¤Ž—Žœȱ ž’•’£Š—˜ȱ Ž•ȱ Š››Ž•˜ȱ Žȱ muestreo truncado. Se imita el procedimiento físico de obtención de la imagen y se ’—Œ˜›™˜›Š—ȱ•Šœȱ–žŽœ›Šœȱ›ŽšžŽ›’Šœȱ™Š›Šȱ•ŠȱŠ–™•’ęŒŠŒ’à—ǯȱœŽȱ™›˜ŒŽœ˜ȱŽœȱø’•ȱŽ—ȱ•Šȱ reconstrucción de detalles de imágenes de baja resolución y para la compresión de las mismas. El método propuesto estudia la conservación de las altas frecuencias en el ™•Š—˜ȱŽȱ–Š¢˜›ȱ›Žœ˜•žŒ’à—ȱ™Š›Šȱ•ŠȱŽ—Ž›ŠŒ’à—ȱŽ•ȱ—øŒ•Ž˜ȱŽȱŠ–™•’ęŒŠŒ’à—ȱ¢ȱ™›ŽœŽ—Šȱ una nueva forma de aplicar directamente el modelo físico de escaneo de la imagen en modo analítico.

Descriptores: ‡ ‡ ‡ ‡ ‡

interpolación reconstrucción de imágenes ampliación muestreo digitalización

Determining Parameters for Images Amplification Pulses Interpolation

Introduction This paper proposes a method for the construction of a ™ž•œŽȱ’—Ž›™˜•Š’˜—ȱꕝŽ›ȱ ’‘ȱ–Š¡’–ž–ȱ›Žœ™˜—œŽȱŠ–™•’žŽȱŠȱ‘’‘ȱœ™Š’Š•ȱ›ŽšžŽ—Œ’Žœǯȱ‘’œȱŠ••˜ œȱŠȱ‹ŽĴŽ›ȱ reconstruction of the details including the use of the Œ‘Š›ŠŒŽ›’œ’Œœȱ˜ȱ‘’œȱ¢™Žȱ˜ȱŠ–™•’ęŒŠ’˜—ȱ’—ȱ˜›Ž›ȱ˜ȱ achieve linear phase response in the edge transitions of the high resolution image. The preservation of edge information in an accurate frequency range gives a har–˜—’˜žœȱ Ÿ’œžŠ•ȱ ŽěŽŒȱ ˜ȱ ‘Žȱ ’–ŠŽȱ Š—ȱ ™›ŽŸŽ—œȱ ‘Žȱ ˜›–Š’˜—ȱ˜ȱ‹•˜Œ”œǯȱ‘’œȱ¢™Žȱ˜ȱꕝŽ›ȱ’œȱžœŽȱ’—ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ǰȱ ’–ŠŽȱ ›ŽŒ˜—œ›žŒ’˜—ȱ Š—ȱ œ’–ž•Š—Ž˜žœ•¢ȱ facilitates the compression process. —ȱ œ˜–Žȱ Š™™•’ŒŠ’˜—œǰȱ Šœȱ Š—ȱ ’–Š’—ȱ Š–™•’ꎛȱ ˜›ȱ ™ŠŒŽȬž•’™•Ž¡Žȱ™’ŒŠ•ȱ›Š—œ–’œœ’˜—ȱǻ£ž›ȱet alǯǰȱ 2012) it is necessary to focus/collimate the light beam to ‘ŽȱŒŽ—Ž›ȱ˜ȱ‘Žȱ‹ž•”ȱŠ–™•’ꎛȱ›˜–ȱ‘Žȱ’–Š’—ȱœ¢œŽ–œȱǻȱŗȱŠ—ȱȱŘǼȱ˜ȱ’ž›ŽȱŗȱŠ—ȱ‘Ž—ȱ˜ȱŒ˜ž™•Žȱ‹ŠŒ”ȱ ˜ȱ‘Žȱ˜ž™žȱꋎ›ǯȱ‘ŽȱŠ–™•’ęŒŠ’˜—ȱŽěŽŒȱ˜ȱ‘Žȱ‹ž•”ȱ could be seeing as the electrical output of a more dense Š—ȱœŽ—œ’‹•Žȱ™‘˜˜ȱŽŽŒ˜›ȱ£˜—Žȱ’—Ž›ŠŽȱ˜ŸŽ›ȱŠȱ™Žriod of time. During this period the light beam activates ‘Žȱ›Žœ˜•ž’˜—ȱŒŽ••œǰȱŠ—ȱ‘Ž—ȱ‘Žȱœ’–ž•Š’˜—ȱ˜ȱŠȱ™‘˜˜ȱ ŽŽŒ˜›ȱœŠ–™•’—ȱ£˜—Žȱ’œȱžœŽž•ȱ˜ȱœ’–ž•ŠŽȱ‘ŽȱŠ–™•’ęŒŠ’˜—ȱŽěŽŒœǯ The medical-image analysis requires an understan’—ȱ˜ȱœ˜™‘’œ’ŒŠŽȱœŒŠ——’—ȱ–˜Š•’’ŽœǰȱŒ˜—œ›žŒ’—ȱ Ž˜–Ž›’Œȱ –˜Ž•œǰȱ ‹ž’•’—ȱ –Žœ‘Žœȱ ˜ȱ ›Ž™›ŽœŽ—ȱ ˜–Š’—œǰȱ Š—ȱ ˜ —œ›ŽŠ–’—ȱ ‹’˜•˜’ŒŠ•ȱ Š™™•’ŒŠ’˜—œǯȱ ‘ŽœŽȱ˜ž›ȱœŽ™œȱ˜›–ȱŠ—ȱ’–ŠŽȬ˜Ȭ–Žœ‘ȱ™’™Ž•’—ŽȱǻŽŸ’ne et alǯǰȱŘŖŗŘǼǯ The proposed method could be used as an edgepreserving interpolation method after the denoise proŒŽœœȱ˜›ȱ—˜’œ¢ȱ’–ŠŽœǯȱ—ȱœ˜–ŽȱŒŠœŽœȱ‘Žȱ’–ŠŽȱ’œȱꛜȱ ŽŒ˜–™˜œŽȱžœ’—ȱ‘Žȱ‹’•ŠŽ›Š•ȱꕝŽ›ȱ’—˜ȱ‘ŽȱŽŠ’•ȱŠ—ȱ base layers which represent the small and large scale ŽŠž›Žœǰȱ ›Žœ™ŽŒ’ŸŽ•¢ǯȱ ‘Žȱ ŽŠ’•ȱ •Š¢Ž›ȱ ’œȱ ŠŠ™’ŸŽ•¢ȱ smoothed to suppress the noise before interpolation and an edge-preserving interpolation method is applied ˜ȱ‹˜‘ȱ•Š¢Ž›œǰȱ’ȱ’œȱŽěŽŒ’ŸŽȱ˜ȱŽ–™•˜¢ȱŽ—˜’œ’—ȱ™›’˜›ȱ ˜ȱ‘Žȱ’—Ž›™˜•Š’˜—ȱǻ ˜—ȱet alǯǰȱŘŖŗŖǼǯ

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72

˜–Žȱ –Ž‘˜œȱ ˜ȱ ’—Ž›™˜•Š’˜—ǰȱ •’”Žȱ ‘Žȱ Œž›ŸŠž›Žȱ ’—Ž›™˜•Š’˜—ȱ –Ž‘˜ȱ ǻǼǰȱ œž¢ȱ ‘Žȱ ŽŽȱ Œ˜–™˜œ’tion of the low resolution image to interpolate the curŸŠž›Žȱ˜ȱ‘Žȱ‘’‘Ȭ›Žœ˜•ž’˜—ȱ’–ŠŽȱ˜–Š’—ǯȱ‘Žȱȱ constructs the high-resolution image by solving a linea›’£Žȱ Œž›ŸŠž›Žȱ ŽšžŠ’˜—ǰȱ ’—Œ˜›™˜›Š’—ȱ ‘Žȱ ’—Ž›™˜•ŠŽȱŒž›ŸŠž›ŽȱŠœȱŠ—ȱŽ¡™•’Œ’ȱ›’Ÿ’—ȱ˜›ŒŽȱǻ Š”›Š—ȱet alǯǰȱ ŘŖŗŗǼǯȱ ‘Ž›ȱ ›Ž›Žœœ’˜—Ȭ‹ŠœŽȱ ’–ŠŽȱ ’—Ž›™˜•Š’˜—ȱ Š•˜›’‘–œȱ‘ŠŸŽȱ‹ŽŽ—ȱ™›˜™˜œŽȱ’—ȱ‘Žȱ•’Ž›Šž›Žǰȱ’—ȱ ‘’Œ‘ȱ ‘Žȱ˜‹“ŽŒ’ŸŽȱž—Œ’˜—œȱŠ›Žȱ˜™’–’£Žȱ‹¢ȱ˜›’—Š›¢ȱ•ŽŠœȱ œšžŠ›ŽœȱǻǼǯȱ ˜ ŽŸŽ›ǰȱ’ȱ‘Šœȱ‹ŽŽ—ȱœ‘˜ —ȱ‘Šȱ’—Ž›™˜•Š’˜—ȱ ’‘ȱȱ–Š¢ȱ‘ŠŸŽȱœ˜–Žȱž—Žœ’›Š‹•Žȱ™›˜™Ž›ties from a robustness point of view: even small Š–˜ž—œȱ˜ȱŠ¢™’ŒŠ•ȱŸŠ•žŽœȱŒŠ—ȱ›Š–Š’ŒŠ••¢ȱŠěŽŒȱ‘Žȱ Žœ’–ŠŽœȱǻ’žȱet alǯǰȱŘŖŗŗǼǯ ‘’œȱ’—ŸŽœ’Š’˜—ȱ›’Žœȱ˜ȱꗍȱ‘Žȱ•’–’œȱ˜ȱšžŠ•’¢ȱ’—ȱ the high resolution image. The curvature content in the high-resolution image domain is studied using classical ‘Ž˜›¢ȱ ›Žœ˜ž›ŒŽœǯȱ ȱ ›Žœ›’ŒŽȱ ’—Ž›™˜•Š’˜—ȱ œ¢œŽ–ȱ ’œȱ –ŠŽȱ  ’‘ȱ œ™Š’Š•ȱ Š—ȱ  ’—˜ ȱ ꕝŽ›œȱ ’—ȱ ˜›Ž›ȱ ˜ȱ ’—crease the high frequency content. The proposed method constructs an interpolation kernel using the high frequency content at high-resolution image domain as Š—ȱŽ¡™•’Œ’ȱ›’Ÿ’—ȱ˜›ŒŽȱ˜›ȱ‘ŽȱŸŠ›’Š’˜—ȱ˜ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ȱ”Ž›—Ž•ȱ™Š›Š–ŽŽ›œǯȱ’‘ȱ‘’œȱ™›˜ŒŽœœǰȱ‘Žȱ˜™’–Š•ȱ •˜ Ȭ‘’‘ȱ›Žœ˜•ž’˜—ȱ™Š’›ȱ’œȱ˜ž—ȱžœ’—ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ȱ”Ž›—Ž•ȱ’ŸŽ—ȱ‹¢ȱ‘Žȱ˜ž›’Ž›ȱ›Š—œ˜›–ȱ˜ȱ‘Žȱ›ž—ŒŠŽȱ œŠ–™•’—ȱ Š››Š—Ž–Ž—ȱ ǻŠ™˜ž•’œǰȱ ŗşŜŜǼǯȱ —ȱ ‘Žȱ ™›˜ŒŽœœȱ ꕝŽ›œȱ Šœȱ žĴŽ› ˜›‘ȱ ǻ›ŠĴǰȱ ŘŖŖŗǼȱ Š—ȱ Š——¢ȱ ǻŗşŞŜǼȱŠ›ŽȱžœŽȱ’—ȱ˜›Ž›ȱ˜ȱž’Žȱ‘ŽȱŒ˜—œ›žŒ’˜—ȱ˜ȱ‘Žȱ interpolation kernel and to raise the high frequency content in the high resolution image.

Image sampling system In a physical image sampling system the sampling Š››Š—Ž–Ž—ȱ œ‘˜ž•ȱ ‹Žȱ ˜ȱ ꗒŽȱ Ž¡Ž—ǯȱ ‘Žȱ œŠ–™•’—ȱ ™ž•œŽœȱŠ›Žȱ˜ȱꗒŽȱ ’‘ȱŠ—ȱ‘Žȱ’–ŠŽȱŒŠ—ȱ‹Žȱœž‹ȱœŠ–™•Žȱ ’‘ȱœ™ŽŒ›Š•ȱ˜ŸŽ›•Š™ǯȱŽŒŠžœŽȱ˜ȱ‘ŽœŽǰȱœ™ž›’˜žœȱ spatial frequency components will be introduced into ‘Žȱ ›ŽŒ˜—œ›žŒ’˜—ǯȱ ‘’œȱ ŽěŽŒȱ ’œȱ ŒŠ••Žȱ Š•’Šœ’—ȱ Ž››˜› ǻ›˜ —ǰȱŗşŜşDzȱ Ž•–œȱŠ—ȱ‘˜–ŠœǰȱŗşŜŘǼȱŠ—ȱ‘Ž›Ž˜›Žȱ ’ȱ’œȱ—ŽŒŽœœŠ›¢ȱ˜ȱŽ¡™•˜›Žȱ‘ŽȱŒ˜—œŽšžŽ—ŒŽœȱ˜ȱ‘Žȱ—˜—Ȭ ideal sampling. —ȱ‘ŽȱŽ¡Š–™•Žȱ˜ȱ’ž›ŽȱŘǰȱŠȱ‘’—ȱ‹ŽŠ–ȱ˜ȱ•’‘ȱ˜Žœȱ through a photographic transparency of an ideal imaŽǯȱŠœœ’—ȱ•’‘ȱ’œȱŒ˜••ŽŒŽȱ’—ȱŠȱŒ˜—Ž—œŽ›ȱ•Ž—œȱŠ—ȱ’œȱ sent directly to a photo detector. The electrical output from the photo detector is integrated over a period of time during which the light beam activates a resolution cell. In this type of system is considered that even lens

Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

Morera-Delfín Leandro

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with perfect focus produces some blurring because of ‘Žȱ’웊Œ’˜—ȱ•’–’ȱ˜ȱ’œȱ˜™Ž—’—ȱǻȂŽ’••ǰȱŗşŜřǼǯ The sampled image is

‘Žȱ˜ž›’Ž›ȱ›Š—œ˜›–ȱ˜ȱ‘Žȱ›ž—ŒŠŽȱœŠ–™•’—ȱŠ››Š—Ž–Ž—ȱǻŠ™˜ž•’œǰȱŗşŜŜǼȱ’œ

Fp  x, y FI  x, y ˜ S  x, y ȱ

ȱ

ȱ

ǻŗǼ

ȱ

ǻŘǼ

where the arrangement of samples is

¦ ¦ P( x  j'x, y  k 'y) ȱȱ J

S ( x, y )

K

j J k K

Š—ȱ ’œȱ Œ˜–™›’œŽȱ ˜ȱ ǻŘJȱ Ƹȱ ŗǼȱ ǻŘK + 1) identical pulses PǻxǰȱyǼȱŠ››Š—Žȱ’—ȱŠȱ›’ȱœ™ŠŒ’—ȱǻxǰȱy). Symmetrical •’–’œȱ˜ȱœž––Š’˜—ȱŠ›ŽȱŒ‘˜œŽ—ǰȱ˜›ȱ—˜Š’˜—Š•ȱœ’–™•’Œ’¢ǰȱ ŽȱŠœœž–Žȱ‘Šȱ‘ŽȱœŠ–™•’—ȱ™˜’—œȱŠ›ŽȱœŒŠ•Žǯ

³ ³ f

f

f f

P ( x, y )dxdy

1ȱ ȱ

ȱ

ǻřǼ

˜›ȱ ™ž›™˜œŽœȱ ˜ȱ Š—Š•¢œ’œȱ ’ȱ ’œȱ Šœœž–Žȱ ‘Šȱ ‘Žȱ œŠ–™•’—ȱ ž—Œ’˜—ȱ ’œȱ Ž—Ž›ŠŽȱ ‹¢ȱ Šȱ ꗒŽȱ Š››Š¢ȱ ˜ȱ ’›ŠŒȱ deltas DTǻxǰȱyǼȱ™Šœœ’—ȱ‘›˜ž‘ȱŠȱ•’—ŽŠ›ȱꕝŽ›ȱ ’‘ȱ’–pulse response Pǻxǰȱy) then

S ( x, y ) DT ( x, y ) P( x, y ) ȱ

ȱ

ȱ

ǻŚǼ

¦ ¦ G ( x  j'x, y  k 'y) ȱȱ J

DT ( x, y )

K

ǻśǼ

j J k K

Taking Equations 1 and 2 yields

¦ ¦ F  j'x, k 'y P( x  j'x, y  k 'y) ȱ J

FP ( x, y )

K

j J k K

I

ȱ

ǻŜǼ

The spectrum of the sampled image is given by

§ 1 · FP ( wx , wy ) ¨ 2 ¸ F ( wx , wy ) ¬ª DT ( wx , wy ) ˜ P( wx , wy ) ¼º ȱ ǻŝǼ © 4S ¹ where PǻwxǰȱwyǼȱ’œȱ‘Žȱ˜ž›’Ž›ȱ›Š—œ˜›–ȱ˜ȱPǻxǰȱy).

DT ( wx , wy )

1· 1· ­ § ½ ­ § ½ sin ® wx ¨ j  ¸ ˜ 'x ¾ ˜ sin ® wy ¨ k  ¸ ˜ 'y ¾ 2 2 ¹ ¹ ¯ © ¿ ¯ © ¿ȱ 'x ½ 'y ½ ­ ­ sin ® wx ˜ ¾.sin ® wy ˜ ¾ 2 ¿ 2 ¿ ¯ ¯

ǻŞǼ

‘’œȱž—Œ’˜—ǰȱŽŸŠ•žŠŽȱŠȱ‘Žȱ•’–’ȱ˜›ȱ•Š›ŽȱŸŠ•žŽœȱ˜ȱJ and Kǰȱ’œȱŒ˜—ŸŽ›Žȱ’—˜ȱŠ—ȱŠ››Š¢ȱ˜ȱ’›ŠŒȱŽ•Šœǯ —ȱŠ—ȱ’–ŠŽȱ›ŽŒ˜—œ›žŒ’˜—ȱœ¢œŽ–ǰȱŠ—ȱ’–ŠŽȱ’œȱ›Žconstructed by interpolation of their samples. The interpolation waveforms selected as Sine function or ŽœœŽ•ȱŽ—Ž›Š••¢ȱŽ¡Ž—ȱ˜ŸŽ›ȱ‘ŽȱŽ—’›Žȱꎕȱ˜ȱ‘Žȱ’–ŠŽȱǻ›ŠĴǰȱŘŖŖŗǼǯȱȱ‘ŽȱŠ››Š—Ž–Ž—ȱ’œȱŠȱ’œŒ˜—’—ž˜žœȱ œŠ–™•’—ǰȱ ‘Žȱ ›ŽŒ˜—œ›žŒŽȱ ’–ŠŽȱ  ’••ȱ Š’•ȱ —ŽŠ›ȱ ‘Ž’›ȱ ŽŽœǯȱ ˜ ŽŸŽ›ǰȱ ‘Žȱ ’œŠ—ŒŽȱ Ž››˜›ȱ ’œȱ —Ž•’’‹•Žȱ ’—ȱ Š‹˜žȱŞȱ˜ȱŗŖȱœŠ–™•Žœȱ˜ȱ¢šž’œȱǻ‹›Š–Š’ŒȱŠ—ȱŠžŽ›ŠœǰȱŗşŞŘǼǯ

Modeling the interpolation system In modeling the system a test image is taken for ampliꌊ’˜—ȱ‹¢ȱ™ž•œŽœȱ’—Ž›™˜•Š’˜—ǯȱ‘ŽȱŠ’–ȱ’œȱ˜ȱŠŒ‘’ŽŸŽȱŠȱ —Ž ȱ‘’‘Ž›ȱ›Žœ˜•ž’˜—ȱ’–ŠŽȱŒ˜—Š’—’—ȱ‘Žȱ‘’‘Žœȱędelity with the original image. The process does not ŒŠžœŽȱ ™’¡Ž•œȱ ‹•˜Œ”ȱ ŽěŽŒȱ Š—ȱ –Š’—Š’—œȱ ‘Žȱ Œ˜—’—ž’¢ȱ ˜ȱ ™‘ŠœŽȱ ’—ȱ ‘Žȱ ˜ž›’Ž›ȱ ›Š—œ˜›–ȱ ˜ȱ ‘Žȱ ›Žœž•ǯȱ ‘’œȱ allows a smooth visual path. It contains the details in •˜ Ȭ›ŽšžŽ—Œ¢ȱŠ›ŽŠœȱŠ—ȱŠŒ’•’ŠŽœȱ‘ŽȱŒ˜–™›Žœœ’˜—ǯȱȱ Œ˜—Ÿ˜•ž’˜—ȱ ”Ž›—Ž•ȱ ǻ‹›Š–Š’Œȱ Š—ȱ ŠžŽ›Šœǰȱ ŗşŝŞDzȱ ›ŠĴȱet alǯǰȱŗşŞŘDzȱ‹›Š–Š’ŒȱŠ—ȱŠžŽ›ŠœǰȱŗşŞŘǼȱ’œȱŽ—Ž›ŠŽȱ ‹¢ȱ ‘Žȱ ˜ž›’Ž›ȱ ›Š—œ˜›–ȱ ˜ȱ ‘Žȱ ›ž—ŒŠŽȱ œŠ–™•’—ȱŠ››Š—Ž–Ž—ȱǻŞǼǯ The space of the band limited functions in the frequency range Z  > S , S @ is œ™Š——Žȱ‹¢ȱ‘Žȱ’—ę—’Žȱǻ¢Žȱ countable) set of sine functions shifted by integers. ‘žœȱŠ—¢ȱœžŒ‘ȱ‹Š—ȱ•’–’Žȱž—Œ’˜—ȱǻǼȱŒŠ—ȱ‹Žȱ›ŽŒ˜—œtructed from its samples at integer spacing.

Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

73

Determining Parameters for Images Amplification Pulses Interpolation

¦ g (n) ˜ sin c(t  n) ȱ ȱ

n f

g (t )

ȱ

n f

ǻşǼ

ȱ

šžŠ’˜—œȱǻŝǼȱŠ—ȱǻşǼȱ¢’Ž•

§ 1 · GP ( wx , wy ) ¨ 2 ¸ G ( wx , wy ) ¬ª DT ( wx , wy ) ˜ P( wx , wy ) ¼º © 4S ¹ ȱȱȱȱȱȱǻŗŖǼ

˜ ȱŒ˜—œ’Ž›ȱŠȱœ™Š’Š•ȱ•’—ŽŠ›ȱ˜™Ž›Š˜›ȱ O ^` ˜ that produces an output image array

Q  m1 , m2 O ^F  m1 , m2 ` ȱ ȱ

ȱ

ǻŗŗǼ

The term O{Gǻt1ǰȱt2} for ti = mi – niȱƸȱŗȱ’œȱ‘Žȱ›Žœ™˜—œŽǰȱŠȱ ‘Žȱ˜ž™žȱŒ˜˜›’—ŠŽǰȱ˜ȱŠ—ȱ’—™žȱ˜ȱ˜—Žȱž—’ȱŠ–™•’žŽȱŠȱŒ˜˜›’—ŠŽȱǻn1ǰȱn2). It is called the impulse response function arrayȱ˜ȱ‘Žȱ•’—ŽŠ›ȱ˜™Ž›Š˜›ȱŠ—ȱ’œȱ ›’ĴŽ—ȱŠœ

G (m1  n1  1, m2  n2  1; m1 , m2 ) O ^G  t1 , t2 ` for 1 d t1 , t2 d L

The impulse response array can change form for each ™˜’—ȱ ǻm1ǰȱ m2) in the processed array Qǻm1ǰȱ m2Ǽǯȱ ˜••˜ ’—ȱ‘’œȱ—˜Š’˜—ǰȱ‘ŽȱꗒŽȱŠ›ŽŠȱœž™Ž›™˜œ’’˜—ȱ˜™Ž›Š’˜—ȱ’œȱŽę—ŽȱŠœ Q  m1 , m2

ȱ

ȱ

¦ ¦ F  n , n ˜ H (m  n  1, m

n1 f n2 f

n1 f n2 f

1

2

1

1

2

 n2  1; m1 , m2 )

n1 f n2 f

n1 f n2 f

1

2

1

 n1  Lc , j2  n2  Lc ; j1 , j2 )

ȱ

ǻŗřǼ

The limits of the summation are

( L  3) / 2 d ji d L  ( N  1) / 2 and Lc ( L  1) / 2 ȱȱȱȱȱȱȱȱȱȱǻŗŚǼ

—ȱ’ž›ŽȱŚȱ‘ŽȱŽ¡Š–’—Š’˜—ȱ˜ȱ‘Žȱ’—’ŒŽœȱ˜ȱ‘Žȱ’–™ž•œŽȱ ›Žœ™˜—œŽȱ Š››Š¢ȱ Šȱ ‘Ž’›ȱ Ž¡›Ž–Žȱ ™˜œ’’˜—œȱ ’—’ŒŠŽœȱ that M = N + L ȮȱŗǰȱŠ—ȱ‘Ž—ŒŽȱ‘Žȱ™›˜ŒŽœœŽȱ˜ž™žȱŠ››Š¢ȱ Q is of larger dimension than the input array F. —ȱ‘Žȱ’—Ž›™˜•Š’˜—ȱœ¢œŽ–ǰȱ‘Žȱ’—™žȱŠ››Š¢ȱȱ’œȱ‘Žȱ •˜ ȱ›Žœ˜•ž’˜—ȱ’–ŠŽȱ˜ȱǻNxǰȱNyǼǯȱ‘Ž—ǰȱžœ’—ȱǻŝǼǰȱǻŞǼȱŠ—ȱ ǻŗřǼǰȱ‘Žȱ˜ž™žȱŠŠȱŠ››Š¢ȱQCǻj1ǰȱj2) is obtained. Using ǻŗŖǼǰȱ‘ŽȱœŠ–™•Žœȱ˜ȱDTǻΝxǰȱΝy) for the interval in which Νȱ [–SǰȱSǾȱŠ›Žȱ’—œŽ›Žǯȱ’ž›Žȱřȱœ‘˜ œȱ‘˜ ȱ˜›'x = 1 and 'yȱƽȱŗǰȱLy = 2 Ȋ K and Lx = 2 Ȋ J.

Ly

Amp ˜ K ' inc ˜ co

(15)

Lx

Amp ˜ J ' inc ˜ co

(16)

where J’ and K’ are the dimensions of the low resolu’˜—ȱ’–ŠŽǰȱ‘’œȱ’œǰȱ‘Žȱ’—™žȱŠ››Š¢ȱǯǻŘŚǼ

Lcx ( Lx  1) / 2 ȱ ȱ

ȱ

Lcy ( Ly  1) / 2 ȱ ȱ

ȱ

ǻŗŘǼ

‘’œȱŽ¡™›ŽœœŽœȱ‘ŽȱꗒŽȬŠ›ŽŠȱœž™Ž›™˜œ’’˜—ȱ˜™Ž›Š’˜—ȱ ’—ȱ‘Žȱ•ŽȬ“žœ’ꮍȱ˜›–ȱ’—ȱ ‘’Œ‘ȱ‘Žȱ’—™žȱŠ—ȱ˜ž™žȱ arrays are aligned at their upper left corners. It is often —˜Š’˜—Ȭ ’œŽȱ Œ˜—ŸŽ—’Ž—ȱ ˜ȱ ž’•’£Žȱ Šȱ Žę—’’˜—ȱ ’—ȱ which the output array is centered with respect to the ’—™žȱŠ››Š¢ǯȱ‘’œȱŽę—’’˜—ȱ˜ȱŒŽ—Ž›Žȱœž™Ž›™˜œ’’˜—ȱ is given by

¦ ¦ F n ,n ˜ H ( j

Qc  j1 , j2

Qc  j1 , j2

¦ ¦

Zx K '/ 2 Z y J '/ 2 Zx  K '/ 2 Z y  J '/ 2

ȱ

ǻŗŞǼ

F Z x ,Z y ˜ DT ( j1  Z x  Lcx , j2  Z y  Lcy ; j1 , j2 )

ȱ

ǻŗşǼ

‹Š’—’—ȱ ‘Žȱ ’–™ž•œŽȱ ›Žœ™˜—œŽȱ ž—Œ’˜—ȱ Š››Š¢ȱ ’—ȱ ž—Œ’˜—ȱ˜ȱ‘ŽȱŽ¡™•˜›Š’˜—ȱ™Š›Š–ŽŽ›œȱinc and co.

)LJXUH5HODWLRQVKLSVEHWZHHQLQSXWGDWDRXWSXWGDWD DQGLPSXOVHUHVSRQVHDUUD\VIRUILQLWHDUHDVXSHUSRVLWLRQ FHQWHUHGDUUD\GHILQLWLRQWDNHQIURP3UDWW

74

ǻŗŝǼ

Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

Morera-Delfín Leandro

DT (Z x ', Z y '; j1 , j2 )

­° § L 1· ­ ½ °½ § L 1· sin ® j1  Z x  Lcx ˜ ¨ x ¸ ˜ 'x ¾ ˜ sin ® j2  Z y  Lcy ˜ ¨ y ¸ ˜ 'y ¾ 2 2 © ¹ °¯ °¿ ¯ ¿ © ¹ 'x ½ 'y ½ ­ ­ sin ® j1  Z x  Lcx ˜ ¾.sin ® j2  Z y  Lcy ˜ ¾ 2¿ 2 ¿ ¯ ¯

ǻŘŖǼ

( Amp ˜ K ' inc ˜ co  1) · § Amp ˜ J ' inc ˜ co  1 · °½ ( Amp ˜ J ' inc ˜ co  1) · § Amp ˜ K ' inc ˜ co  1 · °½ °­§ °­§ sin ®¨ j1  Zx  ¸˜¨ ¸ ˜ 'x ¾ ˜ sin ®¨ j2  Z y  ¸˜¨ ¸ ˜ 'y ¾ 2 2 2 2 °¯© ° ° ¹ © ¹ ¿ © ¹ © ¹ °¿ ¯ DT (Zx ', Z y '; j1 , j2 ) ( Amp ˜ J ' inc ˜ co  1) · 'y ½° ­°§ ( Amp ˜ K ' inc ˜ co  1) · 'x ½° °­§ sin ®¨ j1  Zx  ¸ ˜ ¾ .sin ®¨ j2  Z y  ¸˜ ¾ ǻŘŗǼ 2 2 2 ¹ 2 ¿° ¹ ¿° ¯°© ¯°©

ŸŠ•žŠ’—ȱ‘ŽȱŽ¡™›Žœœ’˜—ȱ˜›ȱΝx  [–SǰȱSǾǰȱΝy  [–SǰȱS] Š—ȱ žœ’—ȱ ǻŗşǼȱ  Žȱ ˜‹Š’—ȱ ‘Žȱ ˜ž™žȱ Š››Š¢ȱ Qcǻj1ǰȱ j2Ǽǰȱ  ‘’Œ‘ȱ‘Šœȱ’–Ž—œ’˜—œȱǻMxǰȱMyǼǰȱ ‘Ž›Ž

M x N x  Lx and M y

N y  Ly ȱ

ǻŘŘǼ

Then the inverse transformation of the output array gives the high resolution image ȱQS ( x, y )

Zx

¦

¦

M x / 2 Zy M y / 2

Zx  M x / 2 Z y  M y / 2

Qc Z x , Z y ˜ exp( j (Z x x  Z y y ))

ǻŘřǼ

—ȱ ˜›Ž›ȱ ˜ȱ Žę—Žȱ ‘Žȱ ™Š›Š–ŽŽ›œȱ inc and co of equa’˜—œȱ ǻŗśǼȱ Š—ȱ ǻŗŜǼǰȱ ‘Žȱ ‘’‘ȱ ›ŽšžŽ—Œ¢ȱ Žœ’–ŠŽȱ ˜ŸŽ›ȱ QSǻxǰȱ yǼȱ ’œȱ Š™™•’Žǯȱ™™•¢’—ȱ Šȱ žĴŽ› ˜›‘ȱ ‘’‘ȱ ™Šœœȱ ꕝŽ›ȱǻ›ŠĴǰȱŘŖŖŗǼȱ˜ȱ•Žœœȱ‘Š—ȱśȱ™’¡Ž•ȱŽ•Ž–Ž—œ QSf (Z xi , Z yi )

1

1 ª º 2 2 2 «§§ M y · § M x · · » « ¨¨ ¨ 10 ¸  ¨ 10 ¸ ¸¸ » ¹ ¹ » © ¹ © « 1 © 1 « » « Z yi 2  Zxi 2 2 » « » «¬ »¼

2 ˜5

˜ QS Z xi , Z yi

ǻŘŚǼ

™Š’Š•ȱꕝŽ›œȱ˜›ȱ‘ŽȱŽŽȱŽŽŒ’˜—ȱ˜ŸŽ›ȱ‘Žȱ‘’‘ȱ›Žœ˜•ž’˜—ȱ ’–ŠŽȱ Š›Žȱ žœŽȱ ’—ȱ Š—˜‘Ž›ȱ ŒŠœŽǯȱȱ Š——¢ȱ ¢™Žȱ ꕝŽ›ȱ’œȱŠ™™•’ŽȱǻŠ——¢ǰȱŗşŞŜǼȱ’—ȱ˜›Ž›ȱ˜ȱŽœ’–ŠŽȱ‘Žȱ amount of edge content and to restrict the parameters Ampǰȱinc and coȱ˜ȱŽšžŠ’˜—œȱǻŗśǼȱŠ—ȱǻŗŜǼǯȱ—ȱ‘’œȱŒŠœŽȱ‘Žȱ ’–™•Ž–Ž—Š’˜—ȱ˜ȱ‘ŽȱŠ——¢ȱ˜™Ž›Š˜›ȱ’ŸŽœ

QSCanny ( x, y ) Canny ^QS ( x, y )`

ǻŘśǼ

Then the appropriate values of Ly and Lx can be found in both cases by QSf Z x , Z y , Lx , Ly Qc  j1 , j2 ȱ

ȱ

ǻŘŜǼ

¦

¦

¦

¦

x M x / 2 y M y / 2 Zx M x / 2 Z y M y / 2

QSop ( Lx , Ly )

ȱ

ȱ

x  M x / 2 y  M y / 2 Zx  M x / 2 Z y  M y / 2

QSf Z x , Z y , Lx , Ly

˜ exp(  j (Z x x  Z y y ))

ǻŘŝǼ

MAX (QSop ( Lx , Ly )) ȱ ȱ

ȱ

ȱ

ǻŘŞǼ

§ w 2QSop ( Lx , Ly ) · ¨¨ ¸¸ 0 ȱ © wLx wLy ¹

ȱ

ȱ

ǻŘşǼ

žŽȱ ˜ȱ ‘Žȱ Œ˜–™•Ž¡’¢ȱ ˜ȱ ‘Žȱ ˜›–ž•Š’˜—ǰȱ —˜ȱ Š—Š•¢’Œȱ œ˜•ž’˜—ȱ‘Šœȱ‹ŽŽ—ȱ˜ž—ǰȱ‹žȱŠȱŸŠ›’Š’˜—Š•ȱŠ™™›˜ŠŒ‘ȱ‘Šœȱ ‹ŽŽ—ȱŽŸŽ•˜™ŽǯȱŽę—’—ȱŠȱę¡ŽȱŸŠ•žŽȱ˜›ȱ‘Žȱ•Š›Žȱ›Žœ˜•ž’˜—ȱ–˜’ꎛȱAmpȱƽȱŗǯŗǰȱ‘Žȱ–Š¡’–ž–ȱ›Žœ™˜—œŽœȱ˜ȱ ‘’‘ȱ ›ŽšžŽ—Œ’Žœȱ Š›Žȱ ˜ž—ȱ ‹¢ȱ ‘Žȱ ˜ž™žȱ ˜ȱ ‘Žȱ ꕝŽ›ȱ only for some combination values of the scan parameters inc and co. The high frequencies response of the ‘’‘ȱ ›Žœ˜•ž’˜—ȱ ’–ŠŽȱ žœ’—ȱ ‘Žȱ žĴŽ› ˜›‘ȱ ꕝŽ›ȱ ˜ȱ ‘ŽȱŽšžŠ’˜—ȱǻŘŖǼȱ’œȱœ‘˜ —ȱ’—ȱ’ž›Žȱśǯ ’ž›ŽœȱŜȱ˜›ȱŞȱœ‘˜ ȱ‘˜ ȱŠȱ™Ž›’˜’Œ’¢ȱŽ¡’œœȱ’—ȱŽŠŒ‘ȱ ›˜ Dzȱ ˜›ȱ Ž¡Š–™•Žǰȱ ’ȱ incȱ ƽȱ ŖǯŚȱ ‘Žȱ ‹Žœȱ ›Žœž•ȱ Š™™ŽŠ›œȱ ŽŸŽ›¢ȱśȱꐞ›ŽœȱŠ—ȱ‘Žȱ›Žœ˜•ž’˜—ȱ˜ȱ‘Žȱ˜ž™žȱ’–ŠŽȱ ’—Œ›ŽŠœŽœȱŠŒŒ˜›’—ȱ˜ȱŽšžŠ’˜—ȱǻŘŘǼǯȱ˜›ȱŠȱ˜˜ȱ›Žœž•ǰȱ if incȱƽȱŖǯŚȱ‘Ž—ȱco = co0 + n Ȋȱśȱ ‘Ž—ȱco0ȱƽȱŚǰȱŽšžŠ’˜—œȱǻŗśǼȱ Š—ȱǻŗŜǼȱŽȱ‘ŽȱŽ¡ŠŒȱ’–Ž—œ’˜—œǯȱžȱ‘Žȱ‹Žœȱ›Žœž•ȱ˜›ȱ ‘ŽȱžĴŽ› ˜›‘ȱž’ŽȱŒ˜–Žœȱ›˜–ȱŠ‹•Žȱŗǰȱcoȱƽȱŗşǰȱinc = ŖǯŚȱ ‘Ž›Žȱ‘’œȱ™ŠĴŽ›—ȱŒ˜–‹’—Š’˜—ȱ’ŸŽœȱžœȱ‘Žȱ–Š¡’–ȱ Œ˜—œŽ›ŸŠ’˜—ȱŸŠ•žŽȱ˜ȱ‘Žȱ›ŽšžŽ—Œ¢ȱ›Š—Žȱ˜ȱ‘ŽȱꕝŽ›ȱ ǻ’ž›ŽȱşǼǯ

Ly Lx

Amp ˜ K ' 0.4 ˜  co0  n ˜ 5

Amp ˜ J ' 0.4 ˜  co0  n ˜ 5

—ȱ˜›Ž›ȱ˜ȱ˜‹Š’—ȱŠȱ—Ž ȱŸŽ›’ŒŠ•ȱ›Žœ˜•ž’˜—ǰȱ›˜–ȱŗŞŘȱ˜ȱ ŗŞŖŖȱ™’¡Ž•œ

My

N y  Ly ǰ Ly M y  N y

Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

75

Determining Parameters for Images Amplification Pulses Interpolation

)LJXUH*UDSKLFSDUDPHWHUVVFDQIRUWKHRXWSXWRIWKHILOWHUW\SH¶%XWWHUZRUWK¶3UDWW RUGHUIRU OHVVWKDQSL[HOHOHPHQWVLQFUHPHQWinc GLIIHUHQWLDWHGSHUFRORUVDQGVDPSOHQXPEHUco LGHQWLI\WKHQXPEHU RIPHDVXUHPHQWVWDNHQIRUHDFKLQFUHPHQW 7DEOH+LJKIUHTXHQF\PDJQLWXGHVDIWHUWKHDGMXVWPHQWSURFHVVRIWKHDPSOLILFDWLRQNHUQHO 6XPRIXQLWVRIOHDVWHQHUJ\FKDQJHSHUSL[HOUHVXOWRIWKHKLJKSDVVILOWHUW\SH%XWWHUZRUWK 3UDWW ǰRUGHUIRUOHVVWKDQSL[HOHOHPHQWV

co

inc=0.1

inc=0.2

’—ŒƽŖǯř

’—ŒƽŖǯŚ

’—ŒƽŖǯś

1

řŞřşŘŞśŜŝǰŜŞŝŜŚś

ŘŝŖŖşśŖřŝǰśŖŞŘŜŜ

ŗŜşŗşŞŖřşǰşŗŘŖşŖ

ŗŖŘغ،ŝŘǰşŘřŚŝś

ŗŘśŗŘŝŜŝŜǰşśŝŘřŘ

2

ŘŜśřřŜŞŘŝǰŗŖśŞśŘ

şŚŜřřśŚşǰŘŖşŚřřś

ŗŘşŝŝŘŝŜśǰŝśŖŖřŜ

şŖŜśśřŘśǰŚŜŗŗŞŘŖ

ŘŝŖŗřŞŗřŝǰŗŝŚŘŞŗ

ȱř

ŗśŝřşşŞŞŘǰŝŞŗŞśŗ

ŗŘŜŘśřřŞŜǰŚŜŞŝřŗ

ŗŜŘŚřśŘśŗǰŚŘśŘŝŜ

śŖŘŚřŗŜŚśǰşŜŚşŘŚ

ŞŖŘŗşŚřşşǰŚŝŖŘŞŞ

ȱŚ

ŝŞŖŜřŚŘŗǰřŘŚŖŝşŗ

ŞŜşřŘŗŝŗǰŖŗŖśŘŞş

śŖŘŖŘŖŜŚŞǰŜŗśŝśŘ

ŞŘşŝŚŝŘśśǰşŞŘşŗŘ

śŗŚŜŝŗśśŚǰśŖŞřŝř

ȱś

ŗŖśŗşśŗŜŖǰŞśŜŜŖŘ

ŘŜśśŗśŜŞřǰŞŚŗŝŖŝ

ŝşŞŞŚŞŜŚřǰŗřŝşŞŜ

śŗřŘřŜŞřşǰŘřŞŞśŘ

ŗŘřŝśřŝřŖǰřşŖśŜŚ

ȱŜ

ŗŗśŜŖŗŚŘŝǰśŚŘŜŗş

śŖŗŝŖŗŖřŞǰŚŞřŜŖŝ

ŝřřşşşśřŖǰşśŘŜřş

şşŝŝŚŝŜŗǰŜşřŗřŗŝ

ŘŜşşŘŞŜŜŖǰŗŗřŖřŖ

ȱŝ

ŗŖŖśŞşşşśǰŚŜşŜśŝ

ŝ؜،śŞřŞǰŗŞşŘŚř

řşŖŜŝŗŗŜŜǰŖŖřŖşŚ

şŖŚŗŖŝŘşǰśŖřśśŖŜ

ŞŖşśŞşŜŜśǰŚřŗśŞŖ

ȱŞ

ŝŗŚŞřŞŖŖǰşŖŝŖŖŞŖ

ŞŘŝŘşŝřşŗǰśŖşŞŞŜ

şśŜŚŞŜŗşǰŘşŞřŝřŞ

śŖŝŞśśśŝŜǰŝşśŞřŞ

śŗŞŜŘŘŚŜŝǰŞşŜśŜŖ

ȱş

ŗśŘřśŚŖŘŜǰşşŜśŘŗ

ŝřŚřŝŝŚŖŝǰřŞŗŗŚŘ

ŗŗřŜśŘŘŗŜǰŞśŝŘśŘ

ŞřŜŖřŘŖŘřǰŞśŗŝŚş

ŗŘřŘřřŜřŘǰřŗşśřŜ

10

ŘŜŖřŘśŖşŘǰŞŜśśřŝ

śŖşşŖŞŚśŝǰřśŖŖŞŝ

ŘŜŞŜŗغتǰşśŞŜśř

śŗŜŚŝŝŚŜŞǰśŞŜşŗŗ

ŘŝřŘŖŝŚŜřǰşŘřŖŚŚ

11

řŝşŖŝŗŚśŗǰřşŘŜŘş

ŘŜŞŗşŘşŖşǰŚŗŚŖřş

ŜŘŞřŘśşśŝǰŞŞŞśŝŝ

şŜŜşŖřŜşǰşŖŝŚŜŘŚ

ŞŗśŝŜŚŞşŘǰŚśŝŝśŘ

12

śŖŖřřŘŚŝŘǰŜŖŘŚŖř

ŝŞşśřŖŝŜǰśŗŘŚŞŝŗ

ŞřŜŚŜşŝşŞǰřśśŝŚŚ

ŞŞŞŖşśŘŗǰŖşŞŖŗŝş

śŘŗŞşŖśŚřǰśŜśŝřŝ

ŗř

ŜŘŘŖŗŖřşśǰŝřŖŜŘŝ

ŗŗŜşřřŝŜŖǰŖŘŜŞřŞ

ŜřŜşŗřŜŚŝǰŘŜřŞŖŗ

śŗřřŘŖŚŗŞǰŚśŝśřŖ

ŗŘŖŚŝŘřŞśǰŞŞşŖřŚ

ŗŚ

ŝŘŝŖŜŞřŗŝǰŚŖŖŚŘŚ

ŝŘŘśśşśŚǰŚŚřŞŞŗř

ŘŝŗřŗŗśşŘǰŚŘřśŚś

ŞŚśŗŖśŘŞśǰŞŚŚŝŗŞ

ŘŝŚŜřŖŘśŘǰŖŚŜŞŘŖ

ŗś

ŞŖŘřŖŖŖşŖǰřŞŗŜŞř

ŘŜřřŘŗşŝŗǰřŜŗŘŖŖ

ŗŖŝŜŜŗřřśǰŚŘŝŝřş

śŘŗŖşşŘŜŜǰŖŚřśŚş

ŞŘŚŝŞřŚŜśǰŚśşŘŞŜ

ŗŜ

ŞřŖŞŘŞśŜŞǰŖŝśŗśŜ

śŖŜŖśŘşşŜǰŜŜŗŖŝŘ

ŝřŖřŗŞŘřǰŜşŗŚŝşŖ

ŞŖşśřŞŝŖǰŜŗŚřśŜŝ

śŘŜřŚśŞŘşǰŘŚŞśŖŚ

ŗŝ

ŞŖŝŜŞŚřŘşǰřŜŞŗŝŜ

ŝřśŗŗŚŝŖŖǰśřŚśşŞ

řŞŝśŝŝŝŚŞǰŞŖŗŜŖŘ

ŝřŝřřśŗŗǰŞřŗŚşŘŜ

ŗŗŖŗŜŜŜşŖǰřŚŘŜşś

ŗŞ

ŝřŚřŝŝŚŖŝǰřŞŗŗŚŘ

ŞřşŝŘŞŖŝŝǰŞřŜřŞŗ

ŝŚřŘŖśŖŞśǰŚŝŚşşŝ

śŗŝŘŚřřŜŘǰŖŜŚŖŝŝ

ŘŝŗşŚśřŞŝǰŖŚŚřŚŜ

ŗş

ŜŘşşŝřŘŗŝǰŚŘŞřşŘ

ŝŚŘŚŗŘřŖŞǰŖŘŗŜŘř

ŞŘŚşŞŚŗřřǰŝŞşřŞş

ŞśŜŝŝŘŗŜşǰřŖŖŗśŚ

ŞřŜŖŗŜŘŜŘǰŘřřşŗŗ

20

řŞřşŘŞśŜŝǰŜŞŝŜŚś

ŘŝŖŖşśŖřŝǰśŖŞŘŜŜ

ŗŜşŗşŞŖřşǰşŗŘŖşŖ

ŗŖŘغ،ŝŘǰşŘřŚŝś

ŗŘśŗŘŝŜŝŜǰşśŝŘřŘ

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Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

Morera-Delfín Leandro

n

1618  1.1 ˜ 182 4 0.4 ǰȱnȱƽȱŝŖŞǯŗȱŠ—ȱcoȱƽȱŝŖŞǯŗȊśȱƸȱŚ 5

’ž›ŽœȱŝȱŠ—ȱŗŖȱœ‘˜ ȱ‘˜ ȱŠȱ•˜œœȱ˜ȱ‘Žȱ‘’‘ȱ›ŽšžŽ—Œ¢ȱ content occurs. The application of the interpolation ker—Ž•ȱ  ’‘ȱ ‘Žȱ žĴŽ› ˜›‘ȱ ꕝŽ›ȱ ž’Žȱ ™›˜žŒŽœȱ ‘’œȱ ŽěŽŒǯȱ‘’œȱꕝŽ›ȱ˜Žœȱ—˜ȱ’ŸŽȱ‘Žȱ–Š¡’–ž–ȱŸŠ•žŽœȱ˜ȱ frequency of contour information. ‘Žȱ Š——¢ȱ –Ž‘˜ȱ ꗍœȱ ŽŽœȱ ‹¢ȱ •˜˜”’—ȱ ˜›ȱ ‘Žȱ •˜ŒŠ•ȱ–Š¡’–ž–ȱ˜ȱ‘Žȱ’–ŠŽȱ›Š’Ž—ǯȱ‘Žȱ›Š’Ž—ȱ’œȱ ŒŠ•Œž•ŠŽȱžœ’—ȱ‘ŽȱŽ›’ŸŠ’ŸŽȱ˜ȱŠȱ Šžœœ’Š—ȱꕝŽ›ǯȱ‘Žȱ method uses two thresholds to detect strong and weak ŽŽœǰȱŠ—ȱ’—Œ•žŽœȱ‘Žȱ ŽŠ”ȱŽŽœȱ’—ȱ‘Žȱ˜ž™žȱ˜—•¢ȱ if they are connected to strong edges. This method is therefore less likely than the others to be deceived by —˜’œŽǰȱŠ—ȱ–˜›Žȱ•’”Ž•¢ȱ˜ȱŽŽŒȱ›žŽȱ ŽŠ”ȱŽŽœǯ

ȱR  x, y ȱ

Z xs ˜ Z ys sin Z xs ˜ x sin Z ys ˜ y ˜ȱ ȱ ȱ S2 Zxs ˜ x ˜ Z ys ˜ y

ȱȱȱȱȱȱȱȱȱȱȱȱȱȱȱǻřŖǼ

The classical separable sine impulse response function Š››Š¢ȱǻ›ŠĴǰȱŘŖŖŗǼȱ’œ ˜–™Š›Žǰȱ’—ȱ’ž›Žȱŗřǰȱ‘Žȱ›Žœž•œȱ˜ȱ‘Žȱœ’—ŽȱœŽ™Š›Š‹•Žȱ’—Ž›™˜•Š’˜—ȱǻřŖǼȱŠ—ȱ‘ŽȱŠ™™•’ŒŠ’˜—ȱ˜ȱ‘Žȱ’–™ž•œŽȱ ›Žœ™˜—œŽȱž—Œ’˜—ȱŠ››Š¢ȱǻŘŖǼȱ›Žœ›’ŒŽȱ‹¢ȱ‘ŽȱŠ——¢ȱꕝŽ›ȱ˜ŸŽ›ȱ‘ŽȱꛜȱŒ˜•ž–—ȱ˜ȱ‘Žȱ‘’‘ȱ›Žœ˜•ž’˜—ȱ’–ŠŽǯ ’ž›Žȱ ŗŗȱ œ‘˜ œȱ ‘Žȱ ™Ž›’˜’Œ’¢ȱ Š—ȱ ‘Žȱ ™ŽŠ”œȱ žœŽž•ȱ ˜›ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ǯȱ‘Žȱ–Š™ȱ˜ȱꐞ›ŽȱŗŘȱ‘Šœȱ•˜ŒŠ•’£Žȱ Šȱ£˜—Žȱ ‘Ž›ŽȱŽ¡’œȱŠȱ–Š¡’–ž–ȱŸŠ•žŽȱ˜‹Š’—Žȱ›˜–ȱ‘Žȱ Š‹•ŽȱŘǰȱ ’‘ȱco = 10 and incȱƽȱŖǯśǯȱ’ž›ŽȱŗŚȱœ‘˜ œȱ‘˜ ȱ the high resolution image maintains the original con›Šœǯȱ‘’œȱꐞ›Žȱ‘Šœȱ‹ŽĴŽ›ȱŒ˜—œŽ›ŸŽȱŒ˜—˜ž›ȱ’—˜›–Štion than the one obtained with the classic interpolation žœ’—ȱ‘Žȱ’–™ž•œŽȱ›Žœ™˜—œŽȱž—Œ’˜—ȱŠ››Š¢ȱǻřŖǼǯ

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Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

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Determining Parameters for Images Amplification Pulses Interpolation

)LJXUH0DSRIKLJKUHVROXWLRQLPDJHV REWDLQHGE\WKHLQWHUSRODWLRQNHUQHOYDU\LQJ SDUDPHWHUV

—ȱ’ž›ŽȱŗśǰȱŠ—ȱ’—ŽŽ›ȱ—ž–‹Ž›ȱ–ž•’™•’Ž›ȱ’œȱžœŽȱ’—ȱ order to construct a more dense mesh of the interpola’˜—ȱ”Ž›—Ž•ȱǻŘŖǼǯȱ‘Žȱ™Š›Š–ŽŽ›œȱ’—ȱŠ‹•ŽȱŘȱ˜›ȱ‘Žȱ–Š¡’mum high frequency response in the high resolution image are used. —ȱ‘ŽȱŒŠœŽȱ˜ȱ’ž›Žȱŗśǰȱ›˜–ȱŽšžŠ’˜—ȱǻŗśǼ

Lx 13 ˜ (1.1 ˜ J ' 0.5 ˜10) —ȱ›˜–ȱŽšžŠ’˜—ȱǻŗŜǼ

Ly 13 ˜ (1.1 ˜ K ' 0.5 ˜10)

Discussion and analysis of results The investigation shows the main aspects of the inter™˜•Š’˜—ȱ ’‘ȱ›Žœ›’Œ’˜—ȱ˜›ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ǯȱ˜›ȱ’—œŠ—ŒŽǰȱ  ‘’Œ‘ȱ ’œȱ ‘Žȱ ˜Š•ȱ ’—ȱ ‘Žȱ Š–™•’ęŒŠ’˜—ȱ ™›˜ŒŽœœǵȱ ‘ŠȱꕝŽ›ȱŒ˜ž•ȱ‹Žȱ‘Žȱž’Žȱ˜›ȱ‘Žȱ™Š›Š–ŽŽ›œȱ’—ȱ‘Žȱ construction of the response impulse function or ampliꌊ’˜—ȱ”Ž›—Ž•ǵȱ‘ŽȱžĴŽ› ˜›‘ȱꕝŽ›ȱŠ”Žœȱ’—ȱŒ˜—œ’Ž›Š’˜—ȱ Šȱ ›Š—Žȱ ˜ȱ ‘’‘ȱ ›ŽšžŽ—Œ’Žœǰȱ ’—ȱ ‘Žȱ ŒŠœŽȱ ˜› 'xȱƽȱśȱ™’¡Ž•œǰȱ‹žȱ‘Žȱ–Š’—ȱ›˜ž‹•Žȱ’—ȱ‘Žȱ’—Ž›™˜•Š’˜—ȱ’œȱ ‘ŽȱŒ˜—˜ž›ȱŒ˜—œŽ›ŸŠ’˜—ȱ˜›ȱ‘’‘ȱ›ŽšžŽ—Œ’Žœǰȱ‘ŠȱŠ›Žȱ just over Sǯȱ‘Žȱ›Žœž•œȱ˜ȱŠ–™•’ęŒŠ’˜—ȱž’Žȱ‹¢ȱžterworth give possible combination of parameters for ŽšžŠ’˜—œȱǻŗśǼȱŠ—ȱǻŗŜǼȱ’—ȱŠȱ›Š—Žȱ˜ȱ›ŽšžŽ—Œ’Žœǯȱ—ȱ˜›Ž›ȱ˜ȱ‘ŠŸŽȱ‹ŽĴŽ›ȱ›Žœž•œȱ’ȱ’œȱ—ŽŒŽœœŠ›¢ȱ˜ȱžœŽȱŠȱꕝŽ›ȱ directly related with the contour information of the image. The edge detectors are developed with a robust ŒŠ™ŠŒ’¢ȱ ˜›ȱ ‘Žȱ ŽŽŒ’˜—ǯȱ—ŽȱŠ–˜žœȱŽŽŒ˜›ȱ’œȱ‘Žȱ Š——¢ȱꕝŽ›ȱǻŠ——¢ǰȱŗşŞŜǼǯ‘Ž—ȱ‘Žȱ‘’‘ȱ›ŽšžŽ—Œ’Žœȱ˜ȱ the interpolation output are related to the impulse res™˜—œŽȱž—Œ’˜—ȱŠ››Š¢ȱžœ’—ȱ‘ŽȱŠ——¢ȱꕝŽ›ǯȱŽȱ˜ž—ȱ the adequate parameters inc and co in order to make the ’—Ž›™˜•Š’˜—ȱ ”Ž›—Ž•ǰȱ ’—Œ›ŽŠœŽȱ ‘Žȱ ‘’‘ȱ ›ŽšžŽ—Œ¢ȱ Œ˜—tent in the high resolution image.

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‘Ž›ȱŠœ™ŽŒȱ’œȱ‘ŽȱŒ˜—›˜•ȱ˜ȱ‘ŽȱŒ˜˜›’—ŠŽœȱ˜ȱŠ–™•’ęŒŠ’˜—ȱ˜ȱ‘ŽȱꕝŽ›ǯȱȱ’œȱœ‘˜ —ȱ’—ȱ’ž›ŽœȱŜȱ˜›ȱŞȱ‘˜ ȱ ™Ž›’˜’Œ’’ŽœȱŽ¡’œȱ’—ȱŽŠŒ‘ȱinc row. The argument of the sine components of the impulse response function Š››Š¢ȱ’—ȱž—Œ’˜—ȱǻŘŖǼȱŠ›Žȱ—ŽŠ›ȱSȦŘȱ ’‘ȱ‘ŽȱŽ¡™•˜›Š’˜—ȱ parameters incȱƽȱŖǯŚȱŠ—ȱco = co0 + n Ȋȱśȱ˜›ȱ‘ŽȱŒ˜—˜ž›ȱ ’—Ž›™˜•Š’˜—ǯ‘Žȱ Š–™•’ęŒŠ’˜—ȱ œž’Žȱ ž—Ž›ȱ ‘’œȱ Œ˜—’’˜—ȱŠ”Žœȱ‘Žȱ’–Ž—œ’˜—œȱŽœ’›Žȱžœ’—ȱǻŗśǼȱŠ—ȱǻŗŜǼǯȱ ’ž›ŽȱŗŖȱ’œȱŠ—ȱŽ¡ŠŒȱŠ–™•’ęŒŠ’˜—ȱ˜ȱ‘Žȱ™›ŽŒ’œŽȱŒ˜˜›’—ŠŽœǯȱžȱ‘’œȱ™›˜ŒŽœœȱ’ŸŽœȱ•˜œœŽœȱ ’‘ȱ‘Žȱž’Žȱ˜ȱ‘Žȱ žĴŽ› ˜›‘ȱꕝŽ›ǯȱ‘Ž—ȱŒ˜—›˜••’—ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ȱ’ȱ ’œȱ ™›ŽŽ›Š‹•Žǰȱ ˜›ȱ ‘Žȱ Œ˜—œŽ›ŸŠ’˜—ȱ ˜ȱ ‘’‘ȱ ›ŽšžŽ—Œ¢ȱ Œ˜—Ž—ȱ ’—ȱ ‘Žȱ ‘’‘ȱ ›Žœ˜•ž’˜—ȱ ™•Š—Žǰȱ ˜ȱ ꗍȱ Šȱ •˜ŒŠ•ȱ –Š¡’–ž–ȱ žœ’—ȱ ‘Žȱ Š——¢ȱ ꕝŽ›ǯȱȱ  ‘˜•Žȱ —ž–‹Ž›ȱ ’œȱ used as multiplier to increase the interpolation kernel ’–Ž—œ’˜—œǯȱ—˜‘Ž›ȱ’–™˜›Š—ȱ˜™’Œȱ’œȱ‘ŽȱŒ˜–™Š›’œ˜—ȱ applying the classical separable sine impulse response ž—Œ’˜—ȱ Š››Š¢ȱ ǻřŖǼȱ Š—ȱ ‘Žȱ ’—Ž›™˜•Š’˜—ȱ ›Žœž•œȱ  ’‘ȱ ‘Žȱ’–™ž•œŽȱ›Žœ™˜—œŽȱž—Œ’˜—ȱŠ››Š¢ȱǻŘŖǼȱŒ˜—œ›Š’—Žȱ‹¢ȱ ‘Žȱ —Ž’‘‹˜›ȱ ŽŽŒ˜›ȱ ꕝŽ›ȱ Š——¢ȱ ¢™Žǯȱ ȱ ’œȱ œ‘˜ —ȱ ‘˜ ȱ‘Žȱ›Žœ›’Œ’˜—ȱ’—Œ›ŽŠœŽœȱ‘Žȱ›Š—Žȱ˜ȱ‘Žȱ™’¡Ž•œȱŸŠlues doing a contribution to the high frequency conservation in the high resolution plane. In the last case it is Š—Š•¢£Žȱ‘˜ ȱ‘Žȱž’Žȱ˜ȱŠ——¢ȱ’œȱžœŽž•ȱ˜ȱŒ˜—œ›žŒȱ an interpolation kernel in order to obtain a very high ›Žœ˜•ž’˜—ȱ’–ŠŽȱŠ”’—ȱ˜—•¢ȱ–Š¡’–ž–ȱ•˜ŒŠ•ȱŸŠ•žŽœȱ’—ȱ Šȱ›Š—Žǯȱ‘Žȱ™Š›Š–ŽŽ›œǰȱ ‘Ž—ȱ‘Žȱ‘’‘ȱ›ŽšžŽ—Œ¢ȱŒ˜—Ž—ȱ’—Œ›ŽŠœŽœǰȱ›Ž™›ŽœŽ—ȱ‘ŽȱŠŽšžŠŽȱŠ–˜ž—ȱ˜ȱœŠ–pler components in the impulse response array Sǻxǰȱy) ˜ȱŽšžŠ’˜—ȱǻŗǼȱ˜›ȱ‘Žȱ’—Ž›™˜•Š’˜—ǯȱ‘ŽȱœŠ–™•Ž›ȱŒ˜–ponents are more numerous than the samples of the ˜›’’—Š•ȱ’–ŠŽȱ˜›ȱ‘Žȱ’—Ž›™˜•Š’˜—ǰȱ‹žȱ‘Žȱ™˜œ’’˜—ȱ˜ȱ ‘ŽœŽȱŠ›Žȱ’—ȱ‘ŽȱŠŒ’ŸŽȱ£˜—Žȱ˜ȱ‘Žȱ›Žœ˜•ž’˜—ȱŒŽ••ȱ˜ȱ‘Žȱ physical scan model and consequently over the valid ŸŠ•žŽœȱ˜ȱ‘Žȱ™’¡Ž•œȱ’—ȱ‘Žȱ˜›’’—Š•ȱ’–ŠŽȱŠ—ȱ—˜ȱ‹Žween. Then the interpolation kernel increases by a whole number multiplier for the adequate parameters Ampǰȱinc and coȱ’Ÿ’—ȱŠȱ‘’‘ȱ›ŽšžŽ—Œ¢ȱ–Š¡’–ž–ȱ’—ȱŠ—ȱ

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Morera-Delfín Leandro

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inc=0.1 ŗŘŖŜś ŗŘśŜŗ ŗŗŖşŗ ȱȱȱȱşŚŞ ŗŗŘŜŝ ŗŗŝŘŞ ŗŗşŖŞ ȱȱŗŖśŞ ŗŗŝŞŚ ŗŘŚŗŚ ŗŗşŞŚ ŗŘŖşŝ ŗŗşŞŜ ŗŗŝŘŖ ŗŘŖŗŝ ŗŘŖŝŚ ŗŗşŚş ŗŗŞśŜ ŗŘŗŝŗ ŗŘŘŘś

inc=0.2 ŗŘŜřŜ ȱȱȱşŚŜ ŗŘşŖŗ ŗŗŜŞ ŗŘŜřś ŗŘŘŘŜ ŗŗŝŜŜ ŗŘŗŘŝ ŗŗŞśŜ ŗŘŘŘś ŗŘŜŗŖ ȱȱȱşřŖ ŗŘŞŝŗ ŗŖśŝ ŗŘŜŖř ŗŘŗřŖ ŗŗŝŘş ŗŘŗŖř ŗŗŞřŜ ŗŘŘŜŜ

’—ŒƽŖǯř ŗŗřřś ŗřŖŞŚ ŗŗŞŜŗ ŗŘŘŖŜ ŗŘŗřŞ ŗŗşŘŚ ŗŘŘŜŝ ȱȱȱȱşŚř ŗŘŗŜř ŗŘŜśŚ ŗŘŗŘŚ ŗŘŘŖŞ ŗŘŘřŘ ŗŘśşŚ ŗŗřŝŜ ȱȱŗŖŝŜ ŗŘŗŞŜ ŗŗŞŗŜ ŗŘŗŝŞ ŗŘŚŘś

’—ŒƽŖǯŚ 1002 ȱȱŗřŞŘ ŗŘŚŜŜ ŗŘřŘŖ ŗŘŚŘŗ ȱȱȱȱşŗŞ ŗŗŜŜ ŗŘŚŚś ŗŘřŘř ŗŘŚŗŜ ȱȱȱȱşŝŗ ŗŗŚŝ ŗŘśŗŗ ŗŘřśŖ ŗŘŚŘś ȱȱȱȱşŘś ȱȱŗŖŜř ŗŘŘśŘ ŗŘřśŚ ŗŘŚŘŘ

’—ŒƽŖǯś ŗŗŜŞŝ ŗŘŞŜŚ ŗŘŘŝř ŗŘŚŘŗ ŗŗŚŝş ŗŘŝŜŜ ŗŘŘśś ŗŘŚřŞ ŗŗśŚŗ ŗŘŞŝŗ ŗŘřŗŚ ŗŘŚśŝ ŗŗśŝŞ ŗŘŞŘŚ ŗŘŚŖŘ ŗŘŚŘŘ ŗŗŜŖŞ ŗŘŞśŘ ŗŘŘŝř ŗŘŚŘŞ

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Determining Parameters for Images Amplification Pulses Interpolation

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Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM

Morera-Delfín Leandro

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initial range. Using this mode we obtained a very high resolution image. The high frequency content in the ‘’‘ȱ ›Žœ˜•ž’˜—ȱ ™•Š—Žȱ  Šœȱ ›ŽŠ’—Žǯȱ ’ž›Žȱ ŗśȱ œ‘˜ œȱ ‘˜ ȱ ‘’œȱ ŽěŽŒȱ ˜ŒŒž›œȱ ˜ŸŽ›ȱ Š—ȱ Š–™•’ęŒŠ’˜—ȱ ˜ȱ ‘Žȱ ŗśśŖƖȱ’—ȱŽŠŒ‘ȱŒ˜˜›’—ŠŽǯ

’쎛Ž—ȱ ™›˜ŒŽœœŽœȱ ˜ȱ ›Žœ˜›Š’˜—ȱ Š—ȱ ›ŽŒ˜—œ›žŒ’˜—ǰȱ preserving the characteristics of the primary image. ‘Žȱ ™›˜™˜œŽȱ –Ž‘˜ȱ –Š”Žœȱ ŠŽšžŠŽȱ Š–™•’ęŒŠ’˜—ȱ ꕝŽ›œȱ˜›ȱœ™ŽŒ’ęŒȱ™ž›™˜œŽœȱ’—ȱ‘Žȱꎕȱ˜ȱ’’Š•ȱ™‘˜˜ȱ restoration and image and video compression.

Conclusions

References

‘’œȱ™Š™Ž›ȱœ‘˜ œȱŠ—ȱŽěŽŒ’ŸŽȱ Š¢ȱ˜›ȱ˜™’–’£’—ȱŠȱ™ž•œŽȱ’—Ž›™˜•Š’˜—ȱꕝŽ›ȱ’—ȱ˜›Ž›ȱ˜ȱ˜‹Š’—ȱ‘Žȱ‹ŽœȱŸ’œžŠ•ȱ ›Žœž•ȱ˜›ȱ‘Žȱ›Žšž’›ŽȱŠ–™•’ęŒŠ’˜—ȱ›ŠŽǯȱ‘Žȱ™›˜™˜œŽȱ Š–™•’ęŒŠ’˜—ȱ –Ž‘˜ȱ ›ŽŠ’—œȱ ‘Žȱ ‘’‘ȱ ›ŽšžŽ—Œ¢ȱ Œ˜—Ž—ȱ Š—ȱ ˜Žœȱ —˜ȱ ™›˜žŒŽȱ ™’¡Ž•œȱ ‹•˜Œ”ȱ ŽěŽŒǯȱȱ —Ž ȱ way of relationship between the constructions of the ’—Ž›™˜•Š’˜—ȱ ”Ž›—Ž•ȱ Š—ȱ ‘Žȱ ž’Žȱ ˜ȱ Œ•Šœœ’Œȱ ꕝŽ›œȱ ’œȱ shown in order to increase the high frequency content in the high resolution image. The restriction of the inŽ›™˜•Š’˜—ȱ™›˜ŒŽœœȱžœ’—ȱ‘’‘ȱ™ŠœœȱꕝŽ›œȱ’—Œ›ŽŠœŽœȱ‘Žȱ ›Š—Žȱ ˜ȱ ‘Žȱ ™’¡Ž•œȱ ŸŠ•žŽœȱ ˜’—ȱ Šȱ Œ˜—›’‹ž’˜—ȱ ˜ȱ ‘Žȱ high frequency conservation in the high resolution pla—Žǯȱ ȱ Š——¢ȱ ŽŽŒ˜›ȱ ’œȱ –˜›Žȱ ŽěŽŒ’ŸŽȱ ‘Š—ȱ ‘Žȱ žŽ› ˜›‘ȱ ‘’‘ȱ ™Šœœȱ ꕝŽ›ȱ ˜›ȱ ‘Žȱ ‘’‘ȱ ›ŽšžŽ—Œ¢ȱ Œ˜—œŽ›ŸŠ’˜—ȱ’—ȱ‘ŽȱŠ–™•’ęŒŠ’˜—ǯȱ‘ŽȱŒ‘Š›ŠŒŽ›’£Š’˜—ȱ ˜ȱ‘ŽȱꕝŽ›ȱ’œȱ—ŽŒŽœœŠ›¢ȱ˜›ȱ‘ŽȱŽĜŒ’Ž—ȱŠ™™•’ŒŠ’˜—ȱ’—ȱ

‹›Š–Š’Œȱ ǯǯȱŠ—ȱŠžŽ›Šœȱǯǯȱ˜››ŽŒ’˜—ȱ˜ȱŽšžŽ—’Š•ȱ˜—Ÿ˜•ž’˜—ȱ ŽŒ‘—’šžŽœȱ ˜›ȱ –ŠŽȱ ’•Ž›’—ǯȱ IEEE Trans. Acoustics, Speech, and Signal ProcessingǰȱȬřŖǰȱŘǰȱ™›’•ȱŗşŞŘǰȱřŚŜǯȱ ’–™•ŽȬŽ••ȱŽŒŽ™’ŸŽȱ’Ž•ȱ›˜™Ž›’Žœȱ‹¢ȱŽŠ›—’—ȱŠȱ™Š›œŽȱ ˜Žȱ˜›ȱŠž›Š•ȱ–ŠŽœǯȱNatureǰȱŸ˜•ž–ŽȱřŞŗǰȱŗşşŜDZȱŜŖŝȬŜŖşǯ ‹›Š–Š’Œȱ ǯǯȱŠ—ȱŠžŽ›ŠœȱǯǯȱŽšžŽ—’Š•ȱ˜—Ÿ˜•ž’˜—ȱŽŒ‘—’šžŽœȱ˜›ȱ–ŠŽȱ’•Ž›’—ǰȱIEEE Trans. Acoustics, Speech, and Signal ProcessingǰȱȬřŖǰȱŗǰȱŽ‹›žŠ›¢ȱŗşŞŘǰȱ™™ǯȱŗȬŗŖǯ ‹›Š–Š’Œȱ ǯǯȱ Š—ȱ ŠžŽ›Šœȱ ǯǯȱ Žœ’—ȱ ˜ȱ  ˜Ȭ’–Ž—œ’˜—Š•ȱ ȱ’•Ž›œȱ›˜–ȱ–Š••ȱ Ž—Ž›Š’—ȱ Ž›—Ž•œǰȱ›˜Œǯȱȱ˜—Ž›Ž—ŒŽȱ˜—ȱŠĴŽ›—ȱŽŒ˜—’’˜—ȱŠ—ȱ–ŠŽȱ›˜ŒŽœœ’—ǰȱ‘’ŒŠ˜ǰȱ Š¢ȱŗşŝŞǯ ›˜ —ȱ ǯǯǰȱ ›ǯȱ˜ž—œȱ˜›ȱ›ž—ŒŠ’˜—ȱ››˜›ȱ’—ȱŠ–™•’—ȱ¡™Š—œ’˜—œȱ˜ȱŠ—Ȭ’–’Žȱ’—Š•œǯȱIEEE Trans. Information Theoryǰȱ ȬŗśǰȱŸ˜•ž–ŽȱŚȱǻ ž•¢ȱ’œœžŽǼǰȱŗşŜşDZȱŚŚŖȬŚŚŚǯ

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Determining Parameters for Images Amplification Pulses Interpolation

Š——¢ȱ ǯȱ ȱ ˜–™žŠ’˜—Š•ȱ ™™›˜ŠŒ‘ȱ ˜ȱ Žȱ ŽŽŒ’˜—ǰȱ IEEE Trans. ŠĴŽ›—ȱ—Š•¢œ’œȱŠ—ȱŠŒ‘’—Žȱ—Ž••’Ž—ŒŽǰȱȬŞǰȱŸ˜•ž–ŽȱŜȱǻ˜ŸŽ–‹Ž›ȱ’œœžŽǼǰȱŗşŞŜDZȱŜŝşȬŜşŞǯ ˜—Ȭ˜˜ȱ ǯǰȱ ž—Ȭ ¢ž—ȱ ǯǰȱž—Ȭ ¢ž—ȱǯǰȱ ˜—Ȭ”ȱ ǯȱȱ˜ŸŽ•ȱ –ŠŽȱ—Ž›™˜•Š’˜—ȱ–Ž‘˜ȱœ’—ȱ‘Žȱ’•ŠŽ›Š•ȱ’•Ž›ǰȱConsumer Electronics. IEEE Transactions,ȱȱŸ˜•ž–ŽȱśŜȱǻ’œœžŽȱŗǼǰȱŘŖŗŖǯ

Ž•–œȱ ǯǯȱ Š—ȱ ‘˜–Šœȱ ǯǯȱ ›ž—ŒŠ’˜—ȱ ››˜›ȱ ˜ȱ Š–™•’—ȱ ‘Ž˜›¢ȱ¡™Š—œ’˜—œǯȱProc. IREǰȱśŖǰȱŸ˜•ž–ŽȱŘȱǻŽ‹›žŠ›¢ȱ’œœžŽǼǰȱ ŗşŜŘDZȱŗŝşȬŗŞŚǯ

Š”›Š—ȱ ǯǰȱ ˜ž—“˜˜—ȱ ǯǰȱ Ž˜—“Š’ȱ ǯȱ ž›ŸŠž›Žȱ —Ž›™˜•Š’˜—ȱ Ž‘˜ȱ ˜›ȱ –ŠŽȱ ˜˜–’—ǰȱ Image Processing. IEEE TransactionsǰȱŸ˜•ž–ŽȱŘŖȱǻ’œœžŽDZȱŝǼǰȱŘŖŗŗǯ ŽŸ’—Žȱ ǯǰȱŠž•œŽ—ȱǯǯǰȱ‘Š—Ȭ˜—“’ŽǯȱŒŠ——’—ȱ˜Š•’’ŽœȱŠ—ȱ Ž’ŒŠ•Ȭ–ŠŽȱ—Š•¢œ’œǯȱ Computer Graphics and Applicationsǰ IEEEǰȱŸ˜•ž–ŽȱřŘǰȱŘŖŗŘǯ ’žȱ’Š—–’—ǰȱ‘Š˜ȱŽ‹’—ȱǰȱ’˜—ȱž’š’—ǰȱ’ Ž’ȱŠǯȱ–ŠŽȱ—Ž›™˜•Š’˜—ȱ’ŠȱŽž•Š›’£Žȱ˜ŒŠ•ȱ’—ŽŠ›ȱŽ›Žœœ’˜—ǰȱImage Processing. IEEE Transactions,ȱŸ˜•ž–ŽȱŘŖȱǻ’œœžŽȱŗŘǼǰȱŘŖŗŗǯ ȂŽ’••ȱǯǯȱIntroduction to Statistical Opticsǰȱ’œ˜—ȬŽœ•Ž¢ǰȱŽŠ’—ǰȱǰȱŗşŜřǰȱ™™ǯȱŞřǯ £ž›Ȭ‹›Š‘’–ǰȱ‘žȬ ˜—ǰȱŠœœȱǯǰ ž’Š—ȱǯȱ‘’—”ȱžœ’Žȱ‘Žȱ ’‹Ž›DZȱ –Š’—ȱ –™•’ꎛȱ ˜›ȱ ™ŠŒŽȬž•’™•Ž¡Žȱ ™’ŒŠ•ȱ

Transmission. ȱ‘˜˜—’Œœȱ ˜ž›—Š•ȱ–Š’—ȱ–™•’ꎛȱ˜›ȱ™tical TransmissionǰȱŸ˜•ž–ŽȱŚȱǻ’œœžŽȱśǼǰȱŒ˜‹Ž›ȱŘŖŗŘǯ Š™˜ž•’œȱ ǯȱ Systems and Transforms with Applications in Opticsǰȱ Œ ›Š Ȭ ’••ǰȱŽ ȱ˜›”ǰȱŗşŜŜǰȱ™ǯȱŗŖśǯ ›ŠĴȬ’••’Š–ȱ ǯȱDigital Image Processing: PIKS InsideǰȱřrdȱŽǯǰȱ˜™¢›’‘ȱȚȱŘŖŖŗǰȱ ˜‘—ȱ’•Ž¢ȱǭȱ˜—œǰȱ—ŒǯȱœDZȱŖȬŚŝŗȬřŝŚŖŝȬśȱ ǻ Š›‹ŠŒ”ǼDzȱŖȬŚŝŗȬŘŘŗřŘȬśȱǻ•ŽŒ›˜—’ŒǼǯ ›ŠĴȱǯ ǯǰȱ‹›Š–Š’Œȱ ǯǯǰȱŠžŽ›ŠœȱǯǯȱŽ‘˜ȱŠ—ȱ™™Š›Šžœȱ ˜›ȱ–™›˜ŸŽȱ’’Š•ȱ–ŠŽȱ›˜ŒŽœœ’—ǰȱǯǯȱ™ŠŽ—ȱŚǰřřŖǰŞřřǰȱ Š¢ȱŗŞǰȱŗşŞŘǯ

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About the author ŽŠ—›˜ȱ˜›Ž›ŠȬŽ•ę—. Graduated in Telecommunications and Electronics Engineering Š—ȱŠœŽ›ȱ’—ȱ’˜–Ž’ŒŠ•ȱŒ’Ž—ŒŽœǯȱ Žȱ’œȱŠŒžŠ••¢ȱŠȱ›˜Žœœ˜›ȱ˜ȱ˜––ž—’ŒŠ’˜—œȱ ‘Ž˜›¢ȱŠ—ȱ’’Š•ȱ–ŠŽœȱŠ—ȱ’—Š•œȱ›˜ŒŽœœ’—ȱŠȱ‘Žȱ ’‘Ž›ȱ˜•¢ŽŒ‘—’Œȱ—œ’žŽȱ ˜œŽȱ —˜—’˜ȱ Œ‘ŽŸŽ››’Šǰȱ ŠŸŠ—Šǰȱ ž‹Šǯȱ Žȱ ‘Šœȱ “˜‹ȱ Ž¡™Ž›’Ž—ŒŽȱ ’—ȱ —Ž ˜›”ȱ administration and software development for surveillance and investigations Š‹˜žȱŒ˜–™›Žœœ’—ȱŠ—ȱ™ŠĴŽ›—ȱ›ŽŒ˜—’’˜—œȱ˜ŸŽ›ȱ’–ŠŽœȱŠ—ȱŸ’Ž˜ǯ

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Ingeniería Investigación y Tecnología, volumen XVI (número 1), enero-marzo 2015: 71-82 ISSN 1405-7743 FI-UNAM