Abstract: Methods and apparatuses are described for automated cross-browser user interface testing. A computing device captures (i) a first image file corresponding to a first current user interface view of a web application on a first testing platform and (ii) a second image file corresponding to a second current user interface view of a web application on a second testing platform. The computing device prepares the image files, and compares the prepared image files using a structural similarity index measure. The computing device determines that the prepared first image file and the prepared second image file represent a common user interface view when the structural similarity index measure is within a predetermined range. The computing device highlights corresponding regions that visually diverge from each other in each of the prepared image files and transmits a notification message comprising the highlighted image files.

Abstract: An electronic device includes a gamma data source, a signal receiving circuit, a buffer circuit, a counter, a multiplexer, and a gamma processing unit. The signal receiving circuit receives source data, and correspondingly generates a grayscale value. The buffer circuit electrically couples the signal receiving circuit and stores the grayscale value. The counter receives a system clock signal to generate a sequence number. The multiplexer electrically couples the counter and the gamma data source. The multiplexer receives the sequence number. The multiplexer outputs a bit message corresponding to the sequence number in the gamma data source. The gamma processing unit electrically couples the multiplexer and the buffer circuit. The gamma processing unit receives the bit message from the multiplexer. The gamma processing unit receives the grayscale value from the buffer circuit. The gamma processing unit outputs a bit value corresponding to the grayscale value in the bit message.

Abstract: An inspection method includes: a first acquisition step of acquiring information in which information on a color difference with respect to a reference color is assigned to each of a plurality of first divided regions into which an reference image is divided; a second acquisition step of acquiring information in which information on a color difference with respect to the reference color is assigned to each of a plurality of second divided regions into which a captured image is divided; a first comparison step of comparing the information on the color difference of each of the first divided regions with the information on the color difference of each of the corresponding second divided regions; and a second comparison step of comparing the information on the color difference of each of the first divided regions with the information on the color difference of each of the second divided regions in a combination different from the combination in which the comparison is performed in the first comparison step, and

Abstract: This disclosure describes various system and methods for monitoring photons emitted by a heat source of an additive manufacturing device. Sensor data recorded while monitoring the photons can be used to predict metallurgical, mechanical and geometrical properties of a part produced during an additive manufacturing operation. In some embodiments, a test pattern can be used to calibrate an additive manufacturing device.

Abstract: An image processing device obtains input image data representing an input gradation value for each of multiple pixels, and generates output image data representing a dot formation state for each of the multiple pixels by executing a halftone process with respect to the input image data. The halftone process includes a state selecting process selecting, from among Q+1 dot states, the dot formation state of a pixel of interest among the multiple pixels, the Q being an integer of 2 or more. The Q+1 dot states include a non-dot state Q with-dot states. The state selecting process includes a threshold determining process of determining Q threshold values to be associated with the Q with-dot states. The threshold determining process determines the Q threshold values using multiple parameters including the input gradation values of the pixel of interest and Q reference gradation values to be associated with the Q with-dot states.

Abstract: One embodiment provides a method comprising determining one or more coordinates representing one or more skin tones in a device-independent color space. The one or more skin tones can include one or more colors of human skin captured in one or more pieces of content. The method further comprises building a dual-core geometric model based on a distribution of the one or more skin tones in the device-independent color space. The method further comprises providing the dual-core geometric model to an electronic device for use in determining a likelihood that an input color of an input content is a skin tone. The input content is adaptively enhanced, based on the likelihood, for presentation on a display connected to the electronic device.

Abstract: An image reading system includes an image reading apparatus including a transport unit configured to transport a document from a bundle of cut documents one by one, a reading unit configured to read an image of the transported document, and a reading control unit, and an external control unit included in an external device coupled to the image reading apparatus, wherein an inclination correction process that corrects an inclination of the image of the document, a size determination process that determines a size of the image as a standard for the bundle of documents, and a size correction process that corrects the size of the image of the document to a specific size determined by the size determination process are performed.

Abstract: The invention relates to a deformable display, more in particular a flexible, stretchable, and transparent deformable display based on light-emitting elements such as for example light-emitting diodes (LEDs). The invention also relates to the use and applications of such deformable display, including systems and methods making use of such deformable display. In addition the invention relates to a flexible, stretchable and transparent display being deformable in real-time while maintaining deformability.

Abstract: A technique includes printing a first image including a plurality of edge pixels and a second image with a lower ratio of edges pixels than the first image and an average density equal to that of the first image and determining a value of a processing parameter so that the average densities of the first image and the second image match. The technique is configured to realize appropriate pre-print processing while preventing the average densities from being low due to the excessive pre-print processing or being high due to the insufficient pre-print processing.

Abstract: Certain aspects of the present disclosure provide a method for performing parallel data processing, including: receiving data for parallel processing from a data processing requestor; generating a plurality of data sub-blocks; determining a plurality of data portions in each data sub-block of the plurality of data sub-blocks; changing an order of the plurality of data portions in at least one data sub-block of the plurality of data sub-blocks; providing the plurality of data sub-blocks, including the at least one data sub-block comprising the changed order of the plurality of data portions, to a plurality of processing units for parallel processing; and receiving processed data associated with the plurality of data sub-blocks from the plurality of processing units.

Abstract: In this print method, assuming that the number of stacking of ink layers in a thickest portion in a thickly-piled portion is N, in a histogram preparation step ST2, a histogram of luminance values of grayscale image data prepared in an image data preparation step ST1 is prepared, and in luminance value range setting steps ST3 to ST10, a luminance value of predetermined gradations of the histogram is divided into N and N luminance value ranges are set. In the luminance value range setting steps ST3 to ST10, a division position of the luminance value of the predetermined gradations is adjusted while the histogram displayed on a predetermined display is checked.

Abstract: An air conditioner includes a camera to capture an image of a drain pan inside the air conditioner. An air-conditioning management device acquires, from the air conditioner, captured image information indicating an image of the drain pan captured by the air conditioner. The air-conditioning management device compares the captured image information with reference image information indicating a captured image of the drain pan used as a reference to calculate a dirtiness degree. The air-conditioning management device transmits, to a monitoring terminal, information useful for monitoring of the drain pan, such as the capture image information, information indicating the dirtiness degree, and the like.

Abstract: There is provided a video signal processing apparatus, including: a first interface configured to obtain a first video signal imaged within a first dynamic range; a second interface configured to obtain information for specifying a partial area of the obtained first video signal; and an arithmetic processing circuit configured to calculate a representative value of brightness of the first video signal of the partial area, to compress a dynamic range of the first video signal depending on the calculated representative value, and to generate a second video signal, a dynamic range of the second video signal being restricted to a second dynamic range narrower than the first dynamic range.

Abstract: A display device is disclosed. The display device comprises: a storage configured to store output brightness information for each gradation according to brightness information of an image; and a processor configured to acquire target brightness corresponding to brightness information of an input image on the basis of the information stored in the storage, acquire a target light amount on the basis of a light amount of the input image, acquire a plurality of correction effects corresponding to a plurality of correction images by applying a plurality of gradation adjustment curves to the input image, acquire a gradation adjustment curve corresponding to the maximum correction effect among the plurality of correction effects, and adjust and output a gradation for each pixel of the input image on the basis of the acquired gradation adjustment curve.

Abstract: An object of the present invention is to reduce computation for color conversion processing on input RGB color signals with maintaining the hue of a picture expressed by the input RGB color signals. A color conversion processor of the present invention includes: a conversion unit that converts color data including multiple color components, which can be displayed in a first color gamut, to corresponding color data in a second color gamut, which is narrower than the first color gamut; a first color correction unit that corrects the converted color data by using a first color correction value; a first derivation unit that derives a first luminance correction value by multiplication using the first color correction value; and a first luminance correction unit that corrects the color data, corrected by the first color correction unit, by using the first luminance correction value.

Abstract: A method and system for enhancing the visibility of screen images in a high ambient light condition or environment by processing the object images comprising a first stage to increase the brightness of transmission light from the back-light of an LCD screen and a second stage to further enhance the brightness as well as the contrast of the images to be displayed.

Abstract: A color detection system, comprising: a card having printed thereon a color chart comprising multiple colored areas, and at least one grayscale area; a camera configured to acquire an image comprising a pixel representation a sample positioned adjacent to the card; a storage device configured to store reference data corresponding to the color chart and the at least one grayscale area; and a processor configured to: perform a grayscale correction to the pixel representation of the color chart and sample using a grayscale correction transformation between the pixel representation the grayscale area and the corresponding reference data, estimate colorimetric coordinates for the sample by applying to the respective grayscale corrected pixel representation, a transformation between the grayscale corrected pixel representation of the color chart and the corresponding colorimetric reference data, and convert the estimated colorimetric coordinates for the sample to RGB.

Abstract: Apparatus and method for directly curing photopolymer printing plates, such as with UV radiation. Printing plates are cured directly by radiation, such as emitted from a high power UV laser beam. No LAMS layer or film bearing the image information is required on top of the polymer plate. The laser beam may be split into several individually-modulated beams by means of an Acousto Optical Deflector. Each individual beam is capable of curing pixels of the image that are to be transferred to the printing plate. Support shoulders for the printing details, formed by the pixels are determined by the caustic of the UV beam propagation.

Abstract: In a case where image correction is performed to deal with a registration shift and a color shift in an electrophotographic image forming apparatus, it is made possible to obtain a sufficient correction effect even in a high-quality mode. The image forming apparatus having a first printing mode and a second printing mode includes: at least one controller, having at least one processor which executes instructions stored in at least one memory, at least one circuitry or a combination of the at least one processor and the at least one circuitries, being configured to: perform scan line changing processing to shift an image in units of pixels in a sub scanning direction at a predetermined scan line changing point in a main scanning direction in accordance with characteristics of a laser scan line; and correct a tone level value of a pixel of interest in an image after the scan line changing processing by referring to a tone level value of a pixel adjacent to the pixel of interest in a sub scanning direction.

Abstract: A method, and document scanner system, for correcting image values recorded by an optical document scanner with multiple color cameras arranged in a staggered configuration with overlap scan areas includes storing color correction values mapped to respective color index values, where a color correction value is computed from a color difference between a first color value and a second color value of the respective overlap sets, and correcting a third color value which is a color value from the second set of color image values by a color correction value which is looked up via a color index value determined from the third color value. Thereby color variations among staggered cameras are corrected.

Abstract: A scanning device comprises a scan assembly, a control processor and an output interface. The scan assembly comprises two scanner modules for scanning front and reverse sides of a data medium to obtain front side analog data and reverse side analog data. The control processor electrically connected to the scan assembly controls the scan assembly, and executes an analog-to-digital converting process to convert the front side analog data and the reverse side analog data into front side digital data and reverse side digital data. The output interface electrically connected to the control processor outputs even sets of digital data, comprising the front side digital data and the reverse side digital data, to an image forming apparatus, so that the image forming apparatus performs even-set data typesetting and outputting after the even sets of digital data are completely outputted. An image forming system using the scanning device is also disclosed.

Abstract: In an image forming apparatus, a storage portion stores a target adhesion amount of toner to adhere to an image bearing member at each of gradation levels as a target gradation level and a setting value of a control parameter used for control on the image forming portion at the target gradation level. A first control processor makes an adhesion amount sensor measure an actual adhesion amount corresponding to the setting value and thereafter compares a resultant measured value and the target adhesion amount and corrects the setting value if needed in association with at least one of the target gradation levels as a subject. A second control processor makes the image forming portion output a calibration image used for correcting the target adhesion amount using a period when the first control processor performs processing if there arises a need to correct the target adhesion amount associated with the subject.

Abstract: An image reading apparatus is capable of reducing glare that a user feels while PWM-controlling an LED and of detecting an original size in a principal scanning direction with sufficient accuracy. A line sensor receives light emitted from a light source and reflected by an original. When a pressure plate that presses an original to a platen is closed, a control unit controls to supply a line periodic signal of which a period is longer than that for reading an original image to the line sensor, and to supply a lighting control signal of which a duty ratio is smaller than that for reading the original image to the light source in synchronization with the line periodic signal to light the light source and to read an image, and controls to detect length of the original in the principal scanning direction using the image signals from the line sensor.

Abstract: A color processing device includes a compositing unit, a modification unit, and an output unit. The compositing unit composites plural color conversion rules in a color space that includes plural color components through serial processing. The modification unit modifies the color conversion rules after being composited by the compositing unit, by applying an inverse conversion of a specific one-dimensional correlation, which correlates input color values and output color values in the color space, to the composited color conversion rules. The output unit outputs the composited color conversion rules after being modified and the specific one-dimensional correlation.

Abstract: Method comprising, for each pixel, obtaining a pixel expansion exponent value (E?(p)) by low pass filtering luminance values of colors of pixels in its spatial neighborhood, obtaining a pixel luminance-enhancement value (Yenhance(p)) by extraction of high frequencies of luminance values of colors of pixels in its spatial neighborhood, and then inverse tone mapping the luminance (Y(p)) according to the equation Yexp(p)=Y(p)E?(p)×[Yenhance(p)]c.

Abstract: In an example, a method of gamut mapping may include generating a first second-order or higher response-surface regression model that maps color values corresponding to the second color space to color values corresponding to the first color space. The method may include generating predicted color values for measured color values by inputting measured color values into the first second-order or higher response-surface regression model. The method may include generating, based on predicted color values and a plurality of color values, a second second-order or higher response-surface regression model that maps predicted color values output by the first second-order or higher response-surface regression model corresponding to the first color space to color values corresponding to the first color space.

Abstract: An image processing apparatus includes an image information acquisition unit acquiring average brightness of image data and cumulative frequency distribution of pixels related to gradation of the image data; a conversion range determination unit acquiring a conversion threshold value from the average brightness based on a previously set relationship and determining a gradation conversion range where the gradation of the image data is to be converted based on a gradation where a cumulative frequency corresponds to the conversion threshold value in the cumulative frequency distribution; and a gradation conversion unit converting the gradation of the image data in the gradation conversion range.

Abstract: Methods and apparatus for generating a binarization threshold are described. A set of input image pixels is processed. Edges including edge pixels are identified. In various embodiments histograms of pixel values and edge pixel values are generated and jointly used in determining an appropriate binarization threshold. In some embodiments a first histogram of edge pixel values is used to determine an interval of pixel values of interest to limit the set of pixel values used for determining a binarization threshold statistically. In some embodiments a first histogram corresponding to edge pixel values and a second histogram corresponding to image pixel values are generated, subjected to one or more smoothing operations, and truncated, based on information derived from the edge histogram. Characteristics of the histograms including, e.g., minima, maxima, are determined, evaluated, and used to generate a best global threshold.

Abstract: Aspects of the present invention are related to systems, methods and apparatus for boundary detection, in an image, by suppression of contours resulting from background textures and text in the image.

Abstract: An embodiment of an image enhancement method is introduced for use in an electronic device. An image composed of a plurality of pixel values is first received, and each pixel therein is separated into two channels. The image is further divided into a plurality of blocks. A distribution of a selected channel is determined for each block, which is selected from the two channels. For each pixel, the pixel value of the selected channel is adjusted according to at least one distribution corresponding to at least one neighboring block. For each pixel, the pixel value of the other channel is adjusted according to the adjusted pixel value of the selected channel. As a result, the adjusted image is provided.

Abstract: An image is processed in a device-independent color space by applying a modification function to one or more dimensions of that color space. The image is then converted into a target device-dependent color space. In order to minimize unwanted changes of color arising from the modification function taking image color values out of gamut when converted to the target device-dependent color space, the modification function is scaled to limit modified image values to within modification limits that are determined, for each image pixel or group of pixels, in dependence on the target device-dependent color space gamut boundary in the device-independent color space.

Abstract: An image reading apparatus capable of accurately detecting abnormal pixels caused by a contaminant such as dirt, stain, or flaw on a moving document reading glass being read. An image processing unit of the image reading apparatus performs a first abnormal pixel detection process to detect pixel signals representing abnormal pixels and output from image sensors for respective colors when a reading unit reads a guide plate in a main scanning direction in a state before an original passes through a moving document reading position, performs a second abnormal pixel detection process to detect pixel signals representing abnormal pixels and output from the image sensors when an original being conveyed is read by the reading unit in the main scanning direction, and detects a main scanning direction of abnormal pixels based on results of these detection processes.

Abstract: An image forming apparatus includes a base value setting unit and a correction value calculating unit. The base value setting unit sets a base value of an image process condition on the basis of the area ratio of an image to be formed. The correction value calculating unit calculates a correction value obtained by correcting the base value set by the base value setting unit on the basis of at least the pattern of the image. The image is formed based on the correction value calculated by the correction value calculating unit.

Abstract: A reading unit reads a first side of a printed material having first image data printed on the first side and second image data printed on a second side, the second side being a back side of the first side, in order to obtain image data. A density determination unit determines whether a density of the second image data exceeds a predetermined density. A comparison unit compares a density of the image data obtained by the reading unit and a density of the first image data, and produces a comparison result. A determination unit, in a case when a density difference between the density of the image data and the density of the first image data exceeds a threshold as the comparison result of the comparison unit, determines that the printed material includes a defect portion.

Abstract: An image forming apparatus capable of performing gradation correction without causing a user to perform a troublesome operation. The image forming apparatus includes an image reading unit that reads an original. A chart is formed on a recording material, and the recording material having the chart formed thereon is conveyed to the image reading unit via a shared conveying path used for both of reading the original and forming an image on the recording material. The conveyed recording material is read by the image reading unit, and gradation correction is performed based on the chart read by the image reading unit.

Abstract: An image processing device processes image data, which expresses an image and is edited to image data including, in correspondence with each pixel of the image, a pixel value field that holds a pixel value of the pixel and an attribute value field that holds an attribute value. The image processing device includes a first and a second image processing module, and a connection module that is connected to the first and second image processing modules, and moves the image data from one image processing module to the other image processing module. At least one of the image processing modules has a first function to output an image processing result for a pixel value of the input image data as an output pixel value, and a second function to output an image processing result for a pixel value of the input image data as an output attribute value.

Abstract: An image processing apparatus includes a color conversion processing unit, a modulation difference creating unit, a color value modulating unit, and an output unit. The color conversion processing unit performs color conversion from an input color value of a target image to a corresponding output color value to obtain a tentative output color value. The modulation difference creating unit creates modulation differences. The modulation differences include combinations that are obtained by either setting or not setting a value equivalent to the minimum resolution of the tentative output color value for each color component of the tentative output color value. The color value modulating unit performs modulation on the tentative output color value by adding or subtracting a modulation difference selected at random to or from the tentative output color value. The output unit outputs the target image according to the color value obtained after modulation.

Abstract: An image processing system comprising: a color conversion unit configured to perform color conversion from color data in a first color space into color data in a second color space by using a color profile; a correction unit configured to perform correction for the color data in the second color space by using correction data; a printing unit configured to print on a printing medium in accordance with the color data in the second color space that is corrected by the correction unit; a first changing unit configured to change the color profile; a second changing unit configured to change the correction data; and a selection unit configured to select, based on a color measurement result of a color chart for correcting a color of an image printed on the printing medium, one of processes including change of the color profile and change of the correction data.

Abstract: A telepresence device allows a remote meeting participant to establish a telepresence at a remote meeting location. The telepresence device provides a video image of the remote meeting participant on a video display screen at a meeting location To enhance the visual sensation that the remote person is present at the meeting location, the telepresence device may display the silhouette image of the remote meeting participant without the image of the background scene of the remote participant's location (e.g., office, home or home office). Further, the telepresence device may be equipped with a rear-facing camera that captures a video image of the background scene behind the telepresence device at the meeting location. The silhouette image of the remote meeting participant may be superimposed on the video image of the background scene at the meeting location, which is captured by the rear-facing camera.

Abstract: An image forming apparatus and a method of processing an image of the image forming apparatus are provided. The image forming apparatus includes a scanner to scan a document, a detector to detect text and image areas included in the scanned document, a conversion unit to convert the scanned document so as to output one of the detected text and image areas as color data and the other as monochrome data, and an output unit to output the converted document, wherein the conversion unit converts the scanned document to set a color value of the detected text area to a preset color value so as to output the detected text area as monochrome data or set a color value of the detected image area to a preset color value so as to output the detected image area as monochrome data.

Abstract: In an image processing apparatus and an image processing method according to the present invention, a gradation correction table creation unit prints stored chart data on a print paper by an image forming device, determines a target color from the color measuring result of the printed chart by a color measuring device, and creates a gray reproduction estimation model function according to the measured color result of the chart. Next, after the gradation correction table is created such that the gray of each gradation is adequately close to the target color, the gradation correction table is changed such that the color difference of neighboring gradations is included in a predetermined range.

Abstract: A control device performs: acquiring an optical measured scan value; setting a measured offset using the measured scan value by reference to predetermined correlations; and determining a control value using the measured offset amount. The control value is used for controlling an amount of colorant to be ejected during printing. In the predetermined correlations, a first pair of scan values is associated with a first pair of offsets. A second pair of scan values is associated with a second pair of offsets. The first and second pair has a first and second scan difference between the scan values, respectively. The first and second pair of offsets has a first and second offset difference between the offsets, respectively. The first scan difference is different from the second scan difference. The first offset difference is same as the second offset difference.

Abstract: A array including array elements in a number smaller than the minimum number of pixels, having values larger than Th_c, and a plurality of array elements having values smaller than Th_c is generated. A formation array having the same size as that of a latent image is generated from the generated array. An image obtained by replacing the pixel value at a pixel position within the first region in the latent image with the value at the pixel position in the formation array is generated as a configuration image. The pixel value at each pixel position in the configuration image, that corresponds to each pixel position in the input image, is output as the amount of a highly infrared absorbent color material used to print a pixel at this pixel position in the input image.

Abstract: In a printing apparatus which performs printing by using special ink for generating special glossiness on a printed material and color inks, the apparatus includes a controller that, when an image in which an index value indicating a degree of the special glossiness is low and an image in which the index value is high are to be printed adjacently, controls to print the image in which the index value is low by substituting other inks for the special ink.

Abstract: An image processing apparatus includes: a multi-level unit that applies a multi-level process on binary halftone dot image data indicating a binary halftone dot image, to form multi-level image data; an acquiring unit that acquires a number of screen lines of the binary halftone dot image data; a tone correcting unit that determines a tone correction amount corresponding to the number of screen lines of the binary halftone dot image data acquired by the acquiring unit, and that applies tone correction on the multi-level image data produced by the multi-level unit, by using the determined tone correction amount; and a re-binarizing unit that binarizes the multi-level image data in which the tone is corrected by the tone correcting unit, without changing a halftone dot shape of the binary halftone dot image data, to produce re-binarized halftone dot image data.

Abstract: A method includes dividing an input range of color values of a first color space into a plurality of sub-ranges, wherein at least one of the sub-ranges comprises an over-range color value. The method also includes determining for each sub-range a level of accuracy in converting color values within each sub-range. The method further includes determining a processing step to be applied to input color values in each sub-range based on the determined level of accuracy.

Abstract: An image processing device (100) includes a gradation correction value acquiring unit (12) that acquires a gradation correction value representing a ratio of a luminance component of an input image and a luminance component of an output image, a chroma analyzing unit (13) that calculates a chroma correction value, in which the total sum of degrees of chroma discrepancy between an analysis image equal to or different from the input image and a corrected image obtained by correcting a luminance component of the analysis image on the basis of one or more gradation correction values is the minimum, in correspondence with the gradation correction value, and an image output unit (14) that outputs as the output image an image obtained by correcting the input image received by the image input unit (11) on the basis of the gradation correction value acquired by the gradation correction value acquiring unit (12) and the chroma correction value correlated with the gradation correction value.

Abstract: An image forming apparatus for forming images at a plurality of resolution levels including at least one low resolution level and one high resolution level includes a photoconductor, onto which a beam size is set for the low resolution level; and an adjustment unit to conduct an exposure time-based density adjustment using a plurality of half-tone patterns prepared by changing an exposure time per pixel at a timing when a resolution level shifts from the low resolution level to the high resolution level and before actually shifting to an image forming operation executed at the high resolution level.

Abstract: An image reading apparatus is disclosed. The image reading apparatus includes an automatic document feeding unit that transports a document in a predetermined direction while the document floats above a platen glass member and a light source disposed below the platen glass member that irradiates a portion on the document at a predetermined reading location such that reflected light from the portion on the document is converted into image data and a floating distance between the document and the platen glass member is detected to set a value of an image parameter for correcting the image data for the document, based on the detected floating distance between the document and the platen glass member.

Abstract: An image processing apparatus that performs a color conversion of a image expressed by coordinates of a multidimensional color space, including, a storage unit for storing lattice point data in the multidimensional color space, an address generating unit for generating addresses for reading from the storage unit the lattice point data in a first region and a second region in accordance with input values of a color signal, the first region being a region where values of all color components are equal to or larger than a first threshold, the second region being different from the first region, and an interpolating unit for generating a converted color signal by an interpolating process based on the read lattice point data. A lattice-point interval in the first region is narrower than a lattice-point interval in the second region.