def _get_dpi(ctm_shorthand, image_size):
    """Given the transformation matrix and image size, find the image DPI.

    PDFs do not include image resolution information within image data.
    Instead, the PDF page content stream describes the location where the
    image will be rasterized, and the effective resolution is the ratio of the
    pixel size to raster target size.

    Normally a scanned PDF has the paper size set appropriately but this is
    not guaranteed. The most common case is a cropped image will change the
    page size (/CropBox) without altering the page content stream. That means
    it is not sufficient to assume that the image fills the page, even though
    that is the most common case.

    A PDF image may be scaled (always), cropped, translated, rotated in place
    to an arbitrary angle (rarely) and skewed. Only equal area mappings can
    be expressed, that is, it is not necessary to consider distortions where
    the effective DPI varies with position.

    To determine the image scale, transform an offset axis vector v0 (0, 0),
    width-axis vector v0 (1, 0), height-axis vector vh (0, 1) with the matrix,
    which gives the dimensions of the image in PDF units. From there we can
    compare to actual image dimensions. PDF uses
    row vector * matrix_tranposed unlike the traditional
    matrix * column vector.

    The offset, width and height vectors can be combined in a matrix and
    multiplied by the transform matrix. Then we want to calculated
        magnitude(width_vector - offset_vector)
    and
        magnitude(height_vector - offset_vector)

    When the above is worked out algebraically, the effect of translation
    cancels out, and the vector magnitudes become functions of the nonzero
    transformation matrix indices. The results of the derivation are used
    in this code.

    pdfimages -list does calculate the DPI in some way that is not completely
    naive, but it does not get the DPI of rotated images right, so cannot be
    used anymore to validate this. Photoshop works, or using Acrobat to
    rotate the image back to normal.

    It does not matter if the image is partially cropped, or even out of the
    /MediaBox.

    """

    a, b, c, d, _, _ = ctm_shorthand

    # Calculate the width and height of the image in PDF units
    image_drawn_width = hypot(a, b)
    image_drawn_height = hypot(c, d)

    # The scale of the image is pixels per unit of default user space (1/72")
    scale_w = image_size[0] / image_drawn_width
    scale_h = image_size[1] / image_drawn_height

    # DPI = scale * 72
    dpi_w = scale_w * 72.0
    dpi_h = scale_h * 72.0

    return dpi_w, dpi_h
Given the transformation matrix and image size, find the image DPI.

    PDFs do not include image resolution information within image data.
    Instead, the PDF page content stream describes the location where the
    image will be rasterized, and the effective resolution is the ratio of the
    pixel size to raster target size.

    Normally a scanned PDF has the paper size set appropriately but this is
    not guaranteed. The most common case is a cropped image will change the
    page size (/CropBox) without altering the page content stream. That means
    it is not sufficient to assume that the image fills the page, even though
    that is the most common case.

    A PDF image may be scaled (always), cropped, translated, rotated in place
    to an arbitrary angle (rarely) and skewed. Only equal area mappings can
    be expressed, that is, it is not necessary to consider distortions where
    the effective DPI varies with position.

    To determine the image scale, transform an offset axis vector v0 (0, 0),
    width-axis vector v0 (1, 0), height-axis vector vh (0, 1) with the matrix,
    which gives the dimensions of the image in PDF units. From there we can
    compare to actual image dimensions. PDF uses
    row vector * matrix_tranposed unlike the traditional
    matrix * column vector.

    The offset, width and height vectors can be combined in a matrix and
    multiplied by the transform matrix. Then we want to calculated
        magnitude(width_vector - offset_vector)
    and
        magnitude(height_vector - offset_vector)

    When the above is worked out algebraically, the effect of translation
    cancels out, and the vector magnitudes become functions of the nonzero
    transformation matrix indices. The results of the derivation are used
    in this code.

    pdfimages -list does calculate the DPI in some way that is not completely
    naive, but it does not get the DPI of rotated images right, so cannot be
    used anymore to validate this. Photoshop works, or using Acrobat to
    rotate the image back to normal.

    It does not matter if the image is partially cropped, or even out of the
    /MediaBox.