Available in: http://www.redalyc.org/articulo.oa?id=57064009 Red de Revistas Científicas de América Latina, el Caribe, España y PortugalSistema de Información CientíficaJ. S. Pérez Huerta, G. Rodriguez ZuritaImage restoration of blurring due to rectilinear motion: constant velocity and constant accelerationRevista Mexicana de Física, vol. 51, núm. Re4, agosto, 2005, p. 398406,Sociedad Mexicana de Física A.C.México How to cite Complete issue More information about this article Journal's homepageRevista Mexicana de Física,ISSN (Printed Version): [email protected] Mexicana de Física A.C.Méxicowww.redalyc.orgNon-Profit Academic Project, developed under the Open Acces InitiativeINVESTIGACI´ON REVISTA MEXICANA DE F´ISICA 51 (4) 398–406 AGOSTO 2005Image restoration of blurring due to rectilinear motion:constant velocity and constant accelerationJ.S. P´erez Huerta and G. Rodriguez ZuritaFacultad de Ciencias F´ısico-Matem´aticas, Benem´erita Universidad Aut´onoma de Puebla,Apartado Postal 1152, Puebla Pue., M´exico,e-mail: [email protected], [email protected] el 6 de diciembre de 2004; aceptado el 24 de mayo de 2005If a relative motion exists between a given object’s image and a recording medium (such as a photographic film) during exposure, the recordedimage will be blurred. This effect can be expressed by an impulse response in irradiance function and the respective image restoration is madepossible by a post-recording process. General one-dimension motion is considered here. Two cases are discussed: constant velocity motionand constant acceleration motion. Image simulations representing those degradation mechanisms are shown, as well as the correspondingrestored images after using the method which is described in this work. This method is based on Dirac delta function propreties. We alsoshow real digital images affected by those motions. It can be experimentally obtained by using an air-rail. The blurred images are processeddigitally by spatial filters given by the proposed method. Numerical and experimental results are shown.Keywords: Image forming and processing; Fourier optics; modulation and optical transfer functionsSi durante el registro de una imagen (una exposici´on fotogr´afica, por ejemplo) existiera un movimiento relativo entre´esta y el medio deregistro, la imagen grabada resultar´a deteriorada (emborronada). Este efecto se expresa por medio de la funci´on de respuesta impulso enirradiancia, y a partir de´esta es posible restaurar la imagen mediante procesos posteriores al registro. Aqu´ı se considera movimiento rectil´ıneogeneral y se analizan los casos de movimiento uniformemente lineal y movimiento uniformemente acelerado. Se presentan simulaciones deim´agenes afectadas por estos movimientos as´ı como su restauraci´on al aplicar el procedimiento sugerido por el an´alisis descrito, que se basaen propiedades de la funci´on delta de Dirac. Tambi´en se muestran im´agenes reales digitalizadas afectadas por los movimientos mencionadoscon la ayuda de un riel sin fricci´on. Estas im´agenes se procesan digitalmente implementando filtros espaciales sugeridos por el m´etodopropuesto. Se muestran los resultados correspondientes.Descriptores: Formaci´on y procesamiento de im´agenes;´optica de Fourier; funciones de modulaci´on y transferencia´opticaPACS: 42.30.Va; 42.30.Kq; 42.30.Lr1. IntroductionCertain factors that can degrade the recorded image in a pho-tographic or holographic film have been studied for a longtime. Relevant works include studies on transverse vibra-tions of one or several components (object, lens, or imageplane) perpendicular to the optical axis, effects studied by A.W. Lohmann [1]. Uniformly linear motion between the origi-nal image and the film was treated by Edward L. O’Neill [2],the shutter operation was analyzed by Ronald V. Shack [3],longitudinal vibrations were worked on by A.W. Lohmannand D.P. Paris [4], out-of-focus degradation was studied byJ.W. Goodman [9], the penumbra effect in radiograph im-ages due to the nonzero dimension of the source was dealtwith by John B. Minkoff, S.K. Hilal, W.F. Konig, M. Arm,and L.B. Lambert [10]. Nonuniform motion was worked onby Som [13], multiply exposed images were studied by D.P.Jablonowski and S.H. Lee [14]. A feature common to theseworks is the characterization of the image degradation by apoint spread function (PSF), so that the degraded image canbe expressed by the convolution of the ideal or original imagewith the respective PSF.Works about restoration and evaluation images by apost-recording process have been demonstrated. In 1953, A.Mar´echal and P. Croce showed the classical image restora-tion technique by inverse spatial filtering, while P.F. Muellerand G.O. Reynolds restored images affected by turbulence in1966. J.L. Harris showed a formulation different from inversefiltering and experimental results are shown in Ref. 5. C.W.Helstrom showed image restoration by the method of leastsquares, which considers spatial noise in the image [7], A.W.Lohmann and D.P. Paris showed computer generated binaryspatial filters which can represent complex inverse filters [8].J.L. Horner showed restoration of photographic images whennoise is considered [11]. George W. Stroke showed holo-graphic image deblurring [12]. D.P. Jablonowski and S.H.Lee proposed a method of synthesizing the appropriate com-posite gratings, which represents a double convolution, witha set of the Dirac delta function, and experimental results areshown in Ref. 14, L. Celaya and S. Mallick deblurred imagesdegraded by a linear smear by using a Wollaston prism andtwo linear polarizers [16].From the above mentioned deblurring methods, many ofthem are optical analog in nature for historical reasons. Anew consideration of these methods within today’s scope,however, would undoubtedly bring benefits to digital im-age processing after proper updates or adaptations. It isunder these considerations that the present work was done.First, a general description of blurring due to linear motionis proposed taking advantage of Dirac delta function proper-IMAGE RESTORATION OF BLURRING DUE TO RECTILINEAR MOTION: CONSTANT VELOCITY AND CONSTANT ACCELERATION 399ties [13] which have not been used in connection with thisproblem, to our knowledge. To test this formulation, twocases are numerically inspected: uniform velocity and uni-form acceleration. By using a given form of