Abstract: A method and/or system for tracking objects, such as humans, over a wide area (that is, over an area that is delineated by a large spatial domain and/or a long-duration temporal domain) is provided. Such tracking is facilitated by processing, in real-time, near real-time or otherwise contemporaneous with receiving, images captured by each of a plurality or network of slightly overlapping stereo sensors, such as stereo cameras. The method includes and the apparatus is adapted for obtaining a plurality of local-track segments, wherein the plurality of local-track segments correspond to an object captured in images taken by a respective plurality of stereo sensors; and combining the local-track segments to form a global track.

Abstract: A method and an apparatus for calculating source image coordinates. In an embodiment, a variation rate of the source image coordinate with respect to a converted image coordinate is calculated by a recursive computation, and variation rates are accumulated to obtain the source image coordinate. Because the converted image coordinate sequentially increases one-by-one, the variation rate may be obtained by using adding operations of previously obtained source image coordinate, rather than multiplying operations with the all of the converted image coordinates. Therefore, the source image coordinates may simply and effectively be calculated.

Abstract: A system and method for a mosaic rendering of a 3D model is provided. Textures are created using polygon information related to a vertex of each face of an inputted 3D model, a normal, and a face index, and the textures are one-to-one mapped on respective polygons of the 3D model, using a geometrical structure of the 3D mode, thereby rendering a mosaic image showing a volume and a crinkled effect of paper.

Abstract: An automatic correcting device of an identification photo system comprises a skin pigmentation area abstracting device that abstracts a skin pigmentation area from an image of a person, a skin pigmentation correction value calculating device that calculates skin pigmentation correction values according to colors of the skin pigmentation area abstracted by the skin pigmentation area abstracting device and a predetermined skin pigmentation correction target value, and a color correcting device that corrects colors of the skin pigmentation area according to the skin pigmentation correction values calculated by the skin pigmentation correction value calculating device. Thus, the skin pigmentation of the person in the photo can be corrected to the person's true pigmentation.

Abstract: An image processing apparatus includes a geometric converter configured to perform a geometric conversion process to a first frame image among frame images that are sequentially generated through imaging, based on motion information obtained by using a motion detector sensor, a motion vector detector configured to detect a motion vector between a geometric conversion image obtained through the geometric conversion process and a second frame image that has not undergone the geometric conversion process, an image stabilization parameter calculator configured to calculate an image stabilization parameter for different motions by using the motion information and the motion vector, and a stabilized image generator configured to generate a stabilized image through a deformation process to the second frame image or a frame image generated subsequent to the second frame image by utilizing the image stabilization parameter.

Abstract: Embodiments of an apparatus for sample rate conversion are described. Various embodiments include an interpolator configured to interpolate a digital input data stream by using values of an interpolation phase shift control quantity to generate a digital output data stream, a computing stage configured to compute values of the interpolation phase shift control quantity, and an enabling/disabling stage configured to selectively disable the interpolator while keeping the computing stage enabled.

Abstract: An approach to resampling a first image that is sampled on a first sample grid. A filter for applying to the first image is computed. This involves computing a spatially quantized representation of the filter for which a degree of spatial quantization of the filter depends on factors including a measure of scale relating the first sample grid and a desired sample grid. The measure of scale can be computed based on a geometric transformation relating the first sample grid and the desired sample. The filter can be computed by selecting the number of spatial samples of the spatially quantized representation of the filter.

Abstract: A method for creating a 360 degree panoramic image from multiple images includes (1) computing a gross rotation error ?R between a first image and a calculated first image rotated to be stitched to a last image, and (2) spreading the gross rotation error ?R to each pixel on the panoramic image. Spreading the gross rotation error ?R includes (1) computing a rotation angle ?0 and rotational axis n0 from the gross rotational error ?R, (2) determining an angle ? of each pixel, and (3) determining a compensation matrix Rc for each pixel using the following formula: Rc(?)=R(?/2??0).

Abstract: A method of mapping geometrical objects to a digital image. At least one anchor point in a model is selected and mapped to the digital image. A geometric transformation is computed on the basis of at least one correspondence between an anchor object and a corresponding mapped object, and the calculated geometric transformation is applied to at least one geometric object thereby mapping it from the model image to a corresponding mapped position in the digital image.

Abstract: Systems and methods are disclosed that facilitate rapidly warping a two-dimensional image using integer math. A warping table can contain two-dimensional floating point output pixel offset values that are mapped to respective input pixel locations in a captured image. The warping table values can be pre-converted to integer offset values and integer grid values mapped to a sub-pixel grid. During warping, each output pixel can be looked up via its integer offset value, and a one-dimensional table lookup for each pixel can be performed to interpolate pixel data based at least in part on the integer grid value of the pixel. Due to the small size of the lookup tables, lookups can potentially be stored in and retrieved from a CPU cache, which stores most recent instructions to facilitate extremely rapid warping and fast table lookups.

Abstract: When a digitized broadcast is recorded in a storage medium, a required capacity of the storage medium is firstly calculated on the basis of a bit-rate of digitized broadcast data and a recording time which is known in advance. If the remaining capacity of the storage medium is smaller than the required capacity of the storage medium, for example, a frame rate decimation and pixel reduction are performed upon a broadcast to be newly stored, a re-encoding processing is performed for the broadcast and then the resultant broadcast is stored in the storage medium. If the remaining storage capacity of the storage medium is larger than the required capacity of the storage medium and thus remains, a frame interpolation and pixel interpolation are performed for the broadcast to be stored and then a re-encoding processing is performed for the resultant broadcast in order to improve its image quality. As a result, recording of the broadcast data in the storage medium can be ensured.

Abstract: The invention provides an image processing circuit having a capability of performing a reduction (resizing) process and an enlargement process on horizontal-scanning-line data inputted in synchronization with an input horizontal synchronization signal, and subsequently adjusting the horizontal synchronization signals so that the input horizontal-scanning-line data is made transferable to external devices in real time, and an image processing method therefor. A reducing unit thins out n-lines of the input horizontal synchronization signals out of m-lines of the input horizontal synchronization signals HD. When an enlarging unit enlarges the image data by an enlargement ratio k (k: natural number) in the vertical direction, the enlarging unit inserts (k?1) lines of the second horizontal synchronization signals for data transmission EHSYNC2 in the transmitting horizontal-synchronization-signal interval time TC1 between adjacent first horizontal synchronization signals for data.

Abstract: A projection apparatus includes a projection unit, an image processing unit, and a detection module. The projection unit receives an image output signal, and converts the image output signal into an image light beam for projecting a projection image. The detection module detects a placement orientation of the projection apparatus, and generates a placement signal according to the placement orientation detected thereby. The image processing unit is coupled to the projection unit and the detection module, and generates the image output signal to the projection unit. The image processing unit receives an image source signal, and adjusts the image source signal to result in the image output signal such that the projection image corresponding to the image output signal is rotated by a predetermined angle. The projection apparatus automatically detects the placement orientation and corrects the angle of the projection image, and user manual setting is no longer required.

Abstract: A special effect device by which a new special picture effect is to be implemented in accordance with a read address control system. The device includes an address signal generating unit which generates a readout address signal for picture signals stored in a frame buffer so tat the picture signals stored in the frame buffer will be output to each of a plurality of corresponding triangular areas of a preset size fractionated from the picture Signals stored in the frame buffer.

Abstract: The formatting of a video image frame (10) having a first aspect ratio to yield an image frame having a different aspect ratio is facilitated by establishing a pan and scan pixel coordinate (P) within the image that defines an image capture window for formatting purposes. A predictor (22) determines the pan and scan pixel coordinate by examining the image frame to determine the location therein of the most pertinent activity. The predictor (22) determines the location of the most pertinent activity by examining at least one of: video data, audio data, and other data such as closed captioning information.

Abstract: An image acquisition apparatus has an imaging optical system. An imaging section has a plurality of imaging pixels arranged at non-uniform intervals within an imaging surface. The imaging section picks up an object image imaged by the imaging optical system, and converts the object image into image signals. An image restoration processing section has a coefficient memory in which a predetermined coefficient sequence is recorded, and an image restoration processing circuit which carries out computation processing between the coefficient sequence recorded in the coefficient memory and the image signals from the imaging section. The image restoration processing section generates an image in a desired display pixel arrangement.

Abstract: A display control apparatus includes a display controller for receiving image signals corresponding to a plurality of images, to perform image processing on the image signals, and to synthesize images corresponding to the thus-processed image signals. While the images are displayed in a main picture and a sub-picture, the display controller further displays, adjacently to the sub-picture, a list of a plurality of images.

Abstract: The invention relates to a method for detecting an object (1) in relation to a surface (2). A synthetic image (3c, 4c) is produced from at least two distinct viewpoints (3, 4), for each of said viewpoints, by means of a geometrical transformation comprising at least one perspective projection (3a, 4a) of the object on the surface (2), each synthetic image being associated with the viewpoint concerned. By discriminating between the synthetic images, it is possible to detect the presence of and determine the position of the object (1) in relation to the surface (2), by searching for contrast inversions. The invention can be used for surveying a swimming pool.

Abstract: There is provided a multi-screen synthesis apparatus that can execute display of video data and update of an OSD image without causing a user to feel a visual sense of incongruity, and reduce system costs. Periodic video source data and a periodic OSD image data are written into a unified memory reserved for planes. The video source data and the OSD image data are read from the unified memory, based on a synthesis layout, for simultaneous display on a single display in a synthesized state. Video data to be written into the unified memory is decimated in units of a frame on an input video source-by-input video source basis. The decimation of the data is controlled based on display priority of the video data determined based on a multi-screen display layout.

Abstract: Apparatus for interactively generating a display signal. The apparatus comprises: a receiver (2) for receiving a broadcast signal (15), the broadcast signal comprising a plurality of datastreams each including image data defining a background object (22–270), and control parameters (28–39); a foreground computer generated imagery (CGI) device (3) for generating a foreground computer generated object (CGO); a mixer (4) for combining the foreground CGO with background object image data to generate the display signal; and interaction means (3) for receiving the control parameters from the receiver, monitoring the position of the foreground CGO, and adapting the display signal with reference to the monitored position of the foreground CGO and the received control parameters.

Abstract: An exemplary aspect of the invention can provide a method for easily adjusting the distortion of a projected image. In adjusting the amount of distortion correction, values of the amount of distortion correction for a desired outline form to be formed based on desired positions are calculated after the desired positions can be specified. Then whether all the calculated values of the amount of distortion correction are within a predetermined range where an original image is transformable into a distortion correction image can be judged. If all the calculated values of the amount of distortion correction are within the predetermined range, the calculated values of the amount of distortion correction can be set in an image distortion correction part.

Abstract: An enlarged image is displayed using a plurality of television sets. A television set serving as a master device and television sets serving as slave devices convert an input image into partial enlarged images and display the resultant partial enlarged images so that the partial enlarged images displayed on the respective television sets form, as a whole, a complete enlarged full image. The master device and slave devices perform mutual authentication with each other. If the authentication is successfully passed, the operation mode is set so that displaying of an enlarged image is allowed.

Abstract: There is described image-processing methods and image processing apparatus, which enable sharpness, scaling coefficients, and image quality to be adjusted with minimum image noise and minimum image quality deterioration and at high processing speed. One of the image-processing methods for creating processed image data by applying a spatial-filtering processing to source image data, comprises the steps of: setting a predetermined upper-limit value for a variation amount of the source image data, before performing an image-conversion processing through which the source image data are converted to the processed image data by applying the spatial-filtering processing; and performing the image-conversion processing for the source image data within a range of the variation amount limited by the predetermined upper-limit value.

Abstract: Disclosed is a computer-implemented method of enhancing a video image. A sequence of video frames is extracted. Each of the video frames is upsampled. The upsampled video frames are interpolated. The interpolated video frames are aligned and a single image is created from the aligned video frames.

Abstract: The invention relates to a method for changing the image size of video images, in which a decimation of video image signals (V) by an integral decimation factor (MHD, MVD) is carried out, and the decimated video image signals are subsequently read into an image memory for buffering. In order to achieve better possibilities of adjusting the image reduction with a relatively low outlay and high image quality, a fine decimation of the video image signals (V) is additionally carried out before buffering by a fine decimation factor (SHS, SVS) which can be adjusted to non-integral values, and a total decimation factor (MH, MV) relevant to the decimation of the video image signals (V) is formed from the integral decimation factor (MHD, MVD) and the fine decimation factor (SHS, SVS).

Abstract: An image processing apparatus and method for realizing panorama/waterglass functions, which are capable of reducing the size of a memory and simplifying memory control, are provided. In order to write a plurality of consecutive pixel data into memory devices, and simultaneously to read the plurality of consecutive pixel data from memory devices, the consecutive pixel data are separated into a plurality of first-in first-out (FIFO) buffers having a small size, thereby minimizing the size of a memory. Further, an identical clock signal is used, and the separated FIFOs are used when the consecutive pixel data are written to the memory devices and read from the memory devices, thereby simplifying memory control such as address control. Further, when realizing panorama/waterglass functions, the scaling ratio function is variably set, thereby variably realizing memories having sizes which vary according to the scaling ratio function.

Abstract: An image object content generation device has a key generation unit that generates a key that is a reference for distinguishing an object pixel from a background pixel per pixel of an inputted image signal, a frame size changing unit that changes a frame size of the image signal, and a low frequency component passing unit that removes high frequency components of the image signal which was processed in the frame size changing unit. The image object generation device also has a pixel compensation unit that performs a padding processing to the image signal which was processed in the low frequency component passing unit by using Low Pass Extrapolation padding algorithm, and a pixel selecting unit that selects one of the Image signal which was processed in the low frequency component passing unit and the image signal which was processed in the pixel compensation unit.

Abstract: Circuit arrangements and methods for real-time image resizing and image rotation. Line buffers are used for storage of lines of pixel values for both resizing and rotation. A first one of the line buffers receives input pixel values, and the line buffers are coupled in a chain such that line buffer i receives pixel values from line buffer i−1. The lines of pixel values are moved from line buffer i to line buffer i+1 as the pixel values are processed for resizing or rotation.

Abstract: A digital signal processing apparatus for parallel executing a plurality of data processes with a single common command is disclosed, that comprises a plurality of input storing means, each of which is composed of a plurality of storing elements, an input controlling means for controlling the input storing means, a calculating means, having a plurality of element calculating means corresponding to the plurality of the storing elements of the input storing means, for parallel calculating data stored in each storing element of the input storing means, a data storing means, having a plurality of storing elements corresponding to the plurality of element calculating means of the calculating means, for storing calculated result data of the element calculating means corresponding to the storing elements, a plurality of output storing means, each of which is composed of a plurality of storing elements corresponding to the plurality of element calculating means of the calculating means, for storing the calculated res

Abstract: A system, method, and computer program product for producing a show. In an embodiment, the invention is directed to a production system having a first production path, a second production path, and a control system that causes the first production path to generate a show in a first aspect ratio (4:3), and that causes the second production path to generate the same show in a second aspect ratio (16:9). In another embodiment, the invention is directed to producing a show from live material and from archived material. This aspect of the invention operates by producing a first show comprising a plurality of stories, segmenting the first show, and storing the show segments in an archive. Then, the invention produces a second show using live portions as well as show segments retrieved from the archive. The invention is also directed to a media manager that interacts with a server. In some cases, the server is integrated with the production system.

Abstract: The two dimensional sample rate conversion capabilities of a video display system are used to produce three-dimensional effects. Linear and non-linear scaling is applied to a video image to convey a sense of depth. The three dimensional effects are used to increase the visual appeal of existing and new feature sets in display systems. A multi-faceted object representation, such as a representation of a cube or a pyramid, can be used to display different video images on each facet of the representation. By appropriately scaling each image on each facet, an impression of depth is achieved. The images on the different facets can be selected to represent different aspects of a common theme, such as datacast information related to a primary source of information. Channel changing on a television can be presented as a rotation of the multifaceted object.

Abstract: The invention relates to a new method of reconstruction of tridimensional scenes. While conventional methods are often limited to the 3D reconstruction of the bounding volume of the concerned objects, the proposed method of recovery of a 3D geometric model from 2D views taken by one single camera, giving an information even about the parts which are hidden in each view, is implemented according to a first depth labeling step, implemented in a sub-system (11) and including initialization and projection sub-steps followed by a refinement process, and to a second reconstruction step, implemented in a sub-system (12). By means of a close cooperation of the 3D depth maps thus obtained for two views of a scene, a 3D model is identified and extracted.

Abstract: The technique of the present invention effectively relieves potential deterioration of the picture quality due to an interpolating process carried out simultaneously with a skipping process. When a skipping rate of the original image is set in at least one of a horizontal direction and a vertical direction, an image display apparatus of the present invention carries out a skipping process at selected positions in the at least one of the horizontal direction and the vertical direction, and simultaneously carries out interpolation on non-skipped pixels adjacent to skipped pixel, thereby generating an interpolated image. The image display apparatus subsequently carries out a filtering process on the interpolated image, the filtering process being carried out with a spatial low pass filter and a spatial high pass filter which are used substantially in this sequence.

Abstract: A display apparatus comprises driving a circuit having: a field discriminating circuit; a copy/thinning discriminating circuit; a pulse generating circuit for generating copy clock pulse signals in a horizontal period in addition to original clock pulse signals in an operation to display an enlarged picture; a gate-clock generating circuit for generating gate-clock signals obtained as a result of superposing all of the original clock pulse signals on some of the copy clock pulse signals the number of which is equal to a difference obtained as a result of subtraction of a vertical-pixel count of a video signal from a vertical-pixel count of a display unit; and a gate driver for generating a plurality of gate driving signals at a high level for pulses of the gate clock signals with timings different from each other.

Abstract: A method of stabilizing and registering a video image in multiple video fields of a video sequence provides accurate determination of the image change in magnification, rotation and translation between video fields, so that the video fields may be accurately corrected for these changes in the image in the video sequence. In a described embodiment, a key area of a key video field is selected which contains an image which it is desired to stabilize in a video sequence. The key area is subdivided into nested pixel blocks and the translation of each of the pixel blocks from the key video field to a new video field is determined as a precursor to determining change in magnification, rotation and translation of the image from the key video field to the new video field.

Abstract: Method of modifying data for obtaining a scalable video signal composed of a base video signal and a set of enhancement video signals from a non-scalable video signal, in a cost-effective manner.

Abstract: Real time effects are provided by using tracking information to define values for parameters of an effect. The tracking information also may be combined with parameters for various effects that operate on the position, rotation, scale, skew and perspective of objects in an image. The values defined for the parameters of an effect may be represented in a transformation matrix used in a real time digital video effects processor.

Abstract: Apparatus for automatic electronic replacement of a billboard in a video image including an automatic camera orientation measurement apparatus including motion measurement means operative to measure the Field of View (FOV) of the TV camera relative to a known reference position.

Abstract: Existing media signals are processed to create new media content by defining content representations for the existing media and establishing functional dependencies between the representations. The content representations comprise different data types which determine the kinds of operations that can be performed and dependencies that can be established. Among the types of transformation that can be achieved are synchronization, sound substitution, and the creation of parametric special effects. The content representations and their functional dependencies are combined to construct a functional dependency network which causes the desired transformations to occur on input media signals. The inputs to the functional dependency network are parametrically specified by media data types to construct a template that can be used to create adaptive media productions.

Abstract: In an electronic kaleidoscopic apparatus, a kaleidoscopic image forming unit extracts an initial image fragment of a specified region on the basis of image information picked up by a video camera. In response to the shape of the initial image fragment, the kaleidoscopic image forming unit regards the initial image fragment as an original image and repeatedly inverts/duplicates an image signal in this shape for every obtained reflection pattern, thereby forming a kaleidoscopic image and outputting the same to a display unit.

Abstract: A method and apparatus is provided to determine geometric distortion of an electronic imaging system, and an apparatus is provided to correct geometric distortion of the electronic imaging system and generate geometric distortion-corrected images. The novel geometric distortion measurement system comprises a light source, a collimator, a two dimensional steering device including a driver, and a control device for automatic sampling of the geometric distortion mapping of the electronic imaging system, and writes the results of distortion sampling into an associative memory. The sampling comprises multiple examples of the geometric distortion mapping across the image plane. Each example includes an ideal position of image pixel and its actual registration on the image plane. The measuring device take one example at a time until the whole sample is accomplished.

Abstract: Determining a phase shift compensation for automatically centering an image by measuring a start time delay between a horizontal scan line signal and an active video signal, measuring an end time delay between the active video signal and the horizontal scan line signal, and computing a phase shift for a horizontal flyback signal using the start time delay and the end time delay. A system to center an image on a monitor includes a video processing circuit and a rasterizer. The video processing circuit receives a video signal and measures a start time delay and an end time delay between a horizontal scan line signal and an active video signal. The circuit generates a phase shift used to generate a horizontal flyback signal using the measured start time delay and the end time delay. The rasterizer receives the output of the video processing circuit to control a scanning element.

Abstract: A device and method for generation of digital advertisement displays in real time in digital TV-transmissions. In such digital TV-transmissions, a digital coding of the image material is made before the transmission is made; this coding compresses the content of the image as much as possible to minimize necessary bandwidth utilization at transmission of the signals. In connection with this coding and by means of pattern recognition, different patterns are identified for predefined objects in the image material, such as for example identifying an object with an unique coloring. Via the pattern identification, the image coder obtains information via a table of what the original object looks like. Also included in the table is information about which advertisement display is to be placed inside the identified object.

Abstract: Apparatus for automatic electronic replacement of a billboard in a video image including an automatic camera orientation measurement apparatus including motion measurement means operative to measure the Field of View (FOV) of the TV camera relative to a known reference position.

Abstract: A pixel image at broadcast or graphics resolution is spatially transformed to represent the 2D projection of an image that has undergone movement in 3D space or is seen from a different viewer's position. For example, an image on a screen is delineated with four reference points designated by touching a pen on a tablet and then one is dragged with the pen along the screen to change the shape of the delineated image. At the same time, the entire delineated image is spatially transformed to make it continue to fill the entire changing shape and only that shape, until the dragged reference point is pinned to fix its position on the screen. Two or more designated point can be selected to move together as the pen drags them along the screen. The image being spatially transformed can be combined with a second image under the control of a stencil image to form a composite image.

Abstract: A two-dimensional spatial transformation system performs two-dimensional spatial transformation on an original video image in accordance with a two-pass method. Video data representing the original video image are input to a first video memory. Herein, a first pass transformation is performed on the original video image on the basis of transformation parameters used for a first pass, wherein the original video image is transformed to an intermediate video image with respect to a first direction (e.g., horizontal direction on a screen of a display). Video data representing the intermediate video image are transferred to a second video memory. A second pass transformation is performed on the intermediate video image on the basis of transformation parameters used for a second pass, wherein the intermediate video image is transformed to a transformed video image with respect to a second direction (e.g., vertical direction). The transformation parameters contain offset values and contraction rates.

Abstract: A video signal processing apparatus inputting a plurality of video signals, making the plurality of video signals in parallel in order to display them as one image and setting peripheral parts of images which are adjacent to each other when they are placed in parallel as overlapped areas, and being able to execute a necessary correction processing about the video signals corresponding to the overlapped areas comprising: a remote controller being able to set the correction starting position and the correction end position, a correction data memory storing the correction data corresponding to the correction starting position and the correction end position set by the remote controller, and a control portion reading out the predetermined correction data from the correction data memory based on the correction starting position and the correction end position in a case where the correction starting position and the correction end position have been set.

Abstract: In a point symmetry shaping apparatus used for a curved figure, any unrecorded symmetry figures are automatically shaped in a point symmetry manner within a short processing time. In the point symmetry shaping apparatus, a coordinate point which constitutes a symmetry center candidate point is calculated from all of center points of symmetry-judging-line-segments. A first point symmetry center candidate point/center point distance calculating unit calculates a distance between the symmetry center candidate point and the center point of the symmetry-judging-line-segment. A point symmetry judging unit judges a point symmetry of the input curved figure in such a manner that a calculation is made of a fluctuation value contained in the distances between the symmetry center candidate point and the center points of the symmetry-judging-line-segments, and the calculated fluctuation value is compared with a threshold value.

Abstract: A first channel selection signal generating circuit successively indicates selection of channels which can be received by the electronic tuner. A second channel selection signal generating circuit indicates selection of a particular channel. A channel selection signal switching circuit effects switching between these two outputs so as to output one of them to an electronic tuner. A digitizing circuit converts the output from the electronic tuner into its digital signal. A scan line converting circuit outputs a signal which has been formed from the digital signal by reducing the number of horizontal scan lines so that the output is stored into a texture memory, as one sub-frame image for each channel. A graphics engine subjects the image data loaded from the texture memory to a 3D image process and stores it into a video image memory.

Abstract: In an image transformation system, an image formed by the inputted video signal is stored in a memory to be read out by the read address of a predetermined read address generating circuit, so that a predetermined image transformation is performed to the image, by merging the mirror processing upon the orthogonal coordinates and the mirror processing upon the polar coordinates. As a result, a transformation image that looks as if it is seen through a kaleidoscope can be obtained.