fig1 shows a schematic diagram for the virtual keyboard input system using three - dimensional motion detection by variable focal length mmal . the first requirement of the small mobile device 11 is compactness . since the virtual keyboard input system projects a keyboard image 15 , no space for the input device is necessary . the input system 12 has a variable focal length mmal for detection of the three - dimensional movement of the user 16 . the input system 12 gathers the information on the motions in three - dimensions by three - dimensional imaging method using the variable focal length mmal . the three - dimensional imaging using the variable focal length mmal was proposed in u . s . patent application ser . no . 10 / 822 , 414 . the three - dimensional imaging method obtains the depth 14 and in - plane position information of user 16 motion . once the information is collected , the motion can be described as a time dependent movement in three - dimensional coordinate system 17 . with the time dependent motion , the system determines which input action is occurred by the user . then the system gives the correct input to the computer based mobile device 11 . independently , the system has a two - dimensional projection system for guiding the input of the user . this projection system projects a keyboard image 15 or some other guidelines to help the user performing proper action in proper space . the 2d projection system using mmal was proposed in u . s . patent application ser . no . 10 / 914 , 474 and 11 / 208 , 114 . fig2 is a schematic diagram showing the method for finding time - dependent change of the point by the variable focal length mmal and imaging unit . the virtual keyboard input system 21 has a variable focal length mmal 26 . the motion of user 22 from p 0 ( x 0 , y 0 , z 0 , t 0 ) to p 1 ( x 1 , y 1 , z 1 , t 1 ) in real coordinates 23 is captured by the two - dimensional image sensor 27 . the three - dimensional imaging and motion detecting function are achieved by image processing about the two - dimensional images captured by the image sensors . the image sensor takes two - dimensional images of user finger with different focal planes that are shifted by changing the focal length of the mmal . the image processing unit 28 extracts in - focus pixels or areas from the two - dimensional images with different focal planes 24 and generates an all - in - focus image . three - dimensional information 29 of the user finger can be obtained from the focal plane of each in - focus pixel . there are several methods for the image processing unit to obtain an all - in - focus image with depth information . the input system generates a time - dependent motion data of the user and gives the proper action commands to the computer based device with respect to the motions of the user . the input system gives keyboard input to the computer based device by recognizing the motion of user fingers . a pointing tool can be used for indicating the three - dimensional position . the pointing tool can be any object which can be imaged in the image sensor . the fast response of the mmal enables the process for finding the focus of the pointing tool instantaneous . image processing unit processes the taken two - dimensional image to determine whether the object image lies on the focus or not . the tracking of the pointing tool can be accomplished by the fast response of the mmal . the input system gives an output just like the conventional keyboard does . the output of the input system can be serial , ps / 2 , or usb type signal . the input system gives an output of a pointing action . the output of the pointing action is time - dependent three - dimensional position . since the changing of the focal length in the variable focal length mmal is so fast , the real - time three - dimensional movements of the fingers can be detected . the input can be obtained by the position and the movements of the fingers . the input system generates a time - dependent motion data of the user . the auxiliary lens 25 can be applied for improving imaging performance such as better numerical aperture and focusing power of variable focal length mmal 26 . fig3 shows the principle of a mmal 31 . two conditions should be satisfied to build a perfect lens . one is a converging condition that all light scattered by one point of an object should converge into one point of the image plane . the other is the same phase condition that all the converging light at the image plane should have the same phase . to satisfy the perfect lens conditions , the surface shape of conventional reflective lens 32 reflects all the incident light scattered from one point of an object to the other one point on the image plane with the same optical path length traveled . thanks to the periodicity of the light phase , the same phase condition can be satisfied even though the optical path length of the converging light is different . when the difference of the optical path length is exactly the same as the multiples of the wavelength , the reflected beam at the focus meets the phase condition . therefore , the surface shape of the conventional reflective lens 32 satisfying perfect lens conditions can be replaced by rotation and translation of micromirrors . each micromirror 33 rotates to converge into focal point and translates to adjust the phase between the reflected lights from different micromirrors 33 . fig4 illustrates the two - dimensional view of a mmal 41 . each micromirror 42 of the mmal 41 is controlled by electrostatic and / or electromagnetic force made by actuating components 43 . the mechanical structures upholding each micromirror and the actuating components to rotate and translate the micromirrors 42 are located under the micromirrors 42 so that the micromirrors 42 have larger active area . fig5 illustrates how the mmal 51 makes an image . arbitrary scattered lights 52 , 53 from the object are converged into one point p on the image plane by controlling the position of each of the micromirrors 54 . phases of individual lights 52 , 53 can be adjusted to have the same value by translating each of the micromirrors 54 . the required translational displacement is at least half of the wavelength of light . the focal length f of the mmal 51 is adjustable by controlling the rotation and / or translation of the micromirror 54 . the operation of the mmal 51 is possible by controlling only rotation regardless of the phase condition . in this case , the quality of the image generated by the mmal is degraded because of aberration . also translation only without rotation can form a fresnel diffraction lens with aberration . the smaller the sizes of the micromirrors 54 can reduce aberration . even though the focusing ability by either rotation or translation is not powerful , the lens with one degree of freedom in motion has the advantage of simple control and fabrication . fig6 shows two degrees of rotational freedom and one degree of translational freedom of the micromirror 61 . the array including micromirrors 61 with two degrees of rotational freedom 62 , 63 and one degree of translational freedom 64 , which are controlled independently can make a lens with arbitrary aspheric surface . incident lights can be modulated arbitrarily by forming an arbitrary aspheric surface . to do this , it is required that incident lights are deflected to an arbitrary direction by controls of two degrees of rotational freedom 62 , 63 . independent translation 64 of each micromirror is also required to satisfy the phase condition . while the invention has been shown and described with references to different embodiments thereof , it will be appreciated by those skills in the art that variations in form , detail , compositions and operation may be made without departing from the spirit and scope of the invention as defined by the accompanying claims .