U ;cʌ@sddlZddlZddlZddlZddlmZddlm Z ddl m Z m Z m Z dZdZdZd_d d Zd`d d Zd dZdaddZddZdbddZdcddZddZddZddZddZddd!d"Zd#d$Zd%d&Zd'd(Zded)d*Z d+d,Z!d-d.Z"d/d0Z#d1ej$ej$gfd2d3Z%dfd5d6Z&dgd7d8Z'dhd9d:Z(didd?Z*d@dAZ+dBdCZ,dDdEZ-djdGdHZ.dIdJZ/dkdLdMZ0dldOdPZ1dmdQdRZ2dSdTZ3dUdVZ4dWdXZ5dYdZZ6dnd[d\Z7d]d^Z8dS)oN)ndimage)CRSProj TransformeriJxgMXAgHz'D@Tc Csd|kr$|dkr$d}|r$tdt|dt|d}}t|dt|d}}|||d}||d }|rtd |td |td ||dkrt|tjStj|||d } |dkrt| tjSt| |df} t| tjSdS)aCalculate slant range distance from input attribute dict Parameters: atr : dict, including the following ROI_PAC attributes: STARTING_RANGE RANGE_PIXEL_SIZE LENGTH WIDTH dimension : int, choices = [0,1,2] 2 for 2d matrix, vary in range direction, constant in az direction, for radar coord only 1 for 1d matrix, in range direction, for radar coord file 0 for center value Returns: np.array (0, 1 or 2 D) : range distance between antenna and ground target in meters Y_FIRSTrzGinput file is geocoded, return center range distance for the whole areaSTARTING_RANGERANGE_PIXEL_SIZELENGTHWIDTH@znear range : %.2f mzcenter range : %.2f mzfar range : %.2f m)numN) keysprintfloatintnparrayfloat32linspacetile) atr dimension print_msgrange_ndRlengthwidthrange_fZrange_cZrange_xZrange_xyr J/home/exouser/operations/rsmas_insar/sources/MintPy/mintpy/utils/utils0.pyrange_distance!s&    r"c Csd|kr$|dkr$d}|r$tdt|d}t|d}t|d}t|d}t|d}|||} tjt|d |d ||d d ||d tj} tjt|d | d ||d d || d tj} | | d } |r$td | td | td| |dkr4| } n|dkrVtj| | |dtj d} n|d krt|d}|dk rt |d dd}tjt||d |d ||d d |||d tj} n"t tj| | |dtj d|df} nt d|| S)aCalculate 2D matrix of incidence angle from ROI_PAC attributes, very accurate. Parameters: atr : dict - ROI_PAC attributes including the following items: STARTING_RANGE RANGE_PIXEL_SIZE EARTH_RADIUS HEIGHT WIDTH LENGTH #for dimension=2 dem : 2D array for height to calculate local incidence angle dimension : int, 2 for 2d matrix 1 for 1d array 0 for one center value print_msg : bool Returns: inc_angle : 2D np.array, incidence angle in degree for each pixel Example: dem = readfile.read('hgt.rdr')[0] atr = readfile.read_attribute('filt_fine.unw') inc_angle = ut.incidence_angle(atr, dem=dem) rrz9input file is geocoded, return center incident angle onlyrr EARTH_RADIUSHEIGHTr rf@r z&near incidence angle : {:.4f} degreez¢er incidence angle : {:.4f} degreez&far incidence angle : {:.4f} degreer FALSE)rendpointdtyper NFrrz un-supported dimension input: {}) rrrrrpiarccosformatrrr"r Exception)rdemrrrrrHrrZ inc_angle_nZ inc_angle_fZ inc_angle_c inc_angler range_distr r r!incidence_angleJsV      >>       &r3cCst|trt|}tj|tjddtj}t|d}t|d}||ttj|}t||}||}|t|}|S)aCalculate the corresponding slant range distance given an incidence angle Law of sines: r + H r range_dist --------------------- = ----------------- = ------------------ = 2R sin(pi - inc_angle) sin(look_angle) sin(range_angle) where range_angle = inc_angle - look_angle R is the radius of the circumcircle. link: http://www.ambrsoft.com/TrigoCalc/Triangle/BasicLaw/BasicTriangle.htm Parameters: atr - dict, metadata including the following items: EARTH_RADIUS HEIGHT inc_angle - float / np.ndarray, incidence angle in degree Returns: slant_range - float, slant range distance r(r#r$) isinstancestrrrrrr*sinarcsin)rr1r/r0ZR2Z look_angleZ range_angler2r r r!$incidence_angle2slant_range_distances   r:FcCsJd|krtddSt|d|d}t|dt|dtj}|S)zkGet range resolution on the ground in meters, from ROI_PAC attributes, for file in radar coord X_FIRSTz5Input file is in geo coord, no range resolution info.Nrr)r r%)rrr3rrr8r*)rrr1rg_stepr r r!range_ground_resolutions   r=cCsd|krtddSz |d}Wnd}YnX|dkrpt|d}t|d}t|d |||}n|d krt|d }|S) zmGet azimuth resolution on the ground in meters, from ROI_PAC attributes, for file in radar coord r;z7Input file is in geo coord, no azimuth resolution info.N PROCESSORisce)roipacr?r#r$AZIMUTH_PIXEL_SIZEgamma)rrr)rprocReheightaz_stepr r r!azimuth_ground_resolutions      rGc Cstttt|d}t|}t|}|tt||tt|}|tt||tt|}t ||||}||9}||9}|dkrd| krt|dt|dd}nd}t |t tt|}t |t d} | |fS)a}Get the default lat/lon step size for geocoding. Treat the pixel in radar coordinates as an rotated rectangle. Use the bounding box of the rotated rectangle for the ratio between lat and lon steps. Then scale the lat and lon step size to ensure the same area between the pixels in radar and geo coordinates. Link: https://math.stackexchange.com/questions/4001034 Parameters: atr - dict, standard mintpy metadata lat_c - float, central latitude in degree Returns: lat_step - float, latitude step size in degree lon_step - float, longitude step size in degree HEADINGNLAT_REF1LAT_REF3r r) rdeg2radabsheading2azimuth_anglerrGr=cosr8sqrtrrad2degr#) rZlat_caz_anglerFr<x_stepy_step scale_factorlon_steplat_stepr r r!auto_lat_lon_step_sizes$$ rXc Cs|sdSt|tr|g}dd|D}|D]>}tj|r*t|dt||td|W5QRXq*t|dkr~|d}|S)apython equivalent function to Unix utily - touch It sets the modification and access times of files to the current time of day. If the file doesn't exist, it is created with default permissions. Inputs/Output: fname_list - string / list of string NcSsg|]}|dk r|qSNr ).0xr r r! sztouch..aztouch r r) r6r7ospathisfileopenutimerlen)Z fname_listtimesfnamer r r!touchs     rfcCs6tdt|dd|ddkd}|d}|S)zConvert UTM Zone string to EPSG code. Parameters: utm_zone - str, atr['UTM_ZONE'] Returns: epsg - str, EPSG code Examples: epsg = utm_zone2epsg_code('11N') ZutmNS)Zprojzonesouthr )rZ from_dictrZ to_authority)Zutm_zoneZcrsZepsgr r r!utm_zone2epsg_codes   rkc Cstddlm}||}|}|s<|ddkr<||fSt|}td}t ||}| ||\}}||fS)aConvert x, y in the projection coordinates of the file to lon/lat in degree. Similar functionality also exists in utm.to_latlon() at: https://github.com/Turbo87/utm#utm-to-latitudelongitude Parameters: infile - str, GDAL supported file path x/y - scalar or 1/2D np.ndarray, coordiantes in x and y direction Returns: y/x - scalar or 1/2D np.ndarray, coordinates in latitutde and longitude r)gdalunitdegreez epsg:4326) osgeorlOpenZ GetSpatialRefZ IsProjected GetAttrValuer GetProjectionrZ from_proj transform) infiler[yrldssrsZp_inZp_outZ transformerr r r! to_latlon#s    rxcCs6d}d}|d}|d}dgd}ddtt|D}|dd|dd}||}d ||dd|} || |d } ||| d | d} d | t| d | d } | d | d | } t| d|| }|| || d |}t| ||d|}|t|||}t|d||d<t|d|d|d<||d t|d|dd||d<t|d|d<t|d|d<|d|d<|d}|d |d}ttt|d|d|d}t |d|dt |d|d}d t|}||fS)aCalculate satellite height and ellpsoid local radius from orbital state vector. This is a simplified version of the following functions from ISCE-2: + isce.isceobj.Planet.xyz_to_llh() + isce.isceobj.Ellipsoid.localRadius() Parameters: xyz - tuple of 3 float, orbital state vector Reference: height - float, satellite altitude in m radius - float, Earth radius in m g@TXAgk{?rrcSsg|]}dgdqS)rryr rZir r r!r\Tsz'xyz_to_local_radius..r g?g@g@r gUUUUUU??) rangercmathrPatan2degreesatantanradiansrOr8)Zxyzr]e2a2Ze4Zr_llhZd_llhxy2pqr/stuvwkDrEbZlatgargradiusr r r!xyz_to_local_radius@s:  . *,rcCsN|\}}}}|||||g}|||||g}ddddt||Dd}|S)zConvert the input bounding box in SNWE into WKT format POLYGON. Parameters: snwe - list of 4 float, south, north, west and east in degrees/meters Returns: polygon - str, WKT format POLYGON z POLYGON((,cSsg|]\}}|d|qS) r )rZlonlatr r r!r\zsz'snwe_to_wkt_polygon..z)))joinzip)ZsnwerhNWElatslonspolygonr r r!snwe_to_wkt_polygonqs  "rr c Cst|dt|d}}|dkr.dd||f}g}|dk r^t|d} t| }W5QRXd|krt|dV} | d|d|d|d|d f} | d |d|d|d|d f} W5QRXn$d |krt|d } t|d } t|d | |dd}t|d| |dd}|d|d}|d |d}|| |d}|| |d}|d krtj|||d|||df\} } n>|dkrtj|||dd} tj|||dd} nt d |nd}|d7}t |tj | tj d} tj | tj d} ||dkr| j dkrN| dd|} | dd|} n<| j d kr| dd|dd|f} | dd|dd|f} | | fS)aExtract precise pixel-wise lat/lon. For meta dict in geo-coordinates OR geom_file with latitude/longitude dataset Returned lat/lon are corresponds to the pixel center Parameters: meta - dict, including LENGTH, WIDTH and Y/X_FIRST/STEP box - 4-tuple of int for (x0, y0, x1, y1) dimension - int, output lat/lon matrix dimension, 1 or 2 y/xstep - int, number of pixels to skip for each output pixel Returns: lats - 1/2D np.array for latitude in size of (length, _width_) lons - 1/2D np.array for longitude in size of (_length_, width) r r Nrr/latituder ryr longituderY_STEPX_STEPr|r;y?T)rr'zun-supported dimension = {}zCan not get pixel-wise lat/lon!z] meta dict is not geocoded and/or geometry file does not contains latitude/longitude dataset.r4)rh5pyFilelistrrrmgridr ValueErrorr,rrndim)metaZ geom_fileboxrystepxsteprrds_listfrrrWrVZlat0Zlon0Zlat_numZlon_numlat1Zlon1msgr r r! get_lat_lon~sV (6      rc sdkrtdtdtd}}fdddD}fdddD}tj||ftjd }tj||ftjd }t|D]}t|D]}|d ||d |d |||d |d ||||f<|d ||d |d |||d |d ||||f<qq||fS) abGet 2D array of lat and lon for metadata dict in radar-coord. WARNING: This is a rough lat/lon value, NOT accurate! Parameters: meta : dict, including LENGTH, WIDTH and LAT/LON_REF1/2/3/4 Returns: lats : 2D np.array for latitude in size of (length, width) lons : 2D np.array for longitude in size of (length, width) rzJInput file is in geo-coordinates, use more accurate get_lat_lon() instead.r r csg|]}td|qS)z LAT_REF{}rr,rzrr r!r\sz#get_lat_lon_rdc..)r rrycsg|]}td|qS)z LON_REF{}rrzrr r!r\sr4rr r)rr-rrzerosrr}) rrrrrrrr{jr rr!get_lat_lon_rdcs   @DrcCs"d|}|t|dd8}|S)aConvert azimuth angle from ISCE los.rdr band2 into satellite orbit heading angle ISCE-2 los.* file band2 is azimuth angle of LOS vector from ground target to the satellite measured from the north in anti-clockwise as positive Below are typical values in deg for satellites with near-polar orbit: ascending orbit: heading angle of -12 and azimuth angle of 102 descending orbit: heading angle of -168 and azimuth angle of -102 Zv@rround)rR head_angler r r!azimuth2heading_angles rcCs"d|}|t|dd8}|S)zNConvert satellite orbit heading angle into azimuth angle as defined in ISCE-2.rrr)rrRr r r!rNsrNcCs|dkr"|dk rt|}ntd|tt|tt|d|tt|tt||tt|}|S)aProject east/north/up motion into the line-of-sight direction defined by incidence/azimuth angle. Parameters: e - np.ndarray or float, displacement in east-west direction, east as positive n - np.ndarray or float, displacement in north-south direction, north as positive u - np.ndarray or float, displacement in vertical direction, up as positive inc_angle - np.ndarray or float, local incidence angle from vertical, in the unit of degrees head_angle - np.ndarray or float, azimuth angle of the SAR platform along track direction measured from the north with clockwise direction as positive, in the unit of degrees az_angle - np.ndarray or float, azimuth angle of the LOS vector from the ground to the SAR platform measured from the north with anti-clockwise direction as positive, in the unit of degrees head_angle = 90 - az_angle Returns: v_los - np.ndarray or float, displacement in line-of-sight direction, moving toward satellite as positive Typical values in deg for satellites with near-polar orbit: For AlosA: inc_angle = 34, head_angle = -12.9, az_angle = 102.9 For AlosD: inc_angle = 34, head_angle = -167.2, az_angle = -102.8 For SenD: inc_angle = 34, head_angle = -168.0, az_angle = -102.0 Nzaz_angle can not be None!rg)rNrrr8rLrO)enrr1rrRZv_losr r r!enu2loss &"rc Cslt|d}t|d}t|d}t|d}t|d}t|d}|||}|||}||||fS)a Get the 4 corners coordinates from metadata dict in geo-coordinates. Parameters: atr - dict Returns: south, north, west, east - float, in degrees or meters Examples: S, N, W, E = ut.four_corners(atr) SNWE = ut.four_corners(atr) r r rrr;r)rr) rrrrVrWwestnorthrjeastr r r! four_cornerss        rc CsJ|\}}tj| ||| ||f\}}|d|d|dk}|S)z5Get mask of pixels within circle defined by (x, y, r)r)rogrid) r[rurshaperryyxxcmaskr r r!get_circular_mask-s rc Cst|d}t|d}t|tr(|}n t|tr8|}n|dd}dd|D}z>t|d}t|d}tt|d }td |||fWnzt|d}t|d} tt|d }t |t|d t|d }t | t|d t|d}td || |Wn4td|tdtdtdYYdSXYnXtj | ||| ||f\} } | d | d |d k} | S)aReturn Index of Elements within a Circle centered at input pixel Parameters: atr : dictionary containging the following attributes: WIDT LENGTH circle_par : string in the format of 'y,x,radius' i.e. '200,300,20' for radar coord '31.0214,130.5699,20' for geo coord Returns: idx : 2D np.array in bool type mask matrix for those pixel falling into the circle defined by circle_par Examples: idx_mat = ut.circle_index(atr, '200,300,20') idx_mat = ut.circle_index(atr, '31.0214,130.5699,20') r r rrcSsg|] }t|qSr )r7rzr r r!r\Osz circle_index..rr rz+Input circle index in y/x coord: %d, %d, %drrr;rz7Input circle index in lat/lon coord: {:.4f}, {:.4f}, {}z* ERROR: Unrecognized circle index format: zSupported format:z4--circle 200,300,20 for radar coord inputz5--circle 31.0214,130.5699,20 for geo coord input ) rr6tuplerreplacesplitrrrrintr,r) rZ circle_parrrZcir_parZc_yZc_xrZc_latZc_lonrur[idxr r r! circle_index6sD        ""  &rrKcCs.|\}}t|}|t||||}|S)zWrap data into a range. Parameters: data_in : np.array, array to be wrapped wrap_range : list of 2 float, range to be wrapped into Returns: data : np.array, data after wrapping )rrmod)Zdata_inZ wrap_rangeZw0Zw1datar r r!wrapns r@cCsJt|}g}t|dd}t|sF||||N}t|dd}q|S)aHGet all connected component with number of pixels larger than threshold Parameters: mask_in : 2D np.array with zero as background and non-zero as foreground min_num_pixel : int/float, minimum number of pixels to be identified and output Returns: mask_out : list of 2D np.array in np.bool_ format r) min_num_pixel)rrget_largest_conn_componentallappend)mask_inrmask_outZmask_ccr r r!get_all_conn_componentszs    rc Cst|jtj}t|d}tt|dd}||krF|St t|ddd}||k}|rt j ddddgd\}}|d ||d ||d ||At |S) aExtract the largest connected component from an 2D array with zero as background value Parameters: mask_in : 2D np.array with zero as background and non-zero as foreground min_num_pixel : int/float, minimum number of pixels to be identified and output display : bool, display the result or not. Returns: mask_out : 2D np.array in np.bool_ format rr Nry)nrowsncolsZfigsizer)rrrbool_rlabelmaxbincountflattenargmaxpltZsubplotsimshowshow) rrdisplayrlabels num_pixelZ max_labelZfigaxr r r!rs rcCsddlm}t|dk\}}t|dd|ddf}||}t|dk\}}t|dd|ddf}||\} } t| } || } || | } | | }|rt t |d|dt | d| dg| d| dgdt | | |fS)atCalculate the min distance between two regions of pixels marked by mask1 and mask2 Parameters: mask1/2 : 2D np.array in size of (length, width) in np.bool_ format Returns: pts1 : tuple of 2 int, bridge point in mask1, in (x, y) pts2 : tuple of 2 int, bridge point in mask2, in (x, y) min_dist : float, min euclidean distance r)cKDTreergr rz-o) Z scipy.spatialrrwherehstackreshapequeryargminrZfigurerZplotr)Zmask1Zmask2rrrur[Zpts1treeZpts2distrZidx_minrZxy1min_distr r r!min_region_distances"   &rlinearc Csddlm}|j}t|dt|df}ttjd|dd|dddtjd|dd|ddd\}}t|dd|ddf}||||dd||} | S)aaInterpolate input 2D matrix into different shape. Used to get full resolution perp baseline from ISCE coarse grid baseline file. Parameters: inData : 2D array outShape : tuple of 2 int in (length, width) interpMethod : string, choose in [nearest, linear, cubic] Returns: outData : 2D array in outShape r)RegularGridInterpolatorr F)r'rg)methodZ bounds_error) Zscipy.interpolaterrrarangemeshgridrrr) ZinDataZoutShapeZ interpMethodZRGIZinShapeZinPtsrrZoutPtsZoutDatar r r!interpolate_datas  rcCsTddlm}m}|\}}|d||fd}||j|dddtj|tjd}|S)arCreate a 2D mask (numpy array in binary) from a polygon. Link: https://stackoverflow.com/questions/3654289/scipy-create-2d-polygon-mask Parameters: polygon - list of tuples of 2 int, e.g. [(x1, y1), (x2, y2), ...] shape - list/tuple of 2 int, for length and width Returns: mask - 2D np.ndarray in bool in size of (length, width) r)Image ImageDrawLr )outlinefillr4) ZPILrrnewZDrawrrrr)rrrrrrimgmaskr r r! polygon2masks rcCsDtt|d}|ddkr(dS|ddkr8dStdSdS) zHgarrettdreyfus on Github: https://gist.github.com/garrettdreyfus/8153571z (y/n): rruTrFzUhhhh... please enter N)r7inputlowerstrip yes_or_no)ZquestionZreplyr r r!r s   r cCsdd}|D]R}t||}||kr@|t||kr@|dksT||krZ|dkrZtq d}q |S)z)Compare new attributes with exsiting onesFNoneT)rr7next)Zatr_newZatr_origupdatekeyvaluer r r!update_attribute_or_nots  $ rcCsjdd}tj|dr&||rf|Sn@tjdtjD],}|d}tj||}||r8|Sq8dS)zTest if executable existscSstj|ot|tjSrY)r^r_r`accessX_OK)fpathr r r!is_exeszwhich..is_exerPATH"N)r^r_renvironpathsepr r)programrr_Zexe_filer r r!whichs  rcCsd}|dkr(d}|rtdd|ddfSzddlm}m}Wn,tdtd d}d|ddfYSXtt||}|dkrd}td td tn|rtd |z||||fWS||ddfYSXdS) aECheck parallel option based file num and installed module Link: https://joblib.readthedocs.io/en/latest/parallel.html Examples: num_cores, inps.parallel, Parallel, delayed = ut.check_parallel(len(file_list)) Parallel(n_jobs=num_cores)(delayed(subset_file)(file, vars(inps)) for file in file_list) Tr Fz2parallel processing is disabled for one input fileNr)ParalleldelayedzCan not import joblibzparallel is disabled.zNparallel processing is disabled because min of the following two numbers <= 1:z(available cpu number of the computer: {}z¶llel processing using %d cores ...)rZjoblibrrminr^ cpu_countr,)Zfile_numrZmaxParallelNumZenable_parallelrrZ num_coresr r r!check_parallels0  r!cCsL|}|D]6}|ds0d|kr0|d|d7}q|d|7}qt|dS)zprint the command line with "" for arguments containing *. Parameters: script_name - str, e.g. prep_isce.py args - list(str), list of input arguments -*z "rrN) startswithr)Z script_nameargscmdrr r r!print_command_line9sr'LTo l?cCst|}|jdkr*|jd}||d}|dkr@tj|dd}|jdkrht|ddd|jdf}tjt||dd|}|S)aCompute the median absolute deviation of the data along the LAST axis. This function is also included as: scipy.stats.median_abs_deviation() in scipy v1.5.0 https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.median_abs_deviation.html statsmodels.robust.mad() in statsmodels https://www.statsmodels.org/dev/generated/statsmodels.robust.scale.mad.html The differences are: 1. scipy: out = out / scale while mintpy: out = out * scale 2. scipy propagates nan by default, while mintpy omit nan by default. The following two are equivalent for a 2D np.ndarray X: mintpy.utils.utils0.median_abs_deviation(X) scipy.stats.median_abs_deviation(X, axis=-1, center=np.nanmedian, scale=1./0.67449, nan_policy='omit') Parameters: data - 1/2D np.ndarray, input array center - 0/1D np.ndarray or None scale - float, the normalization constant Returns: mad - 0/1D np.ndarray rrrgN)axisr )rrrrr nanmedianrrM)rcenterscaleZntimemadr r r!median_abs_deviationMs     r.@cCs.|dkrt|}t||d}|||}|S)zvcalculate rms_threshold based on the standardised residual Outlier detection with median absolute deviation. N)r+)rr*r.)rr+cutoffr- thresholdr r r!median_abs_deviation_thresholdts    r2cCs~t|}|dk rLt|}|j|jkrDtd|j|j||8}|t|}tt|d|jd}|S)z5Calculate the root-mean-square error between x and y.Nz+Input x & y have different size: {} vs. {}!rr ) rrrsizerr,isnanrPsum)r[ruZrmser r r!root_mean_sq_errors r6cCsBtttt|}t|| }||kr2|}n |d|}|S)z@Return the most significant digit of input number and ceiling it rrfloorlog10rMr)r[digitZx_roundZx_ceilr r r! ceil_to_1s   r<cCs$tttt|}t|| S)z1Return the most significant digit of input numberr8)r[r;r r r! round_to_1sr=cCs"t|ddd}|tjS)zfRound a float up to the next odd integer . Link: https://stackoverflow.com/questions/31648729 rr )rceilastypeint16)r[rur r r!round_up_to_oddsrAcCs&tdtt|}t|}|S)z7Given a number x, find the highest power of 2 that <= xr)rpowerr9log2r@)r[resr r r!highest_power_of_2s rEcCs<ddlm}||}dd||D}|dkr8|d}|S)zReturn the k most common item in the list L. Examples: 5, 8 = most_common([4,5,5,5,5,8,8,8,9], k=2) 'duck' = most_common(['goose','duck','duck','dog']) 'goose' = most_common(['goose','duck','duck','goose']) r)CountercSsg|] }|dqS)rr rzr r r!r\szmost_common..r ) collectionsrF most_common)rrrFcntZitem_mmr r r!rHs  rHcCs*zt|WdStk r$YdSXdS)a Check string is a number. Not using str.isnumeric() because it can not handle floating point nor negative sign. Link: https://elearning.wsldp.com/python3/python-check-string-is-a-number/ Parameters: string - str, a string Returns: True/False TFN)rr)stringr r r! is_numbers rK)rT)NrT)F)N)N)NNrr r )NN)r)rF)F)r)r Tr)Nr()Nr/)N)r )9r^r~rnumpyrZmatplotlib.pyplotZpyplotrscipyrZpyprojrrrSPEED_OF_LIGHTr#Kr"r3r:r=rGrXrfrkrxrrrrrrNrrrrr*rrrrrrr rrr!r'r.r2r6r<r=rArErHrKr r r r!s`   ) B#  + 1 J # 8       ( '