Patent Publication Number: US-7213985-B1

Title: Method for image reproduction and recording with the methods for positioning, processing and controlling

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
   PCT/US03/25111, filed on 11 Aug. 2003 and from the provisional U.S. Patent Application No. 60/402,233, filed on 12 Aug. 2002 entitled “System and its apparatuses for image reproduction and recording with the methods for positioning, processing and controlling”. 

   FIELD OF THE INVENTION 
   The present invention relates to a method to reproduce and to record image with a flexible operation (by hand, robot or vehicle) of head carrier without mechanical-guide-apparatus, and the corresponding apparatuses and methods for positioning, processing, and controlling. The motivation is to build a flexible operation (i.e. without a track guide) for image reproduction and recording system, instead of present conventional image reproduction and recording systems in plurality of uses. Due to the flexibility of this invention in operation, the size of image that will be reproduced or will be recorded can be as large as the wall of a building, or golf course, or cliff of a mountain, or can be as small as any size as long as it still makes sense. Therefore, it can be used for plurality of applications, such as images and patterns on building wall or cliff, golf courses, basketball courts, football/soccer fields, billboards, posters, portraits and paintings, industry design blue prints, industry decorations, decoration arts (such as depositing a pattern on china arts), home painting and wall decorations, archaeological image/pattern taking and museum image/pattern backup, sculptures, etc. It can be used for applications either on any flat surface, or on any curved surface. 
   BACKGROUND OF THE INVENTION 
   The conventional method for image reproduction and image recording s, such as the methods used in printing devices and scanning devices sold in the electronics store and those described in U.S. Pat. Nos. 5,968,271, 5,273,059, 5,203,923, 4,839,666, 5,707,689, 6,369,906, 5,642,948, 5,272,543 [1-8]  etc are based on the track-guided positioning systems. The spraying head or reading (recording) head is driven by electric motors and is limited on a track through the precise mechanical-apparatus for positioning. Therefore, they have limitation in size and service objectives, and they have no flexibility for plurality of applications, such as image on billboards, on the walls, with huge size or on a curved surface, etc. Also the conventional method is mechanical-apparatus based and is complex and costly. Therefore the motivation of this invention is to build the flexible hand-operated, or robot-operated or vehicle carried systems for image reproduction and recording. Due to the flexibility of operation, the image that will be reproduced or will be recorded can be arbitrary large, and can be used for either any flat surface, or any curved surface. 
   SUMMARY OF THE INVENTION 
   The key spirits of present invention is the new method for image reproduction and recording with a flexible hand-operated or robot-operated or vehicle carried head carrier, and the corresponding apparatuses and methods for positioning, processing, and controlling. The systems based on this method are flexible, easy and very convenient to use for a plurality of users from industries, offices and home, home decorations, entertainment and arts, etc., instead of the complex and costly precise mechanical-apparatus based systems in present conventional method for image reproduction. 
   A further object of the present invention is to provide constitutions and apparatuses for head positioning, data processing, and head controlling. 
   To achieve the above objects, the first aspect of the invention provides the method for image reproduction on any surface based on image data stored in computer, by arbitrarily moving the flexible-operation (hand, robot, vehicle) apparatus, i.e. head carrier, on the surface. The systems based on this method could have variation of versions, depending on the methods used for positioning. The positioning methods for image reproduction are classified into two catalogs: the wave-based method and relative-motion-based method. The systems using both methods comprise these apparatuses: head carrier, sprayer/sprayer array, operation unit (OU), and a computer for processing and control. Besides these apparatuses, the wave-based method also includes the communication units (CU) and the relative-motion-based method includes two relative motion detectors (MD). 
   In the relative-motion-based method, operation unit (OU) is also called operation module (OM) for convenient in the description below, so as to avoid the confusion with the OU used in wave-based method. The system operation procedures include: OM executes the commands from computer to read the motion information of head from MD, and organizes this information as time-sequences. Then OM sends these time-sequences to computer by multi paths (in parallel). Computer processes the information for locator positioning and determining the coordinates of each head in the head array. The OM executes the commander from computer to control the action (spraying or reading) of the head in head array. For recording system, the OM takes the image information at each image pixel on sensor array, and organizes this information as time-sequences and sends them to computer. Also, as the alternates, any computer-mouse techniques can be employed as MD. 
   In the wave-based method, the system operation procedures include: operation unit (OU) produces and sends the signal current to the transmitting CU. The transmitting CU radiates and the receiving CU receives the radio frequency (RF), electromagnetic wave, light or ultrasonic signals that carry the information of the phase differences or the time differences. The information is sent back to the OU from the receiving CU. The OU processes and converts the information into the data of phase differences or time differences, and sends the data to computer. Another alternate uses Doppler effect to detect the velocity of the receiving CU, and computer calculates the moving distance by integrating the velocity. 
   Computer processes these data and inverses the position coordinates of the sprayer/sprayer array by using the claimed positioning methods in this invention. According to the position coordinates, computer searches for the nearest pixel to this position in the image data file stored in disk of the computer, takes the color data of this pixel, and sends the data to OU or OM. Then OU or OM sends commands and power to the head to execute the jobs (spray or record). Computer then records the history of the image reproducing or recording process. Any pixel, of which the corresponding image has been generated (sprayed or read) on the image surface, will be marked by the computer, and displayed on the computer screen, and will not be generated again if the head moves back to the same position later. 
   The CU in the wave-based system or MD in relative-motioned-based system is also called locator of head, shortly locator. Usually there are two of them. With the first one, the second CU or MD is used for determining the sprayer array direction, so that the position of each sprayer/reader in the sprayer/reader array is determined. 
   The second aspect of the invention provides the method for recording image. The system based on this method takes the image digital data from any image surface to computer for storing and reproducing, and also by arbitrarily moving the hand-operation or robot-operation or vehicle carried apparatus, on the surface. All apparatuses and procedures in the systems are same as that in the image reproduction system, but use image reader/reader array instead of sprayer/sprayer array. Trigged by a trigger clock, the coordinate information and color data are taken from the image surface at the triggered moment and are sent back to the computer. The computer processes the information and data promptly or stores them into a file for processing lately. The computer inverses the coordinate information into coordinates. The coordinates at the triggered moment may not be just at a pixel on the pre-formatted pixel grids. So then the computer calculates the color values at all pixels on the pre-formatted pixel grids based on the obtained coordinates and color data, by using interpolation method. 
   In the third aspect of the invention provides the theories, concepts, ideas, and methods corresponding to each structure, embodiment, apparatus, and procedure, for positioning, processing and controlling the image reproduction and recording, including hardware signal processing and software data processing. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A better understanding of the invention will be obtained by reading the detail description of the invention below, with reference to the following drawings, in which: 
       FIG. 1  is a view showing the constitution of one of the preferred embodiments for the image reproduction and recording system according to the invention, with the CU (communication unit) on the corners, and the color material tanks on the head carrier or in the cartridge that are build together with the head. 
       FIG. 2  is a view showing the constitution of other preferred embodiments for the system according to the invention: (a) the color material tanks on the ground, (b) three CU on the corners, (c) four CU on the middle edges, (d) two CU on the bottom corners. 
       FIG. 3  is the schematic chart of one of the preferred embodiments for the head carrier with single head according to the invention. 
       FIG. 4  is the schematic chart of one of the preferred embodiments for the head carrier with head array according to the invention. 
       FIG. 5  is the schematic chart of one of the preferred embodiments for the head carrier with sprayer array on ink-jet cartridge according to the invention. 
       FIG. 6  is the schematic chart of the preferred embodiments for the transmitting CU&#39;s: (a) Radio frequency (RF) antenna, (b) single-light-source transmitter, (c) four-light-source transmitter, (d) ultrasonic transmitter. 
       FIG. 7  is the schematic chart of the preferred embodiments for receiving CU&#39;s: (a) RF antenna, (b) single-photon-detector receiver, (c) two-photon-detector receiver, (d) four-photon-detector receiver, (e) corner single-photon-detector, (f) corner single-photon-detector with curved substrate, (g) ultrasonic receiver. 
       FIG. 8  is the schematic chart of one of the preferred embodiments for relation motion detector (MD). 
       FIG. 9  is a schematic block diagram of the control and processing for one of the preferred RF-based system according to the invention. 
       FIG. 10  is a schematic block diagram of the control and processing of another of the preferred RF-based system according to the invention. 
       FIG. 11  is a schematic block diagram of phase processing for the direct-RF-based systems. 
       FIG. 12  is a schematic block diagram of the control and processing of one of the preferred modulation-based systems according to the invention, with FOUR wavelengths/frequencies. 
       FIG. 13  is a schematic block diagram of the control and processing of another of the preferred modulation-based systems, with TWO wavelengths/frequencies. 
       FIG. 14  is a schematic block diagram of the control and processing of another of the preferred modulation-based systems, with four wavelengths/frequencies. 
       FIG. 15  is a schematic block diagram of the control and processing of another of the preferred modulation-based systems, with two wavelengths/frequencies. 
       FIG. 16  is a schematic block diagram of the control and processing of one of the preferred time-based systems with an ultrasonic approach. 
       FIG. 17  is a schematic block diagram of the control and processing of one of the preferred time-based systems with another ultrasonic approach. 
       FIG. 18  is a schematic chart of the contour curves for constant phase differences (hyperbola), and constant phase sum (ellipse). 
       FIG. 19  is a flow chart of the position data processing and control for a single head. 
       FIG. 20  is a flow chart of the position data processing and control for the head array. 
       FIG. 21  is a schematic chart of the wrapping of current-phase-relation of in a digital phase detector (DPD) and the wrapped region in the 2-D phase space. 
       FIG. 22  is a schematic chart of data correlation processing for relative-motion-based system: image correlation conception and simple motion. 
       FIG. 23  is a schematic chart of data correlation processing for relative-motion-based system: complex motion. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   The present invention is to provide a method for image reproduction and recording with the flexibility, easiness, and convenience to use for a plurality of users from industries, offices and homes, and home decorations. The systems based on this method are flexible and consist of an easy hand-operation or robot-operation or vehicle carried apparatus, instead of the complex and costly mechanical apparatus-based systems in present conventional image reproduction and recording systems in plurality of uses. 
   &lt;Dictionary&gt; 
   For convenient in reading this invention, it is necessary to build a ‘dictionary’ for the definitions of some terms, as listed in the following: 
   (1) In “Flexible operations”: “hand-operation” means operation by hand of a human being; both “robot-operation” (such as the ‘spiderman’-like) and “vehicle carried operation” means the powered-apparatus-aided operation, but without mechanical-guide-apparatus (such as track guide for guiding the printing head or scanning head in the conventional printer, or scanner) for positioning, if the operation needs a power that exceeds the power of the human being, or if the environment of operation is not accessible for human being; 
   (2) The term “image generation, or generate image” means reproducing (printing, painting, spraying, and deposition) or recording (scanning, and reading) image or pattern on or/and from any surface. 
   (3) The term “image” in phases “image reproduction or image recording” has dual meanings: (a) any predetermined pattern or deposition to be reproduced, or any pattern or deposition to be recorded, which has already existed and was resulted from human&#39;s arts or natural&#39;s arts; (b) the image stored in computer, which could be recorded by scanner, or taken by digital camera, digital camcorder, etc. 
   (4) The term “head” in this invention means either sprayer for image reproduction or reader for image recording. Some time the “head” also means the part on which the head is installed; 
   (5) The term “sprayer” in this invention means the ink-jet, paint sprayer, or any other devices for material deposition. “Spray” or “spraying” means any action for material deposition; 
   (6) The term “reader” in this invention means any device that takes the image information from a predetermined pattern or deposition, such as the image sensor in an image scanner or in a camera. “read” or “reading” means any action of the reader; 
   (7) The “element” of an array is a general term referring to an element in one-dimensional array in positioning method description and claims. However, in image reproduction or recording system, it refers to a head in head array. 
   (8) The CU or MD built on head carrier is called head “locator” in claimed “image reproduction and recording system”. 
   (9) However “positioning locator” in the claims of positioning methods is a general term and is not necessary only for “image reproduction or recording system”; 
   (10) “Light” or “photon” means visible or invisible, coherent or non-coherent electromagnetic radiation from T-ray to X-ray; 
   (11) “Electromagnetic waves (EMW)” means all electromagnetic radiations from long wave up to 1 THz; 
   (12) “Wave” mean means all EMW and ultrasonic waves; 
   (13) “Information carrier” means RF wave or ultrasonic wave on which the information is ridding; while “carrier wave” means the light wave or millimeter microwave on which the RF is ridding (i.e. RF modulation); 
   (14) The term “in a space” or “in image space” means on 2D flat surface or 2D curved surface, or in our real space (3D). It is a well-known knowledge that 1D, is a line, 2D space is 2D plane and 3D space is our real space; 
   (15) The term “computer” means a programmable device (i.e. a generalized computer) for system and embodiment controlling. 
   (16) ‘phase detector’ means a mixer or a digital phase detector; 
   (17) “hand stick” means a device which provides the power to head-carrier for making head-carrier moving, it could be either hand-hold apparatus or powered-apparatus; 
   &lt;System Constitution&gt; 
   A method for image reproduction and recording is described below, by using some specified systems that are based on this method. The method will be understand clearly and fully by describing the system constitution, system operation, apparatuses, and the methods for positioning, processing and controlling, in detail with references to the accompanying drawings. 
     FIG. 1  is used here to show the constitution of one of the preferred embodiments for the wave-based method for image reproduction and recording according to the invention. The by using this method, one reproduces the image on the image area  10  of a surface based on image data stored in computer  900 , or record the image data from image area  10  into computer  900 , by arbitrarily pushing and pulling the “hand stick”  102  of head carrier  100  (or any hand-hold brush-like body), on the surface. The surface can be any surface, such as curved, sphere or flat surface. The head carrier can be a hand-operational apparatus with a “hand stick”  102 , or can be a powered-apparatus-aided apparatus for huge applications, or can be robot operation, or vehicle carried operation, if the environment of operation is not accessible for human being. 
   For image reproduction, four communication units (CU)  201 ˜ 204 , used as the transmitters/receivers with marks (A 1 , A 2 , B 1 , and B 2 ), are set at the four corners. The CU (details in  FIGS. 3˜7  later) set on the head holder  300  are used as the receivers/transmitters, respectively. The information carrier can be either radio frequency (RF), or RF carried on light from T-ray to X-ray, or ultrasonic wave. However, if RF is directly (i.e. not modulation) used as information carrier, the CU must be set at corners or edges and must be fairly far away from the boundaries of image area  10 , due to the nonlinearity of phase dependence of the near-field. 
   For convenience, here, let us describe this case first: using the CU  201 ˜ 204  as the transmitters and using the single CU (head locator) on head holder  300  as the receiver. The operation unit (OU)  400  produces signals and sends signal to CU  201 ˜ 204 , through cables  51 , 52 ,  61 , 62 . The cables  51  and  52  are split from one source, and have the same length from the splitter  50  to A 1   201  and A 2   202 , so that they have the same time delay. The same is applied for cables  61 , and  62 ; they have the same length from the splitter  60  to B 1   203  and B 2   204 . The CU  201 ˜ 204  transmits the waves. The receivers receive the waves with phase or time information and send the message back to the OU  400  through cable  20 . The hardware in operation unit  400  processes the message and converts the message into phase difference or time difference, and sends these data to computer  900  through cable  40 . From these phase data, computer  900  inverses the coordinates of the position of the head locator (details in FIGS.  3 , 4 , 5 ) on head holder  300  by using positioning theories and formulas of this invention. According to the head position coordinates, computer  900  searches the pixel that is nearest to this position in image data file and takes the color data of this pixel, and sends the data to OU  400  through cable  40 . Then OU  400  sends action commands and power to spray head on head holder  300  through cable  30 . Any pixel on screen of computer  900 , of which the corresponding image has been reproduced on the image area  10 , will be marked by computer  900  and will not be reproduced again if the head on holder  300  moves back to the same position later. 
   For image recording, an image reader or reader array is installed on the head holder  300 . The positioning procedures are the same as that for image reproduction, described above. Triggered by the trigger clock, the coordinate information and color data are taken from the image area  10  at the triggered moment and are sent back to computer  900  through OU  400 . Computer  900  processes the information and data promptly or stores them into a file for overall processing lately. Computer  900  inverses the signal that carries the coordinate information into coordinates. The coordinates at the triggered moment may not be just at a pixel in the pre-formatted pixel grids. So computer  900  then calculates the color values at all pixels in pre-formatted pixel grids from the obtained coordinates and color data, by using interpolation method. 
   The transmitter and receiver can be swapped. The CU  201 ˜ 204 , A 1 , A 2 , B 1 , B 2 , can also be used as receivers (serve as receiving CU), while the CU on the head holder  300  can be used as transmitters (serve as transmitting CU). The details will be described in sections below. 
   The procedures described above are applicable for the all preferred and alternative constitutions described below. 
     FIG. 2  shows another preferred constitutions for 2 dimensional (2-D) applications according to the invention. The color material tanks are necessary for large images and are placed on the head carrier  100  (details in  FIGS. 3 ,  4  and  5 ). However, for huge images, the color tanks  140 , 142  and  144  are placed on the ground or on a support platform. The color materials are transported to sprayers on the head holder  300 , through tubes  130 , 132  and  134 , as shown in  FIG. 2  ( a ). In  FIG. 2  ( b ) is shown an option to use only three CU at three corners, with CU A 1   201  and B 1   203  merged together.  FIG. 2  ( c ) is an option to use four CU  201 ˜ 204  on the middle edges, which provides the simplest positioning theories and formulas. For the time-based positioning, the embodiment shown in  FIG. 2(   d ) is used; here only two CU A 1   201  and A 2   202  on bottom corners are used. 
   For 3-dimensional (3-D) applications, another one or two CU&#39;s need be installed at any points (except too close to the image surface) on z-axis of all the cases described in  FIG. 1  and  FIG. 2 . The z-axis is an axis that is vertical to the 2-D frame plane (image surface), or the z-axis could be one edge of the 3-D frame. For all the cases described above, CU can be either fully or partially at either the middle edges or the corners of the frame, and the color tanks can be either on the head carrier  100 , or on the ground, or on any support platform. 
   The cables used for transmitting the phase-doesn&#39;t-matter signal, color data, and operation commands between operation unit  400  and head  300  can be replaced by wireless communication. 
   For the relative-motion-based system, with the optical-image approach or mouse-technique approaches, there are no CU ( 201 ˜ 204 ) and OU  400 , and the cables between them. Instead of CU and OU, MD and OM are installed together with locator on head holder  300  and arbitrarily moving on the image surface. The OM (not shown in figures) is directly connected with computer  900  through a multi-path cable. Computer  900  periodically sends the commands to OM. OM executes the commands to read the motion information of the locators from MD, and organizes this information as time-sequences. Then OM sends these time-sequences to computer  900  by multi paths in parallel through the cable. Computer processes the information for locator positioning and determining the coordinates of the head in the head array. The OM executes the commander from computer to control the action (spraying or reading) of the head in head array. One of the preferred MD&#39;s comprises a two-dimensional array of camera-image sensors (M by N pixels), two lenses, and one laser. For recording system, OM reads out the image information at each image pixel on sensor array, and organizes this information as time-sequences and sends them to computer  900 , and then computer stores this image information on disk. Also, any computer-mouse techniques can be employed as MD. 
   &lt;Apparatus Constitutions and Operations &gt; 
   Head carrier 
     FIG. 3  shows one of the preferred embodiments for the head carrier with single head according to this invention. The head carrier  100  is composed of a frame  110  (main body of head carrier, any shape), one front wheel  112 , two rear wheels  114 , “hand stick”  102 , head arm  106 , and head holder  300 . The wheels ( 112 ,  114 ) enable the carrier  100  moving on the image area  10  freely, and guarantee a constant fly height  301  for the head  382  (sprayer or image reader) over surface  10 . The “hand stick”  102  is connected with the head carrier  100  by a joint  104 , and the stick  102  can freely rotate about joint  104 . The head arm  106  is connected with the head carrier  100  and can rotate about the axle  105  by hand-operation, for flexible application in various situations. The CU  381  and the head  382  are installed on the head holder  300 . Head holder  300  is supported by head arm  106  at one end of the arm. For small image applications, the color materials are stored in the container built-in with the sprayer or color cartridges. For large image applications, three (or four if an additional black tank is needed for color quality) color tanks  120  (cyan),  122  (magenta), and  124  (yellow) are installed on the head carrier  100 , moving together with the head carrier. The color materials are transported to the head from the tanks ( 120 , 122 , 124 ) through color tubes  130 , 132  and  134 . For huge image applications, the color materials are transported to the head from ground tanks  140 , 142 , 144  ( FIG. 2  ( a )) through color tubes  130 , 132  and  134 . 
     FIG. 4  is used to show one of the preferred embodiments for the head carrier with head array according to this invention. The differences of this head carrier from the one described in  FIG. 3  are in the head holder  300  and head cartridge  385  (instead of single head). A number of heads are built on the head cartridge  385  and form a head (sprayer or reader) array  386 . The image resolution (IR) is determined by head density in head array, which is determined by the number of heads in the array and the array length L 1  ( 391 ). Two CU ( 383 ,  384 ) (i.e. two head locators) are installed on the head holder  300 . The holder extension  303  is needed to hold one of the CU,  384 , so as to extend the distance L 2  ( 392 ) between two locators,  383  and  384 . The purpose of using this extension is to increase the accuracy in position determination of each head in the head array  386 . The extension  303  can be added to either side of the head holder  300 , depends on the convenience. The head holder  300  can rotate about the axle  302  by hand-operation, by 360°, for various situations of application. 
     FIG. 5  is used to show another preferred embodiment for the head carrier with sprayer array built-in an ink-jet cartridge according to the invention. The only difference from the one described in  FIG. 4  is that a color ink-jet cartridge  389  with sprayer array  390  is now used. 
   Communication Units 
   The preferred options for transmitting CU (i.e. transmitters) according to this invention are shown in  FIG. 6 , including (a) Radio frequency (RF) antenna  610 , (b) single light source (Laser or LED)  630 , (c) multi light source  640  (four is shown in figure), and (d) ultrasonic transmitter  620 . 
   The RF antenna  610  is used as the transmitter for RF-based system design. The wavelength of the lowest level RF should equal the size of image area  10 . Here is an example: sizes of 100 meters, 30 meters, 3 meters, 10 centimeters and 1 centimeter are corresponding to RF frequency 3 MHz, 10 MHz, 100 MHz, 3 GHz, and 30 GHz, respectively. If the technique for current-phase unwrapping processing is used, the frequency can be higher. 
   For applications with larger image area, the lower frequency is used. Therefore, the RF can be carried on (i.e. modulates) some extremely higher frequencies—millimeter microwave, where the frequency allocation is empty and the use of these frequencies is unlicensed (such as those at peak absorption of atmosphere), so as to avoid to be interrupted with public communication and military frequencies. In these cases, the same procedures as that used in light-based system described below are applicable, except the generator, transmitter, and receiver of carrier wave. 
   For the light-based systems, the RF is carried on the light wave by amplitude modulation or frequency modulation. The light is emitted from the emitter  632 , called single-light transmitter. For 2D application, by a cylindrical lens  634 , rather than a spherical lens, the light is uniformly diversified to the region with an angle  636  (any angle between 90° and 150° is applicable, but 110° is preferred). The design of lens and of light direction makes the light divergent as less as possible in the direction vertical to the paper plane. The single-light transmitter  636  is used for the system of which the transmitters are installed at the corners of the image plane. Multi-light transmitter  640  is built by number of single emitters  630 , and is used for the systems of which the transmitter is installed on the head holder  300 . The ultrasonic transmitter  620  is employed for the time-based systems. For 3D application, the lens is spherical and the six-light transmitter is used. 
     FIG. 7  is used here to show the preferred embodiments for receiving CU (receivers) according to this invention: (a) RF antenna  710 , (b) single-photon-detector  720 , (c) two-photon-detector receivers  730 , (d) four-photon-detector receiver  740 , (e) corner single-photon-detector  750 , (e) corner single-photon-detector with a curved substrate  760 , and (g) ultrasonic receiver  770 . Due to the reciprocal principle of electromagnetic theory, those described in RF transmitters above are applied for RF receivers  710 . The RF that is carried on an extremely high frequency (millimeter microwave) is demodulated by heterodyne or homodyne techniques. 
   For the systems of which the receiver is on the head holder  300 , the two-photon-detector receiver  730  (three-photon-detector for 3D), or the four-photon-detector receiver  740  (and six-photon-detector for 3D), is used. They are built from a single-photon-detector  720 . The latter is made up of photon sensor (photon detecting material)  728 , light wavelength-selection filter  726 , and cone mirror  724 . The cone mirror  724  reflects the light  722  from all directions to the filter  726  and photon sensor  728 . The current signal is generated from the sensor and is sent to the operation unit  400 . Inside the sensor, a pre-amplifier may already be built in. 
   For the systems of which the receiver is at the corner of the image plane, the single corner photon-detector  750 , or the one with a curved substrate  760  is used. The light  752  from different directions is focused on the photon-sensing material  728  by the lens  754 , so as to increase the sensitivity, as shown in  FIGS. 7  ( e ) and ( f ). Before the sensor, there is also an optical wavelength-selection filter. 
   The ultrasonic receiver  770  is employed if the ultrasonic transmitter  620  is used in the system. 
   Motion Detector and Operation Module 
   For the relative-motion-based system, the head includes a motion detector (MD), an operation module (OM), and a sprayer or/and a reader. The preferred apparatus for the MD is the detector of optical image motion ( 340 ), as shown in  FIG. 8 . The MD is built together with the sprayer head  350  or/and recording head (not shown in the figures). The container  359  in sprayer head  350  is a buffer for ink or paint material, which provides the ink or paint material for the sprayers in sprayer array  352 . The optical image motion detector  340  comprises laser  341 , lenses  342 ,  344 , and camera pixel sensor array  346 . The laser  341  is installed at a focus of the lens  342 , so the light is converted into parallel light beams and projects onto the surface, where is the path of head locator on image area  10 . By lens  344 , the optical image of the object (a ‘micro’ texture)  343  (any patterns, roughness distribution on the surface) appears on the surface  345  of the camera pixel sensor array  346 . The light paths  348  for the image system are shown on the right side. The distance between the object  343  and the center of lens  344  is beyond two focus length of lens  344 , while image  345  of the object is in between one and two of the focus length. The OM with a small volume (not shown in the figures) is installed together with the sprayer/reader and MD. OM executes the commands from the computer to read the motion information from MD, and organizes the information into time-sequences. Then OM sends these time-sequences data to the computer by multi-paths in parallel. OM also executes the commands from the computer to control the action of the head, after the computer finishes the processing. 
   For the recording system, the constitution is the same; the sprayer-array is replaced by the reader-array. 
   &lt;System Operations&gt; 
   The procedures of controlling and processing for one of the RF-based system according to the invention is shown in  FIG. 9 . The RF is directly used as the information carrier. The functions of operation unit (OU)  400 , of computer  900 , and of head  300  are shown in the frames of the left dish-line  401 , the right dish-line, and the top dish-line, respectively. Before the system is going to work, the noise detector  411  searches the low noise RF channels. According to the channel selection  412 , the frequency ω (higher) and Δω (lower) is determined (by using these two frequencies, the frequencies ω 1 , ω 2 , ω 3 , ω 4  for four RF channels are generated). The oscillators  413  and  414  are tuned to these two frequencies, and amplified by amplifiers  415  and  416 . The higher frequency is split into three by splitter  417 . Two of them are sent to mixers  419  and  420  and one of them is sent to frequency doubler  422  and then to a switch  423  (optional). The lower frequency is also split into three by splitter  418 . One of them is sent to mixer  420  directly and the second is sent to mixer  419  after frequency doubler  421 . The third one is sent to a switch  423 , which is connected to phase processor  430 . The two mixers provide the sum and differences of the two inputted frequencies. With filters  424 , four frequencies (ω 1 , ω 2 , ω 3 , ω 4 ) are separated and are sent to four transmitting antennas  211 ˜ 214  at A 1 , A 2 , B 1  and B 2  shown in the previous figures. All four RF channels are amplified by amplifiers,  425 . The receiver  311  receives the four signals from the four transmitters ( 411 – 414 ). After the band amplifier  426 , the amplified four signals are split into four paths by splitter  427 . The band pass filters,  428 , allow only one frequency pass through each one of them. Phase processor  430  decodes the phase differences between A 1  and A 2 , and the phase differences between B 1  and B 2 , if the switch is turned to down side. Or, phase processor  430  decodes the phase sums of A 1  and A 2 , and the phase sums of B 1  and B 2 , if the switch is turned to up side. More details about the phase processor are described later with  FIG. 11 . Phase calibration can be done by either the software in computer  900 , or by the phase calibrator  431  before signal goes into computer  900 . The same procedures for signal processing are applied for the receiver on the second locator shown in  FIG. 4  ( 384 ) or in  FIG. 5  ( 388 ). Computer  900  receives two groups of the phase messages for the positions of the two head locators (i.e. the antenna receivers), ( 432 , 433 ) and ( 444 , 445 ). 
   Computer  900  processes the phase data by inverting the coordinates of the positions of the two locators from the phase data, which is based on the positioning theories and formulas of this invention. According to the coordinates of the two locators, computer  900  calculates the coordinates of each of the head in head array ( 386 , in  FIG. 4 ) by using interpolation method. According to the position of each head, the computer  900  searches the pixel in the image data file that is nearest to this position and takes the color data of this pixel, and sends the data to control unit  429 . Then the control unit  429  sends commands of action and power to head  308  through color cables  306  and power cable  307 . 
     FIG. 10  shows the procedures of controlling and processing for another RF-based system according to the invention. The difference here is that the transmitter and receiver are swapped from the system described in  FIG. 9 . The four RF channels are combined together by combiner,  434 , before being sent to the transmitting antenna  321 . The four receiving antennas receive the signals and send the signals to four band-pass filters,  435 , which allow only one frequency to pass through each one of them. The four channels are then sent to phase processor  430  after being amplified by amplifiers,  436 . 
   One procedures of phase processing for the RF-based systems are shown in  FIG. 11  ( a ), the first two frequencies are conducted to mixer  4301 , which produce another two frequencies—the sum and difference of inputted frequencies. The band pass filters  4303  filter out the sum frequency. At this point, the signal with the difference frequency carries the phase difference between A 1  and A 2 . The digital phase detector (DPD) or mixer  4305  decodes the phase difference by homodyning with the signal from  423 . The phase difference  4315  (A 2 −A 1 ) is sent to the computer. The same is applied for the other two frequencies. The output phase difference  4314  (B 2 −B 1 ) is sent to the computer. 
   Another phase processing procedure for the RF-based systems is shown in  FIG. 11  ( b ). The largest and the smallest frequencies are conducted to mixer  4307 , which also produces two frequencies, the sum and difference. But the band pass filters  4309  filter out the difference frequency, and pass the sum frequency. At this point, the signal with the sum frequency carries the phase sum of A 1  and A 2 . The digital phase detector (DPD) or mixer  4311  decodes the phase sum by homodyning with the signal from  423 . Then the phase sum  4317  (A 2 −A 1 ) is sent to the computer. The same is applied for the two middle frequencies. The phase sum  4316  (B 2 −B 1 ) is sent to the computer. 
     FIG. 12  shows the procedures of control and processing for one of the modulation-based systems according to this invention. In this system, RF is used as modulation. The carrier wave of this RF wave is light or millimeter microwave. For the millimeter microwave carrier, the frequency with peak absorption (60˜70 GHz, 120˜130 GHz, and 170˜180 GHz, for example) is preferred but not limited, where the frequency allocation is empty and the use of frequency is unlicensed, so as to avoid to be interrupted with public communication and military frequencies. Here, the laser, as carrier-wave, is used for illustrations. The laser driver  437  provides four currents to four lasers ( 231 ˜ 234 ) to emit four wavelengths or frequencies Ω 1 , Ω 2 , Ω 3 , Ω 4  of lasers. The lights from all lasers are modulated by RF signals with one frequency ω for one level of RF. The RF signal is generated by the RF oscillator  413  and amplified by amplifier  415 . The RF splitter  438  splits the RF signal into four paths and sends the RF to each laser ( 231 ˜ 234 ), so that the light power or light frequency is modulated. The four-photon-detector receiver  331  converts the light power into RF currents (either coherent or non-coherent detection is used, but here using non-coherent as example). Each of the detectors has a different optical filer ( 726  in  FIG. 7 ) to allow only one of the four frequencies Ω 1 , Ω 2 , Ω 3 , Ω 4  to pass through. The currents are sent back to the four RF band pass filters  439  that allows RF frequency ω pass through. After amplified by amplifiers  440 , the phase differences of first two signals and the last two signals,  433  and  432 , are recovered by DPD  441  and  442 , respectively, and are sent to computer  900 . If the mixer is used at  441  and  442 , the filters  443  are needed, before the signals are sent to computer  900 , for filtering out higher frequency if the phase difference is used, or for filtering out the lower frequency if the phase sum is used. 
   In the cases of using millimeter microwave as the carrier wave, the same procedures for controlling and processing in the light-based systems described above and below are applicable, except for the generator, transmitter and receiver of carrier wave. 
   In the cases of using a mixer at the last step before the message goes into computer  900  (above and below), the output of mixer is not directly the phase difference or phase sum, but is the sinusoidal function of them. So the computer software converts the message into phase difference or phase sum for these cases. 
   The procedures of control and processing for another light-based system, but with two wavelengths, are shown in  FIG. 13 . The transmitters  243 ,  244  at A 1  and A 2  emit the same light wavelength (or frequency Ω 1 ), while transmitters  241 ,  242  at B 1  and B 2  emit the same frequency Ω 2  One of the two receivers,  341 , filters out the second light frequency and detects the two signals that are carried by the first frequency Ω 1  (from A 1  and A 2 ), and then sends the two detected RF signals to a RF band pass filter  448 . The two signals are internally homodyned at mixer  452  after the amplifier  450 . The output from the mixer  452 , after a low pass filter  458 , is a sinusoidal function of the phase difference, which is sent to computer  900 . The other receiver,  342 , filters out the first frequency and the sends the two detected RF signals (from B 1  and B 2 , and carried by the second frequency Ω 2 ) to a RF band pass filter  449 . The dish-line-framed part ( 446 ,  454 ,  455 ,  456 ,  457 ) is an option for using the phase sum. 
   The control and processing of another light-based system, with four wavelengths, is shown in  FIG. 14 . The difference from the system described in  FIG. 12  is that the transmitters and receivers are swapped. The four-light-source transmitter  341  is installed on the head holder  300 . Four corner receivers  241 ˜ 244  are used. 
     FIG. 15  is a schematic block diagram of the control and processing of another light-based system. All the procedures for this system are the same as that in the system described in  FIG. 14 , except that only two wavelengths or frequencies are used. 
   The system with its alternatives described in  FIGS. 9 to 15  is based on the phase measurement approaches, called phase-based system. The system can be also based on the measurement of time difference, called time-based system. The information carrier for the time-based system is, usually, ultrasonic wave, but it can be also any kind electromagnetic wave (light, RF or millimeter microwave) as long as we have fast-enough clocks in the future or for huge image applications. Here, the system is illustrated by an ultrasonic-based approach as shown in  FIG. 16  and  FIG. 17 . The clock  475  periodically sends commands (triggers) to the pulse generator  476 , which generates a pulse-modulated current with an ultrasonic frequency. After the current power is amplified at amplifier  477 , the current is sent to transmitter  371 . The ultrasonic pulse is transmitted out from transmitter  371  and is received by receivers  271  and  272 . In the meantime, the power amplifier  477  has also an output signal for the start trigger  478  to trigger the time counters  480  and  481 , so as to start time-counting at the moment the ultrasonic wave is sent out. After the receiver  271  and  272  receive the pulse, the signal is immediately (speed of electromagnetic field is far greater than the speed of sonic) amplified by the amplifiers  482 , and is sent to triggers  484  and  485  to stop the time-counting. Then the time counters  480 ,  481  send the time differences to computer  900 . The ultrasonic frequency filters  483  are used to distinguish the pulse from the other transmitters ( 384  in  FIG. 4 , or  388  in  FIG. 5 ) on head holder extension  303  in  FIG. 4 , because the two transmitters are driven by different ultrasonic frequencies. 
     FIG. 17  is used to show the control and processing of the time-based system with another ultrasonic-based approach. The difference from the system described in  FIG. 16  is that the transmitter and receiver are swapped. More clearly, the receiving CU  381  is on the head holder  300  rather than the transmitting CU on the head holder. Two ultrasonic pulse generators  488  and  489  are used to produce two driving currents with different frequencies. Therefore, the two ultrasonic pulses with different frequencies are transmitted from the transmitters  281  and  281 . The mixed signal from receiver  381  after amplified by amplifier  498  is split into two paths by splitter  495 . Each of the filters,  496 , or  947 , blocks out the other frequency and sends the pulse to triggers  484  and  485  to stop the time-counting. 
   The Doppler effect is an alternative used for detection of relative motion. Only two transmitting CU (transmitters) at the bottom corners (such as A 1 , A 2  in  FIG. 2(   d )) and two receiving CU (receivers on the head holder) as two locators are used. Instead of producing pulse-modulated ultrasonic wave or electromagnetic wave, the generators  488 ,  489  in  FIG. 17  generate an oscillation current with two frequencies a fair away from each other, and the transmitters  281 ,  282  in  FIG. 17  radiate CONTINUOS ultrasonic waves or electromagnetic wave. Receiver,  381  in  FIG. 17 , is replaced by a Doppler-Frequency-Detector. When receiver  381  is moving around in the two wave-fields, the Doppler frequencies, which carry the information of two velocity components along two directions, are detected. One direction is from one transmitter A 1  ( 281 ) to the receiver  381 ; the other direction is from the other transmitter A 2  ( 282 ) to the receiver  381 . So the angles of two directions are timely changing while receiver  381  is moving. The Doppler frequencies are sent to computer  900 . Computer  900  converts the two Doppler frequencies into velocity components and calculates the two displacement components of the receiver (i.e. locator) by integrating the velocity components. Then from the displacement components, the relative position of the locator is determined. 
   The other alternative positioning method for image reproduction and image recording system is to use any mouse-technique-based positioning method for determining the relative position of the locator. 
   &lt;Computer Processing&gt; 
   INTRODUCTION—Computer processing procedures are classified into two cases: using phase difference, or using phase sum. Also, there are two kinds of dependence of phase on the coordinates of the locator in the image area  10 . For the modulation-based systems described above, the dependence of phase on the coordinates is linear; while for the systems that directly use RF described above, the dependence is nonlinear due to phase nonlinearity of the near-field and the distortion from the boundary conditions. For the case of linear phase dependence and using phase difference, the contour curves for constant phase differences are a class of hyperbola curves, as show in  FIG. 18(   a ). While, for the case of linear phase dependence and using phase sum, the contour curves for constant phase sum are a class of ellipse curves, as show in  FIG. 18(   b ). All the hyperbolas or ellipses have the common foci at four CU&#39;s (A 1 , A 2 , B 1 , B 2 ), this is the general conclusion whenever the CU&#39;s are located at the corners or at the four middle edges. This invention provides the general theories, relations, and formulas for all cases: linear or nonlinear, phase difference or sum. This invention also provides the general calibration method for the case of distortion from the boundary conditions, or nonlinearity. Computer processing is based on these theories and formulas. 
   CALIBRATION and INITIALIZING (1)—The communication units,  381  in  FIG. 3 ,  383  and  384  in  FIG. 4 ,  387  and  388  in  FIG. 5 , are also called head locators as mentioned before. Usually there are two locators in the image reproduction and image recording system. The first locator together with the second locator is used for determining the head array position and direction, so that the position of each head in the head array can be determined by interpolation. For convenient in understanding the computer processing procedures of this invention, here we consider this situation first: using phase difference (rather than sum) and linear (rather than nonlinear) phase dependence, and for only one of the two locators. A schematic block of diagram with illustration is plotted in  FIG. 19 . The procedure starts with initialization, including calibration and initializing the status flag of the image pixels. First of all, check the status (is done or not) of calibration— 911 . If the calibration is not done, put the locator at (0,0), the center of the image area  10 , and read out the voltage (or current) from the receiver for phase differences (PD) (between A 2  and A 1  in a CU pair, and between B 2  and B 1  in other CU pair)— 912 . Usually, at this point, the PD is not zero. The zero-PD calibration at (0,0) can be achieved either by hardware (phase shifter) adjustments, or by computer processing.  FIG. 9  shows an example of phase shifter  431 . By adjusting the phase shifter, the PD at (0,0) can be reduced to zero. If by computer processing, these two non-zero PD&#39;s will be stored for later use— 913 . Next put the locator at any corner of the image area and read out the PD— 914 . Procedure  915  calculates the PD changes and distance differences (DD) when the locator moves from the center to the corner. The first DD is defined as the distance difference of two CU&#39;s (such as A 1 −A 2 ) in a CU pair from the head locator, i.e. r A1 −r A2 , and the second DD is defined as distance difference of two CU&#39;s (such as B 1 −B 2 )) in another CU pair from the head locator, i.e. r B1 −r B2 . Then the calibration coefficient is determined by the ratio of DD over the change of PD— 916 , which is the proportional coefficients between DD and the change of PD, and is used for converting the PD change to DD during the operation. The final step of initialization is to initialize flag of image status by setting all P(i)=0 (i denotes the i-th pixel)— 917 . The computer also figures out the scale transformation between the image area  10  and the image source stored in computer. According to the size of the image area  10 , the computer will produce a frame on the computer screen according to the scale, and the operator can move the frame on the screen to the source area that he will most likely to reproduce. The status of any pixel outside the frame is initially set to 1. However, it is initially set to 0 if the pixel is inside the frame. For any pixel of which the corresponding image has been reproduced on the image area  10 , the status P(i) will be changed to 1 from 0. If the status of a pixel is 1, the image of this pixel will not be reproduced again if head moves back to same place during the head arbitrarily moving. However, multiple reading from same pixel and overwriting the old reading doesn&#39;t matter. 
   CALIBRATION and INITIALIZING (2)—If calibration is done, then jump over the calibration block (the left dish-line frame) and wait for the commands (a trigger) for taking the phase information— 920  that is sent from the phase processor, such as the one  430  in  FIG. 9 . By using the PD at (0,0) and the calibration coefficient, the two DD&#39;s are determined— 921 . 
   COMMON PROCEDURES OF COMPUTER PROCESSING (1)—Procedure  922  and hereafter are the common procedures for different cases: linear or nonlinear phase dependencies, using phase difference, or phase sum, or time difference. Procedure  922  is to solve the roots of an equation that includes the DD data, to outputs the locator position (x, y). The equation is different for the different cases listed at the beginning of this paragraph. Procedure  923  takes image information of the pixel that is the nearest to the head, from the stored image data  924 . Then checks the status flag this pixel— 925 . If the pixel has been sprayed (P (i)=1), the next pixel is checked. If all the pixels have been sprayed (all P(i)=1), the job is down, and stop— 926 . If there is at least one pixel with status flag P (i)=0, then judge how close this pixel is to the head position (x, y)— 927 . If the distance is less than or equal to the criteria (1/20˜1/5 pixel of error is preferred), then procedure  928  takes the color data of this pixel from the image file  924 , and then sends the commands for spraying — 929 . Meanwhile, procedure  928  sets the status flag to 1 for this pixel. If the distance is greater than the criteria, then check the next pixel with status flag 0. If there is no such pixel that satisfies this condition at all, then system will wait for the next trigger for the next chance of meeting a pixel that is spray-able, during head arbitrarily moving— 930 . 
   COMMON PROCEDURES OF COMPUTER PROCESSING (2) As shown in the right dish-line frame, there are alternative procedures ( 932 ,  933 ,  934 ) for improving the efficiency. If there is no such a pixel at all, two fast-response actuators (not shown in Figures) are used to slightly adjust the position head. Procedure  933  finds the pixel in the image source which is the nearest to the head at the moment. The computer predicts how much the head array should be moved, by taking in the account the velocity and inertia of the head motion and the response time of the actuators-driven head, and then send commend to move and rotate the head array to right place — 934 . 
   POSITIONING OF EACH INDIVIDUAL HEAD IN ARRAY—For the case with two locators, the calibration described above is made for each of the two locators first — 935  and  936  in  FIG. 20 . After the two pairs of calibration coefficients are obtained for the two locators, the status flag of each image pixel in the image source is initialized to 0— 937 . Now the computer takes the phase information from the two locators— 938 , then the two pairs of DD (distance differences) for the two locators are calculated by using the calibration coefficients— 939 . In the same way as described for the one-locator (single head) case above, the position coordinates of the two locators, (x (1) , y (1) ) and (x (2) , y (2) ), are obtained— 940 , and the status flag of each pixel is checked— 941 ,  942 ,  943 . If all the pixels have been sprayed/read (all P (i)=1), stop— 944 . Otherwise, the program uses interpolation method to determine the position coordinates of each head along the head array— 946 : x(j)=x(j)=x (1) +D x ×(j−1), y(j)=y (1) +D y ×(j−1), D x =(x (2) −x (1) )/N, D y =(y (2) −y (1) /N. Here N is the total number of head along the head array, and j (=1, 2, . . . , N) denotes every head. Procedure  947  checks every head (i=1, 2, . . . , N) on the array—if the distance between any head and any pixel is less than the criteria? If yes, the computer takes the color data from that pixel and set status to 1— 948 , and then commands that head to spray or to read — 949 . Procedure  951  in the dish-line frame is an alternate for improving the efficiency, if there is no such pixel at all, or if there is only a few (too less) such pixels. In this case, three fast-response actuators are used to slightly adjust the head array position and direction, so that each head on the array can aim at a corresponding pixel. Two motors are installed at one end of the array [at the side of locator  383  ( FIG. 4 ) or  387  ( FIG. 5 )] for controlling array position, and the third motor is installed at the other end of the array [at the side of locator  384  ( FIG. 4 ),  388  ( FIG. 5 )] for controlling the array direction. The third motor drives the head array rotates about axle at the first head. Similar to procedure  933  in  FIG. 19 , the computer find out the pixel in image source which is the nearest to the position (x (1) , y (1) ) of locator  1  or the first head at that moment (locator  1  or the first head has a corresponding relation, not necessarily physically the same). The computer predicts how much the array should be moved so that the first head aims at this pixel, by taking in the account the velocity and inertia of head motion and the response time of actuators-driven head, and then move the array so that the first sprayer aims at that pixel. Meanwhile, the computer predicts how much the array should be rotated to make each of the head in head array aim at a corresponding pixel, by taking in the account the moving trend and inertia. Then the actuator rotates the array by the predicted angle and computer commands the head to spray or to read. 
   INVERTING THE LOCATOR&#39;S POSITIONS BY SOLVING EQUATIONS—For the modulation-based method, the phase has a linear dependence on the distance (r) between the receiver and the transmitter. For the given two pairs of detected phase difference (Δφ A =phase of A 2 −phase of A 1 , and Δφ B =phase of B 2 −phase of B 1 ), or phase sum (Σφ A  and Σφ B ), the position coordinates (x, y) of the locator are the roots of the equations 
                     (       x   ⁢           ⁢   cos   ⁢           ⁢     θ   1       +     y   ⁢           ⁢   sin   ⁢           ⁢     θ   1         )     2     /     a   1   2       -         (         -   x     ⁢           ⁢   sin   ⁢           ⁢     θ   1       +     y   ⁢           ⁢   cos   ⁢           ⁢     θ   1         )     2     /     b   1   2         =         1   ⁢           ⁢   and     -         (       x   ⁢           ⁢   cos   ⁢           ⁢     θ   2       +     y   ⁢           ⁢   sin   ⁢           ⁢     θ   2         )     2     /     b   2   2       +         (         -   x     ⁢           ⁢   sin   ⁢           ⁢     θ   2       +     y   ⁢           ⁢   cos   ⁢           ⁢     θ   2         )     2     /     a   2   2         =   1       ,       with   ⁢           ⁢     b   i       =           c   i   2     -     a   i   2         .             
When A 1 , A 2 , B 1 , B 2  are at the corners, θ 1 , is the angle between the line A 1 -A 2  and the right direction of the horizontal line and θ 2 =90−θ 1 . However, when A 1 , A 2 , B 1 , B 2  are at the middle edges, θ 1 =0 and θ 2 =0. For the phase difference approach, c 1 =D A2-A1  (distance between A 1  and A 2 , same meaning hereafter), c 2 =D B2-B1 , a 1 =c 1 −k A Δφ A , a 2 =c 2 −k B Δφ B . For the phase sum approach, c 1 =0.5D A2-A1 , c 2 =0.5D B2-B1 , a 1 =0.5k A Σφ A , a 2 =0.5k B Σφ B . For the phase difference approach, b i  is a pure real number, and the contour curves for constant phase differences are a class of hyperbola curves, and the right root-pair (x, y) is uniquely distinguished from the four pairs of the roots by checking the signs of the two phase-differences. Here is an example: consider the case A 1 −A 2  is vertical to B 1 −B 2 , as the cases shown in  FIGS. 2(   b ) and ( c ). The phase information (Δφ A &lt;0 and Δφ B &lt;0) corresponds to the solution with (x&gt;0, y&gt;0); (Δφ A &lt;0 and Δφ B &gt;0)   (x&gt;0, y&lt;0); (Δφ A &gt;0 and Δφ B &lt;0)   (x&lt;0, y&gt;0); and (Δφ A &gt;0 and Δφ B &gt;0)   (x&lt;0, y&lt;0). While, for the phase sum approach, b i  is a pure imaginary number, and the contour curves for constant phase sum are a class of ellipse curves, and the right root-pair (x, y) cannot be distinguished from the four pairs of the roots by using the phase information. In this case, the computer program sets the region ID (identification) for the four quarter-regions (left bottom=1, right bottom=2, left top=3, right top=4). The operator inputs the locator region ID from a keyboard when the locator begins to move. The computer then changes the region ID whenever the locator moves across the region boundaries. Therefore, the right root-pair (x, y) is distinguished from the region ID and the moving trend.
 
   INVERTING THE LOCATOR&#39;S POSITIONS BY SURFACE FITTING—The above procedures for modulation-based method are characterized by the linear dependencies of the phase. However, for the direct-RF system, radio frequency (RF) is directly (i.e. not used as modulation) used as the information carrier. The phase has a nonlinear dependence on distance (r) between the receiver and the transmitter due to the near field: φ(r)=kr−tan −1 [(k 2 r 2 −1)/(kr)], k is propagation constant of RF wave. The coordinates of locator can be determined by finding the minimum point of I(x, y)=[φ(r A2 )−φ(r A1 )−Δφ A ] 2 +[φ(r B2 )−φ(r B1 )−Δφ B ] 2 , or I(x, y)=[φ(r A2 )+φ(r A1 )−Σφ A ] 2 +[φ(r B2 )+φ(r B1 )−Σφ B ] 2 . Here (Δφ A  and Δφ B ), or (Σφ A  and Σφ B ), are the detected phase differences, or phase sums, respectively. The minimization for first step starts at the initial point that is defined by the roots of the linear equations from the linear limit (the larger r) of the phase dependence, and the minimization for later steps starts at the previous position of the locator. The boundary condition of the electromagnetic filed may introduce a discrepancy of the phase dependence used in the formula above, which is determined by the environment and cannot be predicted ahead. If the discrepancy is significant, a calibration method is employed. The calibration method is to mesh the image area  10 . Move the locator at each node on the mesh. The computer then records the phase difference and the coordinates of the node. Then the computer uses the surface functions to fit the coordinates versus the phase difference by using numerical methods (such as finite element method). By using these surface functions, the computer determines the coordinates from the phase difference when the locator moves to any position on the image area  10 . 
   PHASE-CURRENT PROCESSING (1)—For both cases of using digital phase detector (DPD) and mixer, the phase shifters built in the operation units are so adjusted that, for the zero phase (i.e. phase difference between two inputs of the DPD, or mixer), the output current is zero. The DPD outputs a linear current that is proportional to the phase (i.e. phase difference between the two inputs of the DPD) in the region (−2π, 2π). However, the curve is wrapped out of this region for every 2π of increase or decrease in phase, as shown in bottom of  FIG. 21 . The mixer outputs a current that is proportional to the sine function of the phase. So the monotonous region is (−π/2,π/2). Out of this monotonous region, there are the other monotonous regions for every π increase or decrease in phase. So usually, if the noise level is low enough, only the middle region is used for both cases. Therefore, the wavelength of the RF modulation or RF-carrier should be the maximum dimension of the image area  10  if DPD is used, or should be 4 times of the maximum dimension of the image area  10  if mixer is used. So, using DPD contrast using mixer, signal to noise ratio (SNR) is 4 times better under same noise level, that means, the resolution is 4 times better. The minimum (or best resolution) is determined by the noise level. 
   PHASE-CURRENT PROCESSING (2)—For higher resolution applications (i.e. using multi-level RF), if the noise level cannot be reduced, the current-phase unwrapping (for each level) needs special treatment. For the case of using sin DPD, as shown in the top of  FIG. 21 , the phase space is divided into (2M−1) 2  regions, with M=3 as an example. This leads to M times better resolution. Each region is assigned to an identification (ID) number (ij) (i, j=1, 2, . . . , 2M−1), (i=1, 2, . . . ) denotes the number of wrapped regions for the phase difference between B 1  and B 2 , and (j=1, 2, . . . ) denotes the number of wrapped regions for the phase difference between A 1  and A 2 . The computer&#39;s program always changes the ID number if the locator moves across the boundary and into a new region. Therefore, before locator starts moving at center region, the computer initializes ID at the center region, that is, set ID=33, for the case as shown in  FIG. 21 . Then the locator is moved to the position where the operator wants to start the work, and the computer follows the regions that the locator passes and promptly changes the ID numbers. Finally, for example, the locator moves to the region  51  through a path, and the computer follows the locator and finally changes the ID number to  51  starting from  33 . Let&#39;s distinguish phase-current and detected-phase-current. The detected-phase-current is the output of the DPD (the solid lines in bottom of  FIG. 21 ). Unlike detected-phase-current, the phase-current is the processed current after unwrapping and is scaled to phase (the dish lines in bottom of  FIG. 21 ), and the phase is the phase-difference used in the next computer processing as described above. For the non-center region, the phase-current should jump a value from the detected-phase-current. As shown in  FIG. 21 , for regions  34  and  35 , the phase-current for phase difference of A 2 −A 1  should shift to  959  and  960 , respectively, from the detected-phase-current ( 955  and  956 ). Or, in other words, in the regions  34  and  35 , the unwrapped phase is determined from phase-current, and the latter is obtained from detected-phase-current by adding 2π and 4π, respectively. The phase unwrapping process of current level is monitored by it&#39;s lower level. If using a very large M, this method can also serve as an alternate for relative-motion-based system that will be described later. 
   PHASE-CURRENT PROCESSING ( 3 )—For the case of using the mixer, the procedures are almost the same as DPD, except the region size (all π×π, rather than 2π×2π, 2π×4π, 4π×2π, and 4π×4π in the DPD case) and the sine dependence of the detected-phase-current on the phase (rather than linear dependence). Therefore, the detected-phase Δφ d  is determined by inverting the sine function from the detected-phase-current. The phase is transformed from Δφ d , such as π−Δφ d  and 2π+Δφ d  for the first right region and the second right region from the center region, respectively, for example. 
   COMPUTER PROCESSING OF TIME-BASED METHOD—For the time-based method (i.e., based on the time measurement), the computer receives two time differences, t A1  and t A2  from the OU  400 , which are the times for the pulse propagation from the CU at A 1  and A 2 , as shown in  FIG. 2(   d ), to the CU as head locator. Then the computer solves the root pair (x, y) from the equations (x−x A1 ) 2 +(y−y A1 ) 2 =(t A1 v) 2  and (x−x A2 ) 2 +(y−y A2 ) 2 =(t A2 v) 2 . Here v is the speed of the pulse propagation. If the origin of coordinate system is defined at the middle of bottom edge, so that y A1 =0, y A2 =0, and at the vertical central-line of the image area  10 , x=0, then the two pairs of roots with negative root of y coordinates are dropped. The positive root of y is used and negative sign for x is used if t A1 &lt;t A2 , and positive sign for x is used if t A1 &gt;t A2 . 
   COMPUTER PROCESSING OF OPTICAL IMAGE-MOTION-DETECTOR BASED APPROACH—For the relative-motion-based method, the head includes a motion detector (MD) and operation module (OM). The preferred apparatus for the MD is the optical image-motion-detector ( 340 ), as shown in  FIG. 8 . The camera pixel sensor array  346  converts the optical image into electrical signals, which are sent to the computer&#39;s memory for digital processing. The head starts moving at the center of the image area  10  after the initial setting of the reference point of the relative motion at this point. At this moment, a picture is taken—the middle panel that are shown in  FIG. 22  ( a ) represents the windows of image  964 . The position of the image  965  is defined as the left bottom corner of the window. At this moment, the image position is at  964 . At the moment of the next trigger, the head is moving to the position  966  marked by the filled circle. The picture-taken frequency should be high enough so that between the two neighboring pictures, the position of locator just changes a distance of a few pixels, even if with the fastest moving. Especially, if the head starts from static state, the position just changes a distance within one pixel. The computer starts the minimization of image-correlation at assumed positions, one of the positions is at  967  for example. The image-correlation is defined as the averaged summation (i.e. integral) of squares of difference (or absolute value of difference) of light intensity at pixels over the common image area (thin dash line) of the two pictures. One of pictures is the previous picture  964 , the other is preset picture  968  but at assumed position  967 . So if the image-correlation is smaller, the assumed position is closer to the actual position  966 . From  FIGS. 22  ( a ) and ( b ), the image-correlation for position  967  ( FIG. 22  ( a )) is larger than that for position  969  ( FIG. 22(   b )). The minimization of image-correlation leads to the maximization of convolution of two pictures, for the latter FFT can be applied. However, for small CCD pixel number, FFT has no benefit, so the following method will be used. 
   HEAD SPEED UP MOTION—For the cases that the head starts moving or restarts to move after the speed is reduces to zero, the computer will determine the relative position of this picture  971  to the previous picture  972  in  FIG. 22  ( c ). However, the computer does not know along which direction the head is moving. The computer calculates the image-correlation at five points (i.e. five assumed positions) and then uses a surface to fit five points of the correlation. Computer finds out the maximum point on the surface (or the minimum point of the ‘negative surface’), which is (or very close to) the actual position  971  of the image at the present moment. Among the five points, one point is called the surface-fitting center  972 , which is, at this moment, at the previous position. The other four points are at the four nearest corners (open circles in (c)) of the surface-fitting center. Hereafter, the term “frame” represents the quadrilateral frame, of which the four corners are at the four outer points with the center at the surface-fitting center, for five-point fitting. And it represents the pentagonal frame, of which the five corners are at the five outer points with the center at the surface-fitting center, for six-point fitting below. It would be lucky if the maximum correlation point is inside the frame (as shown in (c)). If the maximum point is inside the frame but too close to the boundary, one more point  973  on the lower side of the surface is needed, and computer redoes the surface fitting by six points, for better accuracy. For each new position at a new trigger moment, the first surface fitting is made by always using five points. The second-and-after surface fitting are made by always using six points. 
   HEAD SIMPLE MOTION—If the head is moving, the computer stores the history data of the head positions. From these data, the head movement trend (the velocity and acceleration) can be determined. Therefore, the position of next picture at the next triggered moment can be predicted at  974  (by extrapolation), as shown in  FIG. 22(   d ), although the actual position is at  975 . Then the computer finds the nearest pixel to the predicted position  974 , and uses this pixel as the surface-fitting center, and repeats the procedures described in the above section (with  FIG. 22(   c )). If the prediction is accurate enough (i.e. head motion is not complex), the actual position (that is the maximum point) of the picture at this moment should be inside the frame. So the same later procedures described in the above section (with  FIG. 22(   c )) are applied. Otherwise, the computer should finish the following procedures. 
   HEAD COMPLEX MOTION—The extrapolation-predictable motion is called simple motion, otherwise it is called complex motion. If head complex motion, the prediction is not efficient. Therefore, as shown in  FIG. 23 , the actual position (may at A or B) of the picture at this moment is out off the frame of which the center (surface-fitting center) is at  977 . The center  977  is the closest pixel to the predicted position  976 . This means that there is no maximum point in the frame center at  977 . Therefore need re-setting surface-fitting center the computer needs to compare the values of the correlation at the four corners, and picks out the point with lowest correlation value, V c  (i.e. the point  980  for the case shown in  FIG. 23(   a )), and picks up the values of the two neighbored corners, V 1  for  979  and V 2  for  981 . Then the computer defines the two variables: R 1 =min{|V 1 −V c |/V c ,|V 2 −V c |/V c } and R 2 =|V 1 −V 2 /V c , and sets the two criteria&#39;s CR 1  (say 0.5) and CR 2  (say 0.2) which need optimization. Here min{ } means taking the minimum value in the list. If R 1 &gt;RC 1 , R 2 &gt;RC 2  and V 1 &lt;V 2 , then use  983  (in  FIG. 23  ( a )) as the next surface-fitting center. If R 1 &gt;RC 1 , R 2 &gt;RC 2  and V 1 &gt;V 2 , then use  985  (in  FIG. 23  ( b )) as the next surface-fitting center. If R 1 &gt;RC 1 , R 2 ≦RC 2 , then use  987  (in  FIG. 23  ( d )) as the next surface-fitting center. If R 1 ≦RC 1  and V 1 &lt;V 2 , then use  988  (in  FIG. 23  ( c )) as the next surface-fitting center. If R 1 ≦RC 1  and V 1 &gt;V 2 , then use  989  (in  FIG. 23  ( c )) as the next surface-fitting center. This time, we don&#39;t need four points, but three or two points (thin open circles) around the new surface-fitting center will be added. The surface-fitting is carried out by six (rather than five) points, the center and the newly added points plus some old points. If using the old points [ 980  and  979  for the case in (a),  980  and  981  for the case in (b),  977 ,  980 , 979  for the case in (c), and  980 ,  979  or  981  for the case in (d)], the point  984  in (a) or  986  in (b) is not necessary. If a maximum point is found in the new frame, the actual position of the head at the present moment is obtained. Otherwise, the computer repeats the procedures until the actual position is found. 
   DOPPLER EFFECT METHOD—The Doppler effect of wave is used for positioning. Here use ultrasonic wave as an example. For the system based on ultrasonic Doppler effect, the generators  488 ,  489  in  FIG. 17  generate the oscillation current with two frequencies that are fair away from each other, and the transmitters  281 ,  282  in  FIG. 17  radiate continuous ultrasonic waves. Receiver  381  is replaced by a Doppler frequency detector. When the receiver  381  is moving around in the two ultrasonic fields, the Doppler frequencies are detected. Computer converts the two Doppler frequencies into the velocities (v 1  and v 2 ) that face to the two wave sources, respectively. Then the displacement of the head facing the two sources can be obtained by integrations 
               Δ   ⁢           ⁢     r   1       =         ∫   0     Δ   ⁢           ⁢   T       ⁢       v   1     ⁢           ⁢     ⅆ   t     ⁢           ⁢   and   ⁢           ⁢   Δ   ⁢           ⁢     r   2         =       ∫   0     Δ   ⁢           ⁢   T       ⁢       v   2     ⁢           ⁢     ⅆ   t             ⁢           ,         
respectively. Here ΔT is the time interval of the two neighboring triggers. If the head positions relative to the two sources at the moment of the latest previous trigger are {right arrow over (r)} 10  and {right arrow over (r)} 20 , respectively, the head displacement is
 
             Δ   ⁢           ⁢     r   →       =       Δ   ⁢           ⁢     r   1     ⁢       Δ   ⁢           ⁢       r   →     10         r   10         +     Δ   ⁢           ⁢     r   2     ⁢           ⁢         Δ   ⁢           ⁢       r   →     20         r   20       .               
Then the head positions relative to the two sources at the moment of present trigger can be written as {right arrow over (r)} 1 ={right arrow over (r)} 10 +Δ{right arrow over (r)} and {right arrow over (r)} 2 ={right arrow over (r)} 20 +Δ{right arrow over (r)}, respectively. Now the computer solves the root pair (x, y) from equations (x−x A1 ) 2 +(y−y A1 ) 2 =r 1   2  and (x−x A2 )+(y−y A2 ) 2 =r 2   2 . The next procedures are same as that in the time-based system described above.
 
   JUMP HAPPENS—In relative motion method, if a jump happens to the head carrier during its moving on the image surface due to some reason, the head needs to be put back to the nearest reference point that are previously set during the process, the most important reference point among the reference points is the center of the image area  10 . 
   RECORDING SYSTEM—For the recording system, the constitutions and procedures are the same, except that the sprayer array would be replaced the by reader array.