text: fig1 is a view for explaining a structure of a projection display apparatus of a first embodiment . in this projection display apparatus , lights emitted from a light source 3 are roughly parallel light beams and random - polarized , passed through a polarization converter 2 and then converted into s polarized lights . the parallel light beams made incident on the polarization converter 2 are first made incident on a first lens plate 22 having a plurality of lens elements 22 a arrayed in a matrix form ( e . g ., 4 × 5 ), and then each of the parallel light beams is divided into a number corresponding to the number of lens elements 22 a based on apertures determined by an outer shape of each of the lens elements 22 a . the outer shapes of the lens elements 22 a of the first lens plate 22 are identical to one another , and similar to shapes of light valves 45 r , 45 g and 45 b ( described later ) which are objects to be illuminated . in a focal position of each lens element 22 a of the first lens plate 22 , arranged is a second lens plate 23 which includes lens elements 23 a arrayed in positions corresponding to those of the lens elements 22 a . because of the foregoing arrangements of the first and second lens plates 22 and 23 , the parallel light beams made incident on the respective lens elements 22 a of the first lens plate 22 are converged on a center part of the lens elements 23 a of the second lens plate 23 . then , a luminescent point is formed on the lens elements 23 a . a light emitted from the luminescent point of the lens element 23 a of the second lens plate 23 is made incident on a polarized beam splitter prism array 24 which is arranged in the vicinity of a light exit surface of the second lens plate 23 . this polarized beam splitter prism array 24 includes polarized beam splitters 24 a and 24 b , each of which has a width equal to ½ of a width of the lens element 23 a of the second lens plate 23 . in this embodiment , one polarized beam splitter 24 a is arranged in a place facing a center side of each of the lens elements 23 a , and the other polarized beam splitter 24 b is arranged in a place facing a boundary side of each of the lens elements 23 a . accordingly , the light emitted from the luminescent point on the lens element 23 a is polarized and separated into p and s polarized lights : the p polarized light being made incident on the polarized beam splitter 24 a and passed through a polarizing and separating section of the same ; and the s polarized light being reflected on the polarizing and separating section , made incident on the adjacent polarized beam splitter 24 b , reflected on a polarizing and separating section of the same and then emitted . the p polarized light passed through the polarized beam spliter 24 a is then converted into an s polarized light by a ½ wavelength phase plate 25 arranged on a light exit surface of the polarized beam splitter 24 a . as a result , all the lights from the light source 3 are converted into s polarized lights by being passed through the polarization converter 2 . each light from the light source converted into an s polarized light by the polarization converter 2 is then made incident on a cross dichroic mirror 94 which includes a dichroic mirror 94 a having a b light reflection characteristic and a dichroic mirror 94 b having r and g light reflection characteristics , the dichroic mirrors 94 a and 94 b being arranged in x - shape . here , the light is color - separated into a b light component which advances in a direction perpendicular to an incident optical axis , and a mixed light between g and r light components which advance in a direction opposite the direction of the b light . the b light obtained as a result of the foregoing color - separation by the cross dichroic mirror 94 advances to enter a bending mirror 95 , further advances after changing an optical axis by a right angle and then enters a polarized beam splitter 1 b for the b light . the mixed light of the g and r lights obtained by the color - separation advances after changing an optical axis by a right angle by a bending mirror 96 , and then enters a g light reflection dichroic mirror 98 arranged on the optical axis . the mixed light is then color - separated into an r light component which is directly transmitted , and a g light component which is reflected and advances after changing an optical axis by a right angle . the r and g lights obtained as a result of the color - separation by the dichroic mirror 98 are respectively made incident on polarized beam splitters 1 r and 1 g . the cross dichroic mirror 94 , the bending mirrors 95 and 96 and the g light reflection dichroic mirror 98 constitute a color - separation optical system . polarizing and separating sections of the polarized beam splitters 1 b , 1 g and 1 r are arranged to reflect incident s polarized lights of respective colors . accordingly , incident b , g and r lights are respectively reflected by the polarizing and separating sections 1 b a , 1 g a and 1 r a of the polarized beam splitters 1 b , 1 g and 1 r , and then emitted from the polarized beam splitters 1 b , 1 g and 1 r . in the vicinity of the light exit surfaces thereof , arranged are reflection light valves 45 b , 45 g and 45 r for respective color lights . the s polarized lights of respective colors made incident on the light valves 45 b , 45 g and 45 r are reflected and emitted as mixed lights of modulated light ( p polarized light ) and unmodulated light ( s polarized light ). the modulated and unmodulated lights from the light valves 45 b , 45 g and 45 r are respectively made incident on the polarized beam splitters 1 b , 1 g and 1 r again . the polarized seam splitters 1 b , 1 g and 1 r analyze lights transmitted through the polarizing and separating sections 1 b a , 1 g a and 1 r a as modulated lights ( p polarized lights ). the analyzed lights of respective colors are made incident from different incident surfaces on a cross dichroic prism 99 of a color composing optical system . then , the color composition is completed by a b light reflection dichroic layer 99 b and an r light reflection dichroic layer 99 r which are arranged in an x - shape inside the cross dichroic prism 99 . as a result , a composed light of the b , g and r lights is emitted from a light exit surface of the cross dichroic prism 99 . the composed light emitted from the cross dichroic prism 99 is then made incident on a projection lens 6 , and then projected as a full - color image on a not - shown screen . referring to fig2 a method to prevent a projected image from being a parallelogram in the projection display apparatus of fig1 will be described . in the projection display apparatus shown in fig1 the polarized beam splitters 1 b , 1 g and 1 r are respectively arranged for b , g and r lights . in fig2 the polarized beam splitter 1 b is shown as a representative of the polarized beam splitters . for easy description of directions , x , y and z axes orthogonal to one another are defined as shown . further , since the projection lens 6 of the projection display apparatus is set telecentric with respect to the light valve 45 b side , principal rays l 1 to l 5 are parallel between the light valve and the polarized beam splitter 1 b . deformation of the projected image to be a parallelogram can be understood when the polarized beam splitter 1 b is constituted as follows . specifically , as shown in fig2 a polarized beam splitter 1 b is manufactured in such manner that right - angled isosceles triangle optical prisms 1 a and 1 b having refractive indexes n 1 are prepared , and a polarizing and separating film 1 d ( or 1 e ) is formed on a bottom surface facing one right angle ( apex angle ) thereof , followed by adhering or cementing the optical prisms 1 a and 1 b to each other using adhesive . in this case , it is difficult to set a thickness of an adhesive layer 1 c to be uniform on the full surface . thus , the adhesive layer 1 c is , as shown in the drawing , formed to be wedge - shaped having more thickness toward a y - direction . the polarizing and separating surface is a plane parallel to a ( 101 ) plane if defined by using the foregoing coordinates . next , five light rays l 1 to l 5 ( referred to as l 1 , l 2 , l 3 , l 4 and l 5 , hereinafter ) reflected and emitted from four corners and an approximately central part of a display surface of the reflection light valve 45 b will be described . the light rays l 1 to l 5 are vertically incident onto an upper plane 1 f ( parallel to xy plane ) of the polarized beam splitter 1 b , transmitted through the upper plane 1 f , are directly moved ahead , and then emitted from the prism 1 a and made incident on the adhesive layer 1 c having a refractive index n 2 in accordance with snell &# 39 ; s law ( law of refraction ). the light rays l 1 and l 2 are passed through thin places of the adhesive layer 1 c . thus , these light rays l 1 and l 2 advance with little deviation and become light rays and l 1 ′ and l 2 ′. the light rays l 3 and l 4 are likewise made incident vertically to the upper plane 1 f of the polarized beam splitter 1 b and directly moved ahead . the light rays l 3 and l 4 advance through the prism 1 a to enter the adhesive layer 1 c . but this light incident portion of the adhesive layer 1 c has a largest thickness and thus causes a positional shift of δx in an x - direction . the shifting occurs with the light rays l 3 and l 4 , and light rays l 3 ′ and l 4 ′ parallel to the light rays l 3 and l 4 are emitted . the light ray l 5 emitted from the approximately central part of the light valve 45 b is likewise made incident to the upper plane 1 f of the polarized beam splitter 1 b , and moved ahead through the prism 1 a . as an incident portion of the adhesive layer 1 c for the light ray l 5 has a thickness of ½ of that of the incident portion for the light rays l 3 and l 4 , the light ray l 5 is shifted in the x - direction by an amount of ( δx · ½ ) to enter the prism 1 b . a shifted light ray advances in parallel with the light ray l 5 , and then exits as a light ray l 5 ′. fig2 is a qualitative view illustrating deviation and emission of a light ray emitted from a display section of the light valve 45 b in parallel with a − z direction when the light ray is transmitted through the polarized beam splitter 1 b . in fig2 a dotted line indicates a position corresponding to the display section of the light valve 45 b ( position of a light ray when the ray is not passing through the polarized beam splitter 1 b ). a solid line indicates a position of the light ray after having passed through the polarized beam splitter 1 b . accordingly , it can be understood qualitatively that no changes occur in a length in the y - direction , but the light ray having passed through the thickest portion of the adhesive layer 1 c is deformed to be a parallelogram , which is a result of δx deviation made in the x - direction in proportion to the thickness of the adhesive layer 1 c . therefore , it can be understood that since the adhesive layer 1 c is formed to be wedge - shaped without having uniform thickness as shown in fig2 a projected image is distorted to be a parallelogram . referring to fig3 illustrated is a state of a light ray made incident on the prisms 1 a and 1 b and the adhesive layer 1 c of the polarized beam splitter 1 b when viewed from the y - direction of fig2 . it is now assumed that a light ray has been made incident from an object ( prism 1 a ) having a refractive index n 1 on a position a of an object ( adhesive layer 1 c ) having a thickness d 1 and a refractive index n 2 ( usually n 1 & gt ; n 2 ) by an incident angle θ 1 . in accordance with snell &# 39 ; s law , the following relationship is established with a refractive angle θ 2 : the light ray having advanced through the adhesive layer 1 c with the refractive angle θ 2 is emitted from the adhesive layer 1 c in a position b , refracted with the angle θ 1 , and then emitted into the prism 1 b . a length ab from the position a to the position b is expressed as follows : an amount of deviation δx 1 of the light ray is expressed as follows : δ x 1 = ab × sin ( θ 2 − θ 1 ) ( 3 ) for the light rays l 1 to l 5 of fig2 the foregoing θ 1 may be set to 45 °. by setting θ 1 to 45 ° in the foregoing expressions ( 1 ) to ( 3 ), the amount of deviation δx 1 can be expressed as follows : δ   x1 = d 1  ( n 1 2 · n 2 2 - 0 · 5 · n 1 2 - 1 2 ) ( 4 ) for the light rays l 3 and l 4 of the polarized beam splitter 1 b shown in fig2 if a thickest portion of the wedge shape of the adhesive layer 1 c is d , a maximum deviation amount δx in this case is expressed as follows : δ   x = d  ( n 1 2 · n 2 2 - 0 · 5 · n 1 2 - 1 2 ) ( 5 ) if a pixel pitch of the light valve 45 b is p in the x - direction , the foregoing deviation amount δx should be set to be ( ½ )× p or less , preferably to be ( ⅓ )× p or less . with these values of the deviation amount , no problems will occur in a projected image . in the projection display apparatus of the embodiment , the polarized beam splitter 1 b is manufactured with a precision for satisfying the expression ( 5 ). in this case , although a projected image may be slightly distorted to be a parallelogram , the distortion is about ½ pixel at the maximum . accordingly , a precision necessary for image processing can be achieved . the embodiment has been described by using the adhesive layer 1 c which is changed in thickness in the y - direction . but the present invention is not limited to this . for example , the invention can be applied to an adhesive layer which is changed in thickness in the x - direction . it was described above with reference to fig2 that almost no wedge or gap exists in the adhesive layer 1 c for the light rays l 1 and l 2 . in most actual cases , however , a portion of the adhesive layer 1 c to be adhered has a limited thickness , and portions of the adhesive layer 1 c for the light rays l 3 and l 4 are thicker . in such a case , it is only necessary to determine a difference in thickness between the thickest and thinnest portions of the adhesive layer 1 c . this is because the light rays l 1 to l 5 take shapes where all the solid lines indicating the amounts of deviation in fig2 have been moved in parallel by amounts equal to the foregoing minimum thickness in the x - direction , and the parallel movements can be adjusted by registration adjustment of the light valve 45 b . only the polarized beam splitter 1 b has been described . the same thing can be said for the other polarized beam splitters 1 g and 1 r . in other words , if each of these splitters is manufactured with a precision for satisfying the foregoing expression ( 5 ), distortion of a projected image will be only ½ pixel at the maximum , and thus an image which satisfies a precision for image processing will be obtained . the projection display apparatus has a structure where a light from the light source is color - separated into b , g and r lights by the color - separation optical system , and the polarized beam splitters 1 b , 1 g and 1 r are arranged for the respective color lights . polarization and separation are performed by the polarized beam splitters 1 b , 1 g and 1 r , and lights emitted from the respective color light valves 45 b , 45 g and 45 r are also analyzed by the splitters 1 b , 1 g and 1 r . especially , the lights made incident from the light valves 45 b , 45 g and 45 r on the polarized beam splitters 1 b , 1 g and 1 r are analyzed as polarized lights to be transmitted . the analyzed lights of respective colors are color - composed , and then projected by the projection lens 6 . in this case , as described above , since the lights emitted from the light valves 45 b , 45 g and 45 r for the respective color lights are analyzed as polarized lights to be transmitted by the polarized beam splitters 1 b , 1 g and 1 r for the respective color lights , registration can be enabled among the light valves 45 b , 45 g and 45 r for the respective color lights , and also distortion of a projected image can be reduced . if a refractive index n 1 of each of the prisms 1 a and 1 b as the constituent elements of the polarized beam splitters 1 b , 1 g and 1 r is 1 . 84 and a refractive index n 2 of the adhesive layer 1 c is 1 . 42 , then the following is established : if a pixel pitch p of each of the light valves 45 b , 45 g and 45 r is 40 μm , a difference between a maximum thickness and a minimum thickness of the wedge shape of the adhesive layer 1 c should be limited to about 22 μm or lower ( preferably , 15 μm ). if a pixel pitch p of each of the light valves 45 b , 45 g and 45 r is a small pixel pitch of 10 μm , then ¼ of the foregoing value is necessary for the thickness difference of the wedge shape of the adhesive layer 1 c . accordingly , the difference should be set equal to 5 . 5 μm or lower ( preferably , 4 μm or lower ). if a refractive index n 1 of each of the prisms 1 a and 1 b as the constituent elements of the polarized beam splitters 1 b , 1 g and 1 r is 1 . 84 as in the case of the preceding example and a refractive index n 2 of the adhesive layer 1 c is 1 . 57 , which is larger than that of the preceding example , then the following is established : if a pitch of each of the light valves 45 b , 45 g and 45 r is 40 μm , then a difference between a maximum thickness and a minimum thickness of the wedge shape of the adhesive layer 1 c must be limited to about 59 μm or lower ( preferably , 39 μm or lower ). if a pixel p of each of the light valves 45 b , 45 g and 45 r is a small pixel pitch of 10 μm , then ¼ of the foregoing value is necessary for the thickness difference of the wedge shape of the adhesive layer 1 c . accordingly , the difference must be set equal to 15 μm or lower ( preferably , 10 μm or lower ). but this value is larger compared with that in the first example . a projection display apparatus of a second embodiment functions as follows : first , lights from a light source are polarized and separated by a polarized beam splitter ; one polarized light is made incident on , for example , a phillips color - separation prism to be color - separated into r , g and b lights ; these r , g and b lights are made incident on reflection light valves arranged for the respective color lights to be modulated ; the lights thereby emitted are made incident on the light exit surface of the prism again for color composing ; the lights are made incident on the polarized beam splitter ; only the modulated lights are analyzed ; and then the analyzed lights are projected by a projection lens . in the system for using the analyzed lights in the polarized beam splitter as lights to be transmitted , the number of polarized beam splitters to be used is one . accordingly , image deviation of each light valve never occurs in the polarized beam splitter , and a problem of a distorted projected image can be solved . referring to fig4 illustrated is a structure of the projection display apparatus of the second embodiment . lights emitted from a light source 3 are roughly parallel light beams and randomly polarized . these lights are passed through the same polarization converter 2 as that of the first embodiment so as to be converted into s polarized lights . the lights converted into s polarized lights by the polarization converter 2 are first made incident on a polarized beam splitter 101 . a polarizing and separating section 101 p of the polarized beam splitter 101 is arranged in an s direction for reflecting the s polarized lights . thus , the s polarized lights made incident on the polarized beam splitter 101 are reflected by the polarizing and separating section 101 p and then emitted . the lights are then made incident on phillips prism 7 which constitutes a color - separation and composing optical system . phillips prism 7 is composed of first , second and third prisms 71 , 72 and 73 . a gap is provided between the first and second prisms 71 and 72 . a b light reflection dichroic film 70 b is formed on a surface of the first prism 71 , and an r light reflection dichroic film 70 r is formed on a joint surface between the second and third prisms 72 and 73 . a b light component included among white s polarized lights made incident on the first prism 71 of phillips prism 7 is reflected by the dichroic film 70 b to advance through the first prism 71 . the b light is subjected to a total reflection by an incident surface of the first prism 71 while advancing through the first prism 71 , and advances to exit from the first prism 71 . then , a light valve 41 b for b light arranged in the vicinity of a light exit surface of the first prism 71 is illuminated by the s polarized light . a mixed light component of an r light and a g light which advance after being transmitted through the first prism 71 is made incident on the second prism 72 to advance . then , the mixed light is divided into an r light component and a g light component : the r light being reflected by the r light reflection dichroic film 70 r provided on the joint surface between the second and third prisms 72 and 73 to advance , and the g light being directly transmitted to advance into the third prism 73 . the former r light advances through the second prism 72 , further advances after being totally reflected by a surface abutting the gap with the first prism 71 and exits . then , a light valve 41 r for r light is illuminated . the latter g light directly advances through the third prism 73 to exit from the same . then , a light valve 41 g for g light is illuminated . the lights of respective colors made incident on the light valves 41 r , 41 g and 41 b for the respective colors are subjected to modulation by color signals inputted thereto . these lights are then reflected / emitted as mixed lights of modulated p polarized lights and unmodualted s polarized lights . the lights emitted from the light valves 41 r , 41 g and 41 b for the respective lights are moved ahead in opposite directions on the same optical axis as an incident optical axis to enter phillips prism 7 again . then , a composed light is emitted from the incident surface of the first prism 71 . the color - composed light emitted from phillips prism 7 is made incident on the polarized beam splitter 101 . only the modulated light included in this composed light is analyzed as a light to be transmitted by a polarizing and separating section 101 p of the polarized beam splitter 101 . the unmodulated light is discarded as a reflected light . the analyzed light is made incident on the projection lens 6 . then , a full - color image is projected on a not - shown screen . the polarized beam splitter 101 now in use serves both as a polarization and separation optical system and a light analyzing optical system . this splitter 101 is manufactured with a precision for satisfying the expression ( 5 ) of the first embodiment . as a result , although a projected image may be slightly distorted to be a parallelogram , its distortion does not exceed about ½ of one pixel at the maximum , and thus the image can satisfy a precision necessary for image processing . fig5 is a view for explaining a structure of a projection display apparatus of a third embodiment . in the apparatus of the third embodiment , rather than analyzing a modulated light from a composed light by using one polarized beam splitter , polarized beam splitters for light analyzing are provided for respective color lights . lights emitted from a light source 3 are roughly parallel light beams and random polarized . these lights are passed through the same polarization converter 2 as that of the second embodiment so as to be converted into s polarized lights . the lights converted into s polarized lights by the polarization converter 2 are made incident on a dichroic mirror 81 arranged on an optical axis for transmitting a b light and reflecting g and r lights , and then color - separated into a transmitted b light and a composite light of reflected g and r lights . the latter composite light of the g and r lights advances in a direction orthogonal to the b light to enter a dichroic mirror 82 having a g light reflection property , which is arranged on the optical axis in parallel with the dichroic mirror 81 . then , the composite light is reflected and color - separated into a g light which advances in an orthogonal direction and an r light which is transmitted to advance straight . as can be understood from the foregoing descriptions , the dichroic mirrors 81 and 82 constitute a color - separation optical system for color - separating a light from the light source into r , g and b lights . the b , g and r lights obtained by color - separation are made incident on polarized beam splitters 201 b , 201 g and 201 r arranged for the respective color lights . polarizing and separating sections of the polarized beam splitters 201 b , 201 g and 201 r are arranged in an s direction so as to reflect incident s polarized lights . incident s polarized lights of the respective colors are respectively reflected by the polarizing and separating sections and then emitted from the polarized beam splitters 201 b , 201 c and 201 r . the s polarized lights of the respective colors emitted from the polarized beam splitters 201 b , 201 g and 201 r are then made incident on reflection light valves 45 b , 45 g and 45 r arranged in the vicinity of light exit surfaces . the s polarized lights made incident on the light valves 45 b , 45 g and 45 r are subjected to modulation by color signals inputted thereto . modulated lights by the light valves 45 b , 45 g and 45 r become p polarized lights . these p polarized lights are reflected and emitted together with s polarized light as unmodulated lights , and then made incident from the light exit surfaces in opposite directions on the polarized beam splitters 201 b , 201 g and 201 r again . the modulated p lights included in the lights made incident on the polarized beam splitters 201 b , 201 g and 201 r are transmitted through the respective polarizing and separating sections and analyzed , and then transmitted through the polarized beam splitters and emitted . the b light included in the modulated p polarized lights emitted from the polarized beam splitters 201 b , 201 g and 201 r is made incident on a b light reflection dichroic mirror 86 arranged on an optical axis , reflected by the same and then moved ahead after changing the optical axis by a right angle . the g light is made incident on a g light reflection dichroic mirror 87 arranged on an optical axis in parallel with the dichroic mirror 86 , reflected by the same and then moved ahead after changing the optical axis by a right angle . the g light is then made incident on the dichroic mirror 86 , transmitted and then composed with the b light . the r light is made incident on the dichroic mirrors 87 and 86 , transmitted through these elements to advance , and then color - composed with the g and b lights . as can be understood from the foregoing , the dichroic mirrors 86 and 87 constitute a color composing optical system . a composed light formed by the color composing optical system which includes the dichroic mirrors 86 and 87 is made incident on a projection lens 6 , and a full - color image is projected on a not - shown screen . each of the polarized beam splitters 201 b , 201 g and 201 r serves both as a polarization and separation optical system and a light analyzing optical system , and is manufactured with a precision for satisfying the expression ( 5 ) described above with reference to the first embodiment . therefore , although a projected image may be slightly distorted to be a parallelogram , the distortion does not exceed about ½ of one pixel at the maximum , and thus the image can satisfy a precision necessary for image processing . fig6 is a view for explaining a structure of a projection display apparatus of a fourth embodiment . this projection display apparatus comprises : a light source 3 composed of a lamp and a concave mirror ; a polarization converter 2 for converting a light from the light source 3 into an s polarized light , which is the same as that of the first embodiment ; a polarized beam splitter 301 having a polarizing and separating section 301 p ; a light valve 40 as a reflection type light modulator arranged on an optical axis in the vicinity of a light exit surface of the polarized beam splitter 301 ; and a projection lens 6 for projecting a modulated light transmitted through the polarized beam splitter 301 and analyzed on a screen ( not shown ). in the projection display apparatus , lights emitted from the light source 3 are parallel light beams and random polarized , and passed through the same polarization converter 2 as that of the second embodiment so as to be converted into s polarized lights . the s polarized lights formed by the polarization converter 2 are made incident on the polarized beam splitter 301 . but since the polarizing and separating section 301 p of the polarized beam splitter 301 is arranged so as to reflect an s - polarized lights and transmit p - polarized lights , the s polarized lights are reflected by the polarizing and separating section 301 p , moved ahead after changing an advancing direction by a right angle , and then emitted from the polarized beam splitter 301 . the s polarized lights emitted from the polarized beam splitter 301 and made incident on the light valve 40 are , in a specified region selected by a color signal to the light valve 40 , subjected to modulation , and then converted into p polarized lights where a vibrating direction has been changed by 90 °. in an unselected region , i . e ., in a region not selected by the color signal , the incident s polarized lights are directly reflected / emitted . specifically , a light emitted from the light valve 40 is a mixed light of a modulated p polarized light and an unmodulated s polarized light . this emitted light is made incident on the polarized beam splitter 301 again , and then polarized and separated by the polarizing and separating section 301 p into a modulated p polarized light to be transmitted and an unmodulated s polarized light to be reflected and discarded , in other words analyzed as such . the modulated p polarized light transmitted through the polarized beam splitter 301 and analyzed is made incident on the projection lens 6 , and then projected on a screen . the polarized beam splitter 301 used in the fourth embodiment serves both as a polarization and separation optical system and a light analyzing optical system . this splitter 301 is manufactured with a precision for satisfying the expression ( 5 ) described above with reference to the first embodiment . therefore , although a projected image may be slightly distorted to be a parallelogram , the distortion does not exceed about ½ of one pixel at the maximum , and thus the image can satisfy a precision necessary for image processing . although the preferred embodiments of the present invention have been described in detail , it should be understood that various changes , substitutions and alterations can be made therein without departing from spirit and scope of the invention as defined by the appended claims .