.NET (Core and Framework)

private class Signature
{
    public List<List<List<double>>> lines { get; set; }
}

var signature = serializer.Deserialize<Signature>(json);

{
   lines: [
            [
               [156.44, 96.44],
               [156.44, 95.33],
               [155.33, 94.22],
               [154.22, 93.11],
               [154.22, 90.89],
               [153.11, 89.78],
               ...
            ]
   ]
}

private sealed class SignatureLine
{
    public int lx { get; set; }
    public int ly { get; set; }
    public int mx { get; set; }
    public int my { get; set; }
}

private class Signature
{
    public List<List<List<double>>> lines { get; set; }
    
    public IEnumerable<SignatureLine> SignatureLines
    {
        get
        {
            foreach (List<List<double>> stroke in lines)
            {
                if (stroke.Count == 1) continue;
                
                List<double> lastPoint = stroke[0];
                Debug.Assert(lastPoint.Count == 2);
                
                foreach (List<double> point in stroke.Skip(1))
                {
                    Debug.Assert(point.Count == 2);
                    
                    // For now, just round the floating point values to the nearest integer:
                    yield return new SignatureLine
                    {
                        lx = (int)Math.Round(lastPoint[0]),
                        ly = (int)Math.Round(lastPoint[1]),
                        mx = (int)Math.Round(point[0]),
                        my = (int)Math.Round(point[1])
                    };
                    
                    lastPoint = point;
                }
            }
        }
    }
}

...

var serializer = new JavaScriptSerializer();
var signature = serializer.Deserialize<Signature>(json);
foreach (SignatureLine line in signature.SignatureLines)
{
   signatureGraphic.DrawLine(pen, line.lx, line.ly, line.mx, line.my);
}

signatureImage.Save(Server.MapPath("~/images/s1.jpg"), ImageFormat.Jpeg);

using (MailMessage mail = new MailMessage())
{
    ...
    SmtpClient smtp = new SmtpClient();
    smtp.Send(mail);
}