Introduction to Multivariate Image Analysis (MIA) Table of Contents

Transcription

1 Introduction to Multivariate Image Analysis (MIA) Copyright Eigenvector Research, Inc. No part of this material may be photocopied or reproduced in any form without prior written consent from Eigenvector Research, Inc. Table of Contents Intro to 3-way arrays and simple visualizations and size/shape analyses Practical Multivariate Image Analysis (MIA) Principal Component Analysis (PCA) SIMCA Multivariate Curve Resolution (MCR) Partial Least Squares Discriminant Analysis (PLSDA) 2

2 Eigenvector University Our week long series of courses each spring May 10-15, 2015 Seattle, Washington, USA 6 full days and 3 evenings 15 hands-on courses 8 instructors (Eigenvector staff + Rasmus Bro) User poster session and group meeting EigenU Europe Oct 5-8, 2015 Hillerød, DENMARK 4 full days 7 hands-on courses Training We offer standard and custom courses on a range of chemometric and application topics: Chemometrics Without Equations Series Chemometrics Without Equations Advanced Chemometrics Without Equations Basic Chemometrics Series Linear Algebra for Chemometricians MATLAB for Chemometricians Chemometrics I -- PCA Chemometrics II -- Regression and PLS Clustering and Classification Advanced and Specialty Topics Advanced Preprocessing Applied Multiway Analysis Multivariate Statistical Process Control for PAT Calibration Model Maintenance Calibration Transfer and Instrument Standardization Chemometrics in Mass Spectrometry Chemometrics in Metabolomics Classical Least Squares (CLS) Methods Common Mistakes in Chemometrics Correlation Spectroscopy Design of Experiments for QbD Getting PLS_Toolbox/Solo Models Online Hierarchical and Optimized Models Implementing Chemometrics in PAT Introduction to Multivariate Image Analysis Modeling Fluorescence EEM Data MSPC-Multivariate Statistical Process Control Multi-block, Multi-set, and Data Fusion Methods Multivariate Curve Resolution Non-linear Methods for Calibration and Classification PLS_Toolbox Beyond the Interfaces Robust Methods Variable Selection Bring Your Own Data (BYOD) And we're always adding more

3 Resources Hyperspectral Image Analysis, eds. P. Geladi and H. Grahn, Wiley (2007), ISBN Chemometrics, M.A. Sharaf, D.L. Illman and B.R. Kowalski, Wiley-Interscience (1986) ISBN Multivariate Analysis, K.V. Mardia, J.I. Kent and J.M. Bibby, Academic Press, (1979) ISBN Multivariate Calibration, H. Martens and T. Næs, John Wiley & Sons Ltd. (1989) ISBN Chemometrics: a textbook, D.L. Massart et al., Elsevier (1988) ISBN Chemometrics: A Practical Guide, K.R. Beebe, R.J. Pell, M.B. Seasholtz, Wiley (1998) ISBN Multivariate Data Analysis In Practice, Kim H. Esbensen, CAMO ASA (2000), ISBN A user-friendly guide to Multivariate Calibration and Classification, T. Næs, T. Isaksson, T. Fearn, T. Davies, NIR Publications(2002), ISBN Journal of Chemometrics IEEE Trans. on Geosci. and Remote Sensing Chemometrics and Intelligent Laboratory Systems Analytical Chemistry Analytica Chemica Acta Applied Spectroscopy Critical Reviews in Analytical Chemistry Journal of Process Control Computers in Chemical Engineering Technometrics... 5 Univariate Image Grey scale each pixel is an number defining an intensity level e.g., integer (0 to 255) unsigned 8-bit integer (0 to 4095) double (floating point) x-pixels M x xm y pixels provides spatial information y-pixels

4 Multivariate Image (3 Variables) Red/Green/Blue (RGB) (e.g. JPEG) each layer defines color intensity level much more information-rich Image Analysis Many methods have been developed to examine the spatial structure w/in an image the methods recognize spatial patterns within an image based on the light / dark contrast and continuity of regions edge detection, image sharpening, wavelets particle size distributions, machine vision, medical applications, security, MIA has been traditionally applied to the spectral dimension first followed by spatial analysis some methods that examine both are appearing 8

5 Multivariate Image (4-10 Variables) Measure at several wavelengths (e.g., Landstat) How should we display a seven variable image? thermal SWIR-2 blue green red NIRSWIR-1 9 Multivariate Image (4-10 Variables) Choose 3 of 7 (Landstat) Montana (blue/swir-1/thermal) Paris (NIR/blue/SWIR-1) * * contrast enhanced

6 Hyperspectral Image (>10 Variables) Spectrum at each pixel could be s of variables often floating point double s Mbytes x each pixel is a spectrum Absorbance Wavelength (nm) y ν Pixels Spatial Information Spectral Chemical Information each voxel is a channel in the spectrum 11 File Formats Inherent Image Formats Cameca Ion-Tof BIF/BIF6 Image (BIF,BIF6) ENVI Image Format (HDR) Lispix Raw Formatted Image (RAW) Multi-layer TIFF files (TIFF) Physical Electronics RAW Image (RAW) Image standard (JPG, TIFF, GIF, BMP, PNG) Non-Image Formats (add image context after load) Text (e.g. CSV) Thermo-Galactic SPC (binary) 12

7 Memory Considerations 512 x 512 pixels and 2048 variables = 536 Million data points = 4.3 GB memory (double precision) BEFORE preprocessing! Larger images require 64-bit computers with 4GB or more of memory 13 Multivariate Images Data array of dimension three (or more) where the first two dimensions are spatial and the last dimension(s) is a function of another variable (e.g, spectroscopy). Chemical system(s) of interest include microscopic, medical, machine vision, process monitoring crystallization, stand-off and remote sensing, vapors, liquids, solids (or combination) visible, infra-red, Raman, mass spectroscopy, 14

8 Displaying a Multivariate Image (4-10 Variables) How to choose the 3 variables? In which order should they be displayed? Doesn t choosing ignore potential information in the remaining variables? How could information be extract from the image? What happens when we go to more variables?.... Factor-based techniques use the correlation structure to enhance S/N really good for hyperspectral 15 EVRI Product Outline Modeling & Analysis: Image Analysis: Model Export: Model Application: Matlab-Based PLS_Toolbox MIA_Toolbox Model_Exporter Stand-Alone Solo Solo+MIA Solo+Model_Exporter Solo_Predictor Matlab-Based products provide access to all Graphical User Interfaces (GUIs) plus command-line scripting and programming functionality Stand-Alone products provide access to same GUIs plus basic script operations without needing Matlab

9 EVRI Product Outline Modeling & Analysis: Image Analysis: Model Export: Model Application: Matlab-Based PLS_Toolbox MIA_Toolbox Model_Exporter Stand-Alone Solo Solo+MIA Solo+Model_Exporter Solo_Predictor Exporting of models is for use in high-frequency or low-resource applications such as hand-held instruments Solo_Predictor supports all model types, preprocessing, calibration transfer, and many other PLS_Toolbox/Solo features Map of Eigenvector Software PLS_Toolbox and MIA_Toolbox (in Matlab) Solo+MIA (Stand-alone) Workspace Browser (Starting Point) Trend Tool (Visualization) Image Manager Analysis (Modeling) Texture Analysis Plot Controls & DataSet Editor 18 Particle Analysis

10 Simple Image Analysis Tools TrendTool Univariate Data Investigation Analyze multivariate data using simple univariate measurements Image Manager Data Manipulation and Analysis Concatenating / Manipulating (e.g. rotation) Images Preprocessing 19 TrendTool Display results of univariate calculations on multivariate data Signal at given variable Integrated signal across range of variables Peak position Peak width With or without baselines Ratio of measurements 20

11 Opening TrendTool Workspace Browser Plot Controls Window Image Manager Toolbar 21 TrendTool Windows: Data View Use Data View to: Set analysis markers Choose analysis mode Select references and baseline points Hints: Right-click white space to set marker or use toolbar button Drag markers to move Right-click markers to change types Use toolbar to save or load marker sets 22

12 TrendTool Windows: Trend View Results displayed in Trend View Single marker displays with false-color Multiple markers display in RGB Toolbar Buttons: autoscale image select pixels to display in Data View save or spawn plot of results (respectively) 23 TrendTool Analysis Modes Height gives response at position (single marker) Area gives integrated response between markers Position gives position of peak response between markers Width gives full width at half height between markers "Add Reference" to subtract a single point baseline. Convert reference to baseline (via right-click) to do two-point linear baseline. "Normalize to Region" to normalize all regions to the response of the selected region. 24

13 Opening Image Manager Workspace Browser Plot Controls Window Plot Toolbar 25 Image Manager Overview Load / Import Images Controls Image Manager & Tools Settings Currently Loaded Images List 26

14 Image Groups Grouping allows you to: Combine images into a single DataSet for analysis Apply a univariate operation (rotate, crop, etc) to all images Image Group Controls Example: combining three slabs of RGB image 27 Image Groups 28 click to view

15 Concatenating Images With all 3 images loaded and grouped 29 Concatenating Images: Spatial Domain (768 x 1536) x 1 X, Y, Z, or tile 30

16 Concatenating Images: Variable Domain (768 x 512) x 3 31 Group Manipulation Example: Rotation Hint: to apply an action to only ONE image, click the "Apply Changes to Image Group" button until only one thumbnail is outlined in the image group pane. 32

17 Image-Oriented Preprocessing Image-specific preprocessing operates in pixel-space and are either Intensity or Binary based Intensity-Based Image Correction: Background Subtraction (Flatfield): Rolling-ball background subtraction for images. Min: Min value over neighboring pixels. (filter out high-value pixels) Max: Max value over neighboring pixels. (filter out low-value pixels) Mean: Mean value over neighboring pixels. (filter out low/high pixels) Median: Median value over neighboring pixels. (robust filter of low/high pixels) Trimmed Mean: Trimmed mean value over neighboring pixels. Trimmed Median: Trimmed median value over neighboring pixels. Smooth: Spatial smoothing for images. (a weighted mean) 33 Image-Oriented Preprocessing Binary-Based Image Correction Dilate: Perform dilation on a binary image. Erode: Perform erosion on a binary image. Close (Dilate+Erode): Perform dilation followed by erosion on a binary image. Open (Erode+Dilate): Perform erosion followed by dilation on a binary image. NOTE: Image-Oriented methods may break covariance (add multivariate rank) because variable slabs handled separately Standard variable-space preprocessing can be used too, but are spatially insensitive 34

18 MIA: PCA-Based Methods Many methods are based on the spectroscopic information in an image although spatial information is ignored mathematically images are examined for spatial structure PCA (Principal Components Analysis) Exploratory analysis SIMCA (Soft Independent Method Class Analogy) Classification 35 Image PCA Matricizing PCA: scores, scores images, loadings unusual samples Q and T 2 score-score plots, density plots linking scores and image plane(s) contrast enhancement 36

19 Matricizing (a.k.a. Unfolding) PCA works on X (MxN) but the image is MxxMyxN reshape by matricizing such that each pixel is a row in a Matricized Image new MxMyxN matrix Original Image MxxMyxN x y ν MxMyxN ν 37 PCA Math Summary For a data matrix X with M samples and N variables (generally assumed to be mean centered and properly scaled), the PCA decomposition is X = t p + t p + K + t p + K + t p T T T T K K R R Where R min{m,n}, and the t k p kt pairs are ordered by the amount of variance captured. Generally, the model is truncated to K PCs, leaving some small amount of variance in a residual matrix E: T T T T X = t p + t p + K + t p + E = TP + E where T is MxK and P is NxK. K K 38

20 Properties of PCA X = p T 1 t 1 + p T 2 t t K p T K + E t k,p k ordered by amount of variance captured λ k are the eigenvalues of X T X X T Xp k = λ k p k λ k are variance captured t k (scores) form an orthogonal set T K (MxK) describe relationship between samples pixels (M = M x M y ) p k (loadings) form an orthonormal set P K (NxK) describe relationship between variables 39 PCA Graphically 8 6 PC 2 PC 1 Variable Mean Vector

21 Reshape Scores To Images PCA gives scores T (MxK) which is reshaped to scores images (M x xm y xk) each score vector is a M x xm y scores image Original Scores M x M y xk Scores Images M x xm y xk x y k 41 Plots / Images for PCA scores and loadings plots are interpreted in pairs plot t k vs sample number find relationship between samples pixels each M x M y x1 score vector is reshaped to a M x xm y matrix that can be visualized as a "scores image" showing spatial relationships between pixels p k vs variable number relationship between variables responsible for observations in samples it is useful to plot t k+1 vs. t k and p k+1 vs. p k examine image and score / score plots 42

22 Image PCA Conclusions Image PCA is a useful unsupervised pattern recognition technique for exploring images scores and loadings are useful for determining what original variables are responsible for differences observed in an image score-score plots and linked score plots contrast enhancement might be needed to see small changes Image SIMCA is a useful supervised pattern recognition technique find similar / dissimilar portions of an image very quickly 43 MCR Based on the classical least squares (CLS) model, attempt to estimate C and S given X: X = CS T + E where X is a MxN matrix of measured responses, C is a MxK matrix of pure analyte contributions, S is a NxK matrix of pure analyte spectra, and E is a MxN matrix of residuals. 44

23 MCR Objective Decompose a data matrix into chemically meaningful factors pure analyte spectra pure analyte concentrations Easy to interpret provides chemically / physically meaningful information caveats: rotational and multiplicative ambiguity use of constraints 45 Linear Discriminant Analysis LDA seeks axis (in n-d space) which maximizes ratio of between class to within class variance an axis e.g., PC1 X2 LDA X2 X1 Projection onto axis 46

24 Partial Least Squares Discriminate Analysis (PLS-DA) Exactly as with Linear Discriminant Analysis (LDA), the objective is to determine an axis to project data on that discriminates between classes choose axis so individual distributions are narrow choose axis so centers of distributions are far apart Determine axes from factor-based model of data therefore more stable with high collinearity. Automatically attempts to identify directions of interest! 47 Partial Least Squares Discriminate Analysis (PLS-DA) Use logicals (0,1) in Y-block to indicate if sample belongs to a class or not dummy variables Develop PLS model to predict class block Thresholds must be set between 0 and 1 to indicate if new samples are a member of each class... Can use Bayes theorem to set threshold and include prior probability of each class Regression Vector Threshold 48

25 Image PLSDA and SIMCA Conclusions If classes (regions) are known, PLSDA is a useful supervised pattern recognition technique for exploring images can often bring out more contrast than PCA If only examples of one class are known, then SIMCA (i.e. PCA models) should be used 49 Comments on Presenting Images Images are representations of spatial and chemical information, but they can be mis-used. users can control colors and contrasting and select channels or PCs (or rotations thereof) as a result some things can be highlighted while others can be hidden It is important to report how images were constructed the work must be reproducible 50